An Overview : Peak to Average Power Ratio (PAPR) in OFDM system using some new PAPR techniques (with by Z AL Hashemi

An Overview : Peak to Average Power Ratio (PAPR) in OFDM system using some new PAPR techniques (with matlab code)

Zainab S. H. AL-Hashmi

An Overview : Peak to Average Power Ratio (PAPR) in OFDM system using some new PAPR techniques (with matlab code) By Zainab Saad Hadi AL-Hashmi A graduate of University of Baghdad, College of Engineering Electronic & Communications Engineering Department

‫تغى هللا انشدًٍ انشدٍى‬

‫﴿‬

‫د‬ ‫ة َػ ُْ ُك ُى ان ِّشجْ َ‬ ‫إًَّا ٌُشٌ ُذ هللاُ نٍِ ُْز ِْ َ‬ ‫ظ أ ْْ َم انثَ ٍْ ِ‬ ‫ٌُٔطَِّٓ َش ُك ْى ذَ ْ‬ ‫ط ٍِٓشا‬

‫﴾‬

‫صذق هللا انؼهً انؼظٍى‬ ‫{االدضاب ‪}ٖٖ/‬‬

‫‪i‬‬

‫االْذاء‬ ‫بسم هللا الرّحمن الرّحيم‬ ‫﴿إِ ٌَّ َّ‬ ‫صهُّٕا َػهَ ٍْ ِّ‬ ‫ٌٕ َػهَى انَُّثِ ًِّ ٌَا أٌََُّٓا انَّ ِز َ‬ ‫ُصهُّ َ‬ ‫ٌٍ آَ َيُُٕا َ‬ ‫هللاَ َٔ َي ََلئِ َكرَُّ ٌ َ‬ ‫ٔآل ُي َذ ًَّذ‪.‬‬ ‫َٔ َعهِّ ًُٕا ذَ ْغهًٍِا﴾ انهَّـُٓ َّى َ‬ ‫صمِّ َػهَى ُي َذ ًَّذ ِ‬ ‫أْذٌٓا انى َثً انشدًح ٔشفٍغ االيح دثٍة لهٕتُا ٔشفٍغ رَٕتُا عٍذ‬ ‫انخهك اجًؼٍٍ ٔانًثؼٕز سدًح نهؼانًٍٍ اتً انماعى يذًذ صم هللا‬ ‫ػهٍّ ٔآنّ ٔعهى ٔأْذٌٓا انى تاب يذٌُح انؼهى سعٕل هللا صم هللا ػهٍّ‬ ‫ٔآنّ ٔعهى اتٕ انغثطٍٍ ايٍش انًؤيٍٍُ ٔانى ػرشخ سعٕل هللا‬ ‫فرمثهْٕا يًُ ٌا آل غّ ٔ أَرى انكشاو ال ذشدٌٔ انٓذاٌا‪.‬‬ ‫ٔأدة أٌ أْذٌٓا انى يٍ ستٍاًَ صغٍشا انى جذذً ٔجذي انًشدٕو‬ ‫انغٍذ دغٍ ػهً ػثاط صٌٍٔ ٔانى أيً ٔأْهً ٔأخص تانزكش انغٍذ‬ ‫لاعى دغٍ صٌٍٔ ٔانغٍذ يٍصى دغٍ صٌٍٔ ٔانغٍذ فاظم دغٍ صٌٍٔ‬ ‫ٔانغٍذ ػادل دغٍ صٌٍٔ‪.‬‬ ‫ادة اٌعا اْذٌٓا نكم غانة ظهًّ انًششف أ يٍ ْٕ يغؤٔل ػُّ‬ ‫أ اخز دمّ تاخرصاس اْذٌٓا نكم يظهٕو ٔالٕل صادة انذك‬ ‫عهطاٌ فذافؼٕا ػٍ دمٕلكى‬

‫‪ii‬‬

Acknowledgments praise belongs to God who showed favour to us through His religion, singled us out for His creed, and directed us onto the roads of His beneficence, in order that through His kindness we might travel upon them to His good pleasure, a praise which He will accept from us and through which He will be pleased with us. !Allah send peace and blessings upon Mohammed and his progeny (S.A.W.) Finally I would like to thank my family,

Especially my grandfather Mr. Hassan Ali Zwain, my mother, Mr. Qasim Hassan Zwain and Mr. Maythem Hassan

Zainab saad hadi 2015

iii

Abstract The Orthogonal frequency division multiplexing (OFDM) is multicarrier modulation scheme which has recently become comparatively popular in both wireless and wired communication systems for transfer the multimedia data. OFDM could be used at the core of well-known systems like Asymmetric digital subscriber line (ADSL) internet, digital television/radio broadcasting, wireless local area network (LANs), and Long Term Evolution (LTE). High PAPR is the major drawback of OFDM, which results in lower power efficiency hence impedes in implementing OFDM. The PAPR problem is more significant in the uplink because the efficiency of power amplifier is critical because a mobile terminal has a limited battery power. High peak-to-average power ratio (PAPR) occurs due to large envelope fluctuations in OFDM signal this requires a highly linear high power amplifier (HPA). Power amplifiers with large linear range are expensive, bulky 50% of the size of a transmitter lies and difficult to manufacture. In order to reduce the PAPR, several techniques have been proposed in this thesis, primarily the repeated frequency domain filtering and clipping (RFC) has been proposed and compared with the existing method repeated clipping and frequency domain filtering (RCF). The RFC is better than RCF in performance especially when I ≥ 2, although they have the same complexity and cost. The proposed method is not only improving PAPR but also improving BER. Best case for the bit error rate (BER) is at I =4 and CR =4, where ) improved by (5.7601 dB) Signal to Noise Ratio (SNR) at BER ( and Complementary Cumulative Distribution Function (CCDF) of PAPR was improved by (4.775 dB) and PAPR was improved by (11.4177 dB). The best one improvement in PAPR and CCDF of PAPR So as not to BER deteriorate is at I =4 and CR =1.75. The improvement in PAPR by = (18.2789 dB), CCDF of PAPR = (8.0187 dB), and the SNR at ) by = (0.6101 dB). BER( In addition to (RFC) six new types of companding have been proposed and compared with the μ-law and A-law compandings. all these proposed methods have better performance than the μ-law and A-law compandings, and the best one is Absolute Exponential (AEXP) companding and the iv

best one improvement in PAPR and CCDF of PAPR is at d= 1.1. The improvement in PAPR by = (17.6492 dB), and CCDF of PAPR = (7.2405 ) deteriorated by = (-3.4186 dB). dB), while the SNR at BER( Five types of pre-coding are used in this work and then compared them with each other. The best type of precoding in term of reduced PAPR and BER is the Discrete Fourier Transform (DFT) pre-coder, while the least is the Walsh Hadamard Transform (WHT) pre-coding. Also four new types of hybrids PAPR reduction techniques have been proposed. These methods are: 1. RCF with precodings (WHT, Discrete Cosine Transform (DCT), Discrete Sine Transform (DST),and Discrete Hartley Transform (DHT)). 2. RCF with compandings (the all proposed compandings, μ-law and Alaw compandings). 3. RFC with compandings (the all proposed compandings, μ-law and Alaw compandings). 4.and finally precodings (WHT, DCT, DST,and DHT), with compandings (the all proposed compandings, μ-law and A-law compandings). The best one improvement is at (RFC with AEXP) because the PAPR, CCDF of PAPR, and BER. This improvement in PAPR and CCDF of PAPR is at d = 0.6 and CR =4. The improvement in PAPR by (21.0509dB), CCDF of PAPR = (8.7178 dB), and the SNR at ) by (0.0116 dB). BER( The DHT with tangent Rooting (tanhR) have acceptable results where the PAPR and CCDF of PAPR were improved while BER was degarded. The best one improvement for this case is at k=15, y=.8 and DHT. The improvement in PAPR by = (22.7711 dB), and CCDF of PAPR = (8.9691 ) deteriorated by = (-1.1828 dB). dB), while the SNR at BER( All methods are simulated using matlab.

Contents Chapter One: Introduction

1.1 Introduction

1.2 Literature survey

Chapter Two: LTE and OFAM

2.1. Introduction

2.2. LTE Requirements

2.3. LTE Architecture

2.4. Air interface in LTE

2.5 History of OFDM

2.6 OFDM

2.6.1 Orthogonality of the subcarriers and OFDM

2.6.2 Guard Interval

2.6.3 One-tap equalizer

2.7 OFDM based Multiple Access

2.8 Orthogonal Frequency Division Multiple Access

2.9 SC-FDMA

Chapter Three: Peak To Average Power Ratio Reduction

3.1Definitions of PAPR

3.2 PAPR of OFDM signal

3.3 Oversampling discrete OFDM symbols to find true (continuous) peaks

3.4 Distribution of PAPR

3.5 Identification of the Problem

3.5.1 Nonlinear HPA and DAC

3.5.2 Power Saving

3.6 Factors influencing the PAPR

3.6.1 The number of sub carriers

3.6.2 The order of Modulation

3.6.3 Constellation shape

3.6.4 Pulse Shaping

3.7 The gauge for judgment of the PAPR reduction in OFDM systems

3.8 Fitness function-based approach for determining an appropriate Algorithm

Chapter Four: PAPR Reduction Techniques

4.1There are three different way to divide the PAPR

4.1.1The first way is

4.1.2 The second way

4.1.3The third way

4.1.4 And finally there is Hybrid techniques

4.2 Clipping and Filtering

4.3 Peak Windowing Method

4.4 Envelope Scaling

4.5 Peak Reduction Carrier

4.6 Companding Technique

4.6.1 Square-Rooting Companding Technique ( SQRT) for PAPR Reduction in

OFDM Systems 4.6.2 Exponential Companding Algorithm

4.6.3 Trapezoidal power companding

4.6.4 Hyperbolic tangent (

4.6.5 Error Function (

) companding

) Companding

4.6.6 Logarithm Function (log) Companding

4.7 Coding techniques

4.8 Selective Mapping (SLM)

4.9 Partial Transmit Sequence (PTS)

4.10 Tone Reservation

4.11 Tone Injection

4.12 Interleaving

4.13 Active Constellation Extension (ACE)

4.14 Dummy Sequence Insertion (DSI)

Chapter Five: Simulation Results and Analysis

5.1 OFDM System model

5.2 PAPR techniques used

vii

5.2.1 Repeated clipping and frequency domain filtering (RCF)

5.2.2 Repeated frequency domain filtering and clipping RFC

5.2.3 The OFDM System with discrete time companding

5.2.3.1 A companding

5.2.3.2

Companding

5.2.3.3 Rooting Companding Technique (RCT)

5.2.3.4 New error function Companding (NERF)

5.2.3.5 Absolute Exponential companding (AEXP)

5.2.3.6 Cos companding

5.2.3.7 tangent Rooting Companding (tanhR)

5.2.3.8 Logarithmic Rooting Companding (logR)

101

5.2.4 OFDM System with pre-coding

104

5.2.4.1 Pulse Shaping or Pre-coding

104

5.2.4.2 Discrete Hartley transform (DHT)

105

5.2.4.3 Walsh-Hadamard Transform (WHT)

105

5.2.4.4 Discrete Cosine Transform (DCT):

106

5.2.4.5 Discrete Sine Transform (DST) Precoding Technique

107

5.2.4.6 The Discrete Fourier Transform (DFT) Precoding

107

5.2.4.7 Simulation results and analysis of OFDM system with pre-coding

108

Chapter six: Simulation Results and Analysis of Hybrid PAPR techniques

110

6.1 Hybrid pre-coding with RCF

110

6.2 Hybrids RCF with companding

119

6.2.1 RCF + A companding

119

6.2.2 RCF +

121

6.2.3 RCF + RCT

123

6.2.4 RCF + AEXP

126

6.2.5 RCF + cos

128

6.2.6 RCF + NERF

130

6.2.7 RCF + tanhR

131

6.2.8 RCF +logR

132

6.3 Hybrid RFC with companding

134

6.3.1 RFC + A companding

134

viii

6.3.2 RFC +

137

companding

6.3.3 RFC + RCT

139

6.3.4 RFC + AEXP

141

6.3.5 RFC + cos

143

6.3.6 RFC + NERF

145

6.3.7 RFC + tanhR

146

6.3.8 RFC +logR

147

6.4 Pre-coding + companding

148

6.4.1 Pre-coding + A companding

149

6.4.2 Pre-coding +

152

6.4.3 Pre-coding + RCT

154

6.4.4 Pre-coding + AEXP

156

6.4.5 Pre-coding + cos

159

6.4.6 Pre-coding + tanhR

161

6.4.7 Pre-coding + logR

162

6.4.8 Pre-coding + NERF

163

Chapter seven : Conclusions and future work

165

7.1Conclusions

165

7.2Future work

167

References

168

Appendices

Appendix A : Table of Results

A.1

Appendix B : MATLAB Code

B.1

Chapter One

Introduction Chapter One Introduction

1.1 Introduction: During the last two decades, the demand for multimedia wireless communication services have grown tremendously and this trend are expected to continue in the near future. Orthogonal frequency division multiplexing (OFDM) is one of such multicarrier techniques which have attracted vast research attention from academics, researchers and industries since last two decades. It has become part of new emerging standards for broadband wireless access [1]. Energy efficiency, particularly matters in future mobile communications networks. A key driving factor is the growing energy cost of network operation which can make up as much as 50% of the total operational cost nowadays [2]. The transmitted signal of OFDM exhibits a high Peak-To-Average Power Ratio (PAPR). This high PAPR reduces the efficiency of high power amplifier and degrades the performance of the system [3]. A major source for reducing energy costs is to increase the efficiency of the high power amplifier (HPA) in the radio frequency (RF) front end of the base stations [4]. However, the efficiency of the HPA is directly related to the PAPR of the input signal. The problem, especially, becomes serious in OFDM multicarrier transmission, which is applied in many important wireless standards such as the third Generation Partnership Project (3GPP) Long Term Evolution Advanced (LTE-A). The PAPR problem still prevents OFDM from being adopted in the uplink of mobile communication standards, and, besides from power efficiency, it can also place severe constraints on output power and therefore coverage in the downlink. In the past, there have been many efforts to deal with the PAPR problem resulting in numerous papers and several overview articles, e.g., [5], [6], [7]. PAPR has a deleterious effect on battery lifetime in mobile applications. As handy devices have a finite battery life, it is significant to find ways of reducing the PAPR allowing for a smaller, more efficient HPA, which in turn will mean a longer lasting battery life. In many low-cost applications, the problem of high PAPR may outweigh all the potential benefits of multicarrier transmission systems [6]. A number of promising approaches or techniques have been proposed & implemented to reduce PAPR with the expense of increase transmit signal Power, bit error rate (BER) & computational complexity and loss of data rate, etc. So, a system trade-off is required [8].

1.2 Literature survey: In 1996 Robert [9]. The selected mapping was used for the reduction of PAR. The selected mapping can be used for arbitrary numbers of carriers and any signal constellation. The selected mapping provides significant gains at moderate additional complexity. Actually, it is appropriate for all kinds of multiplex techniques, which transform data symbols to the transmit signal. Even in single carrier systems where 1

Chapter One

Introduction

PAR grows as the roll of factor of the pulse shaping filter decreases, selected mapping can be applied advantageously. The first nonlinear companding transform (NCT) for PAPR reduction was given by Wang et.al in 1999 [10]. It was based on the speech processing algorithm µ-law and it has found better performance than that of clipping technique. The µ-law companding transform mainly focuses on enlarging small amplitude signals while keeping peak signals unchanged, and thus it increases the average power of the transmitted signals and may lead to overcome the saturation region of the HPA to make the performance of the system worse. In order to overcome the problem of µ-law companding (increasing average power) and to have an efficient PAPR reduction. [10] In 2000 Myonghee et.al [11] Hadamard transform is an effective technique to reduce the PAPR of an OFDM system. The PAPR can be reduced in OFDM system without any power increase and side information. The increase of system complexity is not much. As further study, the equalization problem combining with Hadamard transform, which is induced to reduce PAPR, over multipath fading channel, is considered. In 2001 J. Armstrong [12] the clipping and frequency domain filtering PAPR reduction technique has been described in which an interpolated version of the baseband signal is clipped and then filtered with a new form of filter. The filter consists of a forward and an inverse fast Fourier transform (IFFT). It is designed to remove the out-of-band (OOB) noise without distorting the in-band discrete signal. It is shown that significant PAPR reduction can be achieved without any increase in OOB power. Some in-band distortion results, but this will have negligible effect on the overall BER in most systems. In 2002 J. Armstrong [13] the repeated clipping and frequency domain filtering of an OFDM signal can significantly reduce the PAPR of the transmitted signal. This method causes any increase in OOB power. Considerable PAPR reduction can be obtained with only moderate levels of clipping noise. In 2004 Ryu, et al. [14] The Dummy Sequence Insertion (DSI) technique reduces PAPR through increased the average power of the signal. Herein, after switchting the input data stream into parallel through the serial to parallel converter a, dummy sequence is inserted in the input signal. Thus, the average value is raised and the PAPR is reduced later. In 2005 Tao Jiang et.al [15] “exponential companding”. It can adjust the amplitudes of both large and small input signals, while maintaining the average power unchanged by properly choosing transform parameters, so as to make the output signals have a uniform distribution (with a specific degree). The exponential companding schemes can efficiently reduce PAPR for various modulation formats and sub-carrier sizes. 2

Chapter One

Introduction

The exponential companding schemes make less spectrum side-lobes than µ-law companding. Simulation results have shown that exponential companding schemes could provide better system performance in terms of PAPR reduction, power spectrum, BER, and phase error than the µ -law companding scheme. In 2007 Wisam et.al [16] square rooting companding (SQRT) companding a simple method of reducing the PAPR value of OFDM symbol by changing the statistical characteristics of the output signals . This was achieved by applying a non-linear square rooting operation of the OFDM signals. The process changed also the describing parameters of power signals: average and peak power values, and as a result the PAPR value is reduced. This companding more suitable for OFDM applications that do not have sophisticated processor, since it allows significant reduction in PAPR value with very low cost of computational complexity, and only slight performance degradation. In 2008 Pisit et.al [17] the simple PAPR reduction method by using the dummy subcarriers. The features of proposed method is to decide the frequency data for dummy subcarriers theoretically by using the certain number of larger amplitude levels detected in the time domain signal and to achieve the better PAPR performance with less computational complexity. In 2008 Carole et.al [18] they present an incipient PAPR reduction technique which exploits the utilization of used carriers in addition to the phase information of pilot signals in OFDM systems to limit the PAPR while not degrading channel estimation or frequency offset. Compared to conventional techniques like clipping and windowing, this technique introduces significantly lower OOB distortions and provides a lower BER since there is no interference to the original data signals. There is additionally no requisite for side information to be transmitted to the receiver. In 2009 Kazuki and Fumiyuki [19] A tone injection (TI) has been suggested which exploits the property of a nonlinear modulo function. The TI is identically equivalent to the one that superimposes a quadrature amplitude modulation (QAM) signal on the data symbol to reduce the PAPR. Without the transmission of the side information, the TI dramatically reduces the PAPR level. Albeit the TI-OFDM reduces the 1% PAPR level by about 3~4.5dB, the BER performance remarkably degrades. However, the utilization of antenna diversity reception can reduce the BER performance degradation. In 2010 Zhongpeng et.al [20] a combined μ companding transform and hadamard transform technique is suggested to reduce PAPR of OFDM signal .Simulation results shows that the PAPR reduction performance is improved compared with companding transform used only. On the other hand, the BER of system using proposed PAPR reduction scheme is not degraded. 3

Chapter One

Introduction

In 2010 Imran and Varun [21] the PAPR of discrete hartley transform (DHT)Precoded OFDM system for M-ary Quadrature Amplitude Modulation (M-QAM) (where M=16, 32, 64, 256). The Matlab simulation shows that DHT-Precoded OFDM System shows better PAPR gain as compared to OFDM-Original system, Walsh Hadamard transformation (WHT)-Precoder Based OFDM system and selective mapping (SLM) OFDM (with V=2) system respectively. Thus, it is concluded that DHT Precoder Based OFDM System shows better PAPR reduction then WHTPrecoder Based OFDM System, SLM-OFDM System and OFDM-Original system for MQAM. Additionally, the DHT-Precoded OFDM system does not require any power increase, complex optimization and side information to be sent for the receiver. In 2011 Zhongpeng [22] a combined reduction in PAPR of the OFDM signals based on the combination of the discrete cosine transform (DCT) with Îź companding. While taking both BER performance and PAPR performance into account, a united DCT and companding scheme to reduce the PAPR of OFDM signals. In 2011 Hem [23] a combinational method of pre-coding and clipping is proposed to reduce the PAPR of an OFDM system. The proposed technique is better than conventional because it does not require any increase in roll-off factor to reduce PAPR. Thus, it reduces the overhead in comparison to conventional pre-coding method. Increasing the roll-off factor degrades the BER as given in [24]. The clipping after pre-coding reduces PAPR but degrades BER. However, this degradation in not significant in comparison to degradation obtained by increasing roll off factor. In 2012 Malhar and Prof.Abhishek [25] tone reservation includes no of set of reservation of tones. By using this technique reserved tones can be utilized to minimize the PAPR. This method is used for multicarrier transmission and also demonstrated the reserving tones to limit the PAPR. Advantage of this tone reservation is very positive that no process is needed at receiver end. Furthermore there is no need to transmit the side information combined with the transmitted signal. In 2012 Eugen [26] The PAPR reduction technique based on combination of a WHT and a new signal companding function. The numerical results show that the hybrid scheme realizes an improved PAPR reduction than the component methods. The computation complexity increases linearly with the number of considered signal variants because of several signal variants are applied to the precoding block. This problem can be solved, by using few subcarriers as markers. In 2012 Chau, and Hsuan [27] presents a combination scheme, which using a combination of precoding by utilizing least null subcarriers in the frequency domain and nonlinear companding technique by applying proper -Law characteristic in time domain, for reducing PAPR. Simulation results indicate that the proposed scheme 4

Chapter One

Introduction

achieves a advantageous trade-off between OOB power emission in OFDM systems and the reduction of PAPR. In 2013 Sroy et.al [28] an Iterative Clipping and Filtering (ICF) Technique for PAPR Reduction of OFDM Signals: System Using DCT/ inverse discrete cosine transform (IDCT) Transform. The OFDM symbol enters the ICF block with DCT/IDCT technique, then clipping and filtering is iteratively performed. Although we demonstrate that significant PAPR reduction is obtained through Iterative clipping and filtering using fast Fourier transform (FFT)/IFFT transform, but better results are observed applying DCT/IDCT in the classical ICF technique. In 2013 Zihao et.al [29] a trapezoidal power companding method which could significantly reduce the PAPR for a complex OFDM or Filterbank Based Multicarrier Transmission (FBMC) system. The proposed approach provides a convenient way for designing a compander where the trade-offs among several system performances (such as PAPR, power spectral density (PSD) and BER) can be made. In 2013 Mohit et.al [30] the performance of tanh and erf companding is approximately. Log companding is better than the hyperbolic tangent and error function companding . μ-law and A-law companding give the same performance and the μ-law and A-law companding is better than the tanh, log and erf companding. Some more non-linear transform have been suggested in the paper [31, 32, 33, 34, and 35] In 2013 Jaspreet et.al [36] the performance analyzed in terms of PAPR in Orthogonal Frequency Division Multiple Access (OFDMA) by utilizing some pre-coding techniques, called Zadoff-Chu Transform (ZCT) and WHT with the µ-law companding to limit the PAPR of the OFDM signals .These pre-coding techniques produced the lower PAPR as compared to the conventional OFDM system. Furthermore ZCT is better than WHT because it produced the lowest PAPR than WHT. μ -law companding further reduces PAPR of OFDM signal and as with increasing the value the PAPR reduces. The obtained results approved that the proposed method have gotten better results than conventional OFDM. In 2013 Navneet and Lavish [37] The PAPR reduction method is based on combining clipping with WHT. Combined technique is simple to implement and has no limitations on the system parameters such as number of subcarriers modulation order, and constellation type. This system produces the lowest PAPR and is efficient, signal independent, distortion less and do not require any complex optimizations representing better PAPR reduction methods than others existing techniques because it does not require any power increment, complex optimization and side information to be sent to the receiver.

Chapter One

Introduction

In 2013 Mohit et.al [38] To reduce the PAPR of OFDM has been proposed Hybrid Clipping-Companding techniques for PAPR Reduction. the performance of hybrid PAPR reduction scheme with either tanh or erf as companding function is approximately same .Hybrid PAPR reduction scheme with log companding function is better than the last two. Hybrid PAPR reduction scheme with either Îź-law or A-law companding gives the same performance and the Hybrid PAPR reduction scheme with either Îź-law or A-law companding is best. In 2013 K. muralibabu et.al [39] In the proposed scheme, a combined reduction in PAPR of the OFDM system by grouping the sub carrier on the basis of the combination of joining the Discrete Cosine Transform (DCT) with companding technique. The simulation results indicat that the proposed scheme can yield good tradeoff between computational complexity and PAPR reduction performance In 2014 Jijina et.al [40] a comparative study is made on the three typical linear precoding techniques: Hadamard transform precoding, Discrete Sine Transform (DST) precoding and Square root raised cosine function precoding used in OFDMA system. The performance of these different schemes in terms of PAPR reduction is analyzed with the conventional Random Interleaved OFDMA system. Linear precoding schemes are efficient, signal independent, distortion less and do not require complex optimization when compared to the other reduction schemes.

Chapter Two

LTE and OFAM Chapter Two LTE and OFAM

2.1. Introduction: The growth in data intensive mobile services and applications like Web browsing, social networking, video streaming and music has become a driving force for development of the next generation of wireless standards. Thus, new standards are being developed to provide the data rates and network capacity needful to support worldwide delivery of these kinds of rich multimedia application. LTE have been developed to respond to the requirements of this generation and to achieve the aim of realizing global broadband mobile communications [41].

2.2. LTE Requirements: The demand for high speed and widespread network access in mobile communications increases every day as the number of users‟ increases and applications are constantly developed with greater demand for network resources. As a result of this trend, mobile communications has experienced significant developments within the last two decades, which is the result of tremendous research that has been carried out. [42] Requirements and objectives for the LTE Discuss the main requirements for the new LTE system Resulted in a the creation of a formal „Study Item‟ in 3GPP with the specific aim of „evolving‟ the 3GPP radio access technology to guarantee competitiveness over a ten-year time-frame. Depending on the study of this Study Item, the requirements for LTE Release 8 were revised and crystallized. They can be summed up as follows [41,43, and 44]:  High peak data rates and diminished delays, in both connection establishment and transmission latency. These improvements are to be realized through the simplification of the overall system, the decrease of complexity and the automated process of system management (i.e. optimization).  greater flexibility of spectrum usage, in each of the new and pre-existing bands;  Seamless integration with existing systems (Universal Mobile Telecommunications System (UMTS), Wireless Fidelity (Wi-Fi), etc.). Infrastructure-building economy. Although the implementation of every new system brings construction and building costs, LTE should be realized through minimal investment and use as much of the existing mobile communication infrastructure as possible.  Multi-antenna support.  Improved system capacity and coverage  Reasonable power consumption for the mobile terminal. The mobile terminal is being associated with mobile phones and similar devices which have limited battery capacities. Therefore a flexible bandwidth system (with lower frequencies used for uplink transmission) and automated signal power-level optimization have to be included into LTE [45].  Seamless mobility, including between different radio-access technologies;  Simplified network architecture;  Increased cell-edge bit-rate, for unification of service provision;  Increased user data rates;  Reduced cost per bit, implying an enhanced spectral efficiency; 7

Chapter Two 



LTE and OFAM

Packet switched domain utilization. To eliminate additional system complexity, introduced through the support of both the circuit switched and packet switched domain, the circuit switched domain will not be included into the LTE system. The traditional voice and text messaging services must be replaced with systemexternal subsystems (e.g. Information Management System (IMS)). High-level security and mobility. As the mobile communication system is now similar to a data network (e.g. internet), additional emphasis will be set on new security measures in combination with IP (Internet Protocol)-security functions. Mobility efficiency is provided through the use of evolved base stations, i.e. eNodeBs (E-UTRAN Node-B or Evolved Node-B).

These main targets resulted in the creation of additional requirements and spin-off functionalities, whose realizations were researched, developed and evolved by 3GPP and hence introduced in LTE‟s specifications and standardization upgrades. These improvements were further evolved and enhanced in Release 9, which contained additional techniques, functionalities and technology approaches to enable a quick, efficient and low-cost implementation of the LTE system. The following techniques are included:  introduction to Self-Organizing Networks (SON),  improved approach to emergency calls, as they oppose the system‟s security policy,  multiple-eNodeB broadcast signal combination (LTE MBMS),  further improvement of Frequency Division Duplex (LTE-FDD) and Time Division Duplex (LTE-TDD),  improvement of SON technologies and mechanisms, and  Minimization of system drive-tests (MDT). The LTE system and its standardization are 3GPP‟s most significant milestone achieved so far, triggering an increase of participation in their further projects and worldwide acknowledgement of their existing work. Takahiro Nakamura, the 3GPP RAN Chairman, states: “Operators need to work on issues that have been created in signaling and the volume of data being carried. Therefore, further improvements to the 3GPP system are being driven by that data explosion”. A continued evolution of the system is given in Releases 10, 11 and 12, introducing an improved mobile communication standard named LTE-Advanced [45].

2.3. LTE Architecture: The LTE architecture was highly simplified and flattened, as shown in Figure 2.1. The system contains only two types of nodes named Mobility Management Entity/System Architecture Evolution Gateway (MME/SAE GW) and evolved Node-B (eNB) [46, 47]. All LTE network interfaces are based on IP protocols and therefore two major changes were made compared to previous cellular radio architectures. The first significant modify is that the Radio Network Controller (RNC) is removed from the data path and its functions are now situated in eNB [46]. The main benefits of this type of single node access network are the diminished latency and the distribution of the RNC processing overhead into multiple eNBs. The second major change is that there are no nodes for Circuit Switched (CS) domain, such as the Mobile Switching

Chapter Two

LTE and OFAM

Centre (MSC). Therefore speech services are handled as Voice over IP (VoIP) calls in the LTE network [47, 48]. The eNBs are connected to each other via X2 interface and to Evolved Packet Core (EPC) through S1 interface, as also shown in Figure 2.1. The S1 interface supports in addition many-to-many relations between MMEs / SAE Gateways and eNBs [46]. SAE Gateway contains two logical gateway entities named as the Serving Gateway (SGW) and the Packet Data Network Gateway (P-GW). The S-GW is responsible for receiving and forwarding IP packets. Therefore, it can be seen as a local mobility anchor to the eNBs [48]. The P-GW, on the other hand, is responsible for handling the internet protocol functions, like routing, packet filtering, policy enforcement and address allocation [47]. The new system architecture was designed so that it will reduce the overhead from increased traffic. This is achieved because only the MME is responsible for signaling and therefore the user IP packets do not go over MME. This way the network capacity stays on a good level as the signaling and the traffic can grow separately [49]. The main responsibilities of MME are idle-mode User Equipment (UE) reachability including the control and execution of paging retransmission, different type of authentication procedures with Non-Access Stratum (NAS) signaling, roaming, PGW/S-GW selection, tracking area list management and bearer management including dedicated bearer establishment [47,48].

2.4. Air interface in LTE: The air interface and communication environment used in LTE mobile communication systems is called the LTE Radio Access Network. [45] The LTE air interface is based on OFDMA for the downlink. OFDMA is an extension of OFDM for the implementation of a multi-user communication system. For the uplink, a single-carrier frequency-division multiple access (SC-FDMA) technique has been selected. Advantages of this method include the relatively low adjacent channel power, even if the power amplifier is not 100% linear. With SC-FDMA, no exacting requirements are imposed on the linearity of the power amplifier in the mobile handset. As a result, power consumption can be kept within limits. [50] The utilization of OFDM provides considerable advantages over alternative multipleaccess techniques and signals severe departure from the past. Among the benefits are adaptability for broadband data transmission and high spectral efficiency, impedance to Inter Symbol Interference (ISI) resulting from the multipath fading, naturally provide MIMO (Multiple Input Multiple Output) schemes, and provide frequencydomain techniques like frequency-selective scheduling [51]. The design of the time-frequency representation of OFDM to provide high levels of flexibility in allocation of each of the time frames for transmission and the spectra. The spectrum flexibility in LTE supports not only a scalable set of bandwidths, but also a variety of frequency bands. LTE also supplies a small frame size of 10 ms in order to reduce latency. By designate short frame sizes, LTE allows better channel estimation to be carried out the mobile, allowing timely feedbacks needful for link adaptations to be supplied to the base station.[41]

Chapter Two

LTE and OFAM

Figure 2.1: System architecture for LTE Rel-8 network [47].

2.5 History of OFDM: The initial development of multi-carrier communication system was basically done by military systems in the late 1950s and mid-1960s. KINEPLEX, ANDEFT and KATHRYN etc. are the few OFDM based systems utilized by US military systems for high frequency applications [10]. In 1966, the concept of multicarrier communication was first introduced by Chang [60] .He suggested a multicarrier scheme utilizing the parallel data transmission by means of 10 frequency division multiplexing (FDM) with overlapping subcarriers. It was found to be an efficient scheme for bandwidth utilization and to mitigate the effect of multipath propagation. It also eliminated the requirement of high-speed equalization technique. He gave the basic concept of OFDM and outlined a theoretical way to transmit simultaneous data stream trough linear band limited channel without Inter Symbol Interference (ISI) and Inter Carrier Interference (ICI) [61] [62].

Chapter Two

LTE and OFAM

These systems are called classical Multicarrier Modulation (MCM) system and transmitted data on non-overlapped band-limited orthogonal signals. These systems require analog oscillator and filter of much wider bandwidth and sharp cut-off. Therefore, the concept of OFDM was not gained so much attention or popularity because of the difficulty in subcarrier recovery without inter-subcarrier interference by means of analog filters. Due to this reason only, a number of studies in the 1960s were dedicated for MCM employing overlapped band-limited orthogonal signals [63, 64, and 65]. In the year 1967, B. R. Saltzberg suggested a MCM system employing Orthogonal time-staggered Quadrature Amplitude Modulation (O-QAM) on the carriers [63]. The concept of MCM scheme employing time-limited orthogonal signals, which is similar to OFDM, was first given by H. F. Marmuth [66] in 1960. [10] The KINEPLEX system was developed by Collins Radio Company for data transmission at high frequency over multipath fading channel, in this system, 20 tones are modulated by DQPSK without filtering, which resulted in overlapping channels. Initially the implementation of an OFDM system with large number of subcarriers was very complex and expensive because it requires the array of sinusoidal generators and coherent demodulators for parallel operations. In order to avoid the crosstalk between the subcarriers, the system should be free from frequency and timing offsets [62]. A major breakthrough in the history of OFDM came in 1971 when Weinstein and Ebert used Discrete Fourier Transform (DFT) to perform baseband modulation and demodulation which eliminated the need of bank of subcarrier oscillators thus making the operation efficient and simpler [1,67]. In 1979, after evolutionary growth and development in signal processing and VLSI technologies, high speed chips can be built around special-purpose hardware performing the large size Fast Fourier Transform (FFT) (efficient algorithm for DFT) at affordable price [68], [69]. All the proposals of OFDM systems used guard spaces in frequency domain and a raised cosine windowing in time domain to combat ISI and ICI. Another milestone for OFDM history was when Peled and Ruiz introduced Cyclic Prefix (CP) or cyclic extension in 1980 [67,70] .This solved the problem of maintaining orthogonal characteristics of the transmitted signals at severe transmission conditions. The generic idea that they placed was to use cyclic extension of OFDM symbols instead of using empty guard spaces in frequency domain. This effectively turns the channel as performing cyclic convolution, which provides orthogonality over dispersive channels when CP is longer than the channel impulse response [56,70]. Since 1990s, OFDM has been utilized for many broadband communication systems, it includes high-bit-rate digital subscriber lines (HDSL) [71], asymmetric digital subscriber lines (ADSL) [72], very high-speed digital subscriber lines (VHDSL) [72], high definition television (HDTV) terrestrial broadcasting etc. It has also been utilized by many wireless standards like Digital Audio Broadcasting (DAB) [73] The DAB standard was in fact the first OFDM-based standard (project started in 1988 by ETSI and completed in 1995), Digital Video Broadcasting (DVB) [74]. Many standards have been proposed for wireless local area networks (WLANs) operating in ISM band, which are based on spread-spectrum technology. A number of studies regarding the commercial applications of OFDM were made during 1990s like High Bit rate Digital Subscriber Lines (HDSL; 1.6 Mbps), Asymmetric Digital Subscriber Lines (ADSL; 6 Mbps), Very High Speed Digital Subscriber Lines

Chapter Two

LTE and OFAM

(VDSL; 100 Mbps), DAB and High Definition Television (HDTV) terrestrial broadcasting [75]. In 1997, first OFDM-based WLAN standard, IEEE 802.11, was released. IEEE 802.11 can support a data rate up to 2 Mbps. Later on, in 1999, IEEE approved an OFDM based standard 802.11a for supporting a data rate up to 54 Mbps. During this period ETSI has also standardized the HiperLAN/2 standard, which has adopted OFDM for their PHY standards [1]. In 2001, the FCC came with some new rules for modulations scheme operating in the 2.4 GHz range, which allow IEEE to extend 802.11b to 802.11g standard. Now days, it has also been used in WiMAX (IEEE 802.16), and mobile broadband wireless access (MBWA) IEEE 802.10. It is 11 also utilized by 4G wireless communication systems, such as IMT-A. It is also been considered for 3GPP Long Term Evolution, which is under deployment [62].

2.6 OFDM: With the ever growing require of this generation, the necessity for high speed communication has become a top priority. Different multicarrier modulation techniques have developed to meet these demands, a few prominent among them being OFDM and Code Division Multiple Access (CDMA) [52]. The fundamental principle of OFDM is a division of high data rate streams into a number of lower data rate streams and then transmitted these streams in parallel using several orthogonal sub-carriers (parallel transmission). Due to this parallel transmission, the symbol duration increases, thus decrease the prorated amount of dispersion in time resulting from the multipath delay spread. OFDM can be seen as either a modulation technique or a multiplexing technique [10]. OFDM communication systems do not depend on increased symbol rates for achieving higher data rates. That makes the task of managing ISI much easier. Because data is transmitted in parallel instead of serially, OFDM symbols are basically much longer than symbols on single carrier systems of equivalent data rate [53]. The concept of OFDM is very much similar to the well-known and extensively used technique of Frequency Division Multiplexing (FDM). OFDM uses the principles of FDM to allow multiple messages to be sent over a single radio channel. It is however in a much more controlled manner, allowing an improved spectral efficiency [54]. In conventional broadcast, each radio station transmits on a different frequency, effectively using FDM to maintain a separation between the stations. Due to nonorthogonal nature of carrier frequencies in FDM, a large band gap is required to avoid inter-channel interference, which reduces the overall spectral efficiency. The difference between FDM and OFDM is shown in Figure 2.2 [55].

Chapter Two

LTE and OFAM

Figure 2.2: Comparison of FDM and OFDM [55] The sub-carriers are mutually orthogonal (The principle of orthogonality is discussed in next sub-section.) in the frequency domain which alleviates the effects of ISI as indicated in the Figure 2.3. All of these sub-carriers experiences â&#x20AC;&#x17E;flat fadingâ&#x20AC;&#x; because they have a bandwidth less than the Mobile channel coherence bandwidth [56]. Figure 2.4 shows a baseband transceiver structure for OFDM utilizing the Fourier transform for modulation and demodulation. Here the serial data stream is mapped to complex data symbols (Phase Shift Keying (PSK), QAM, etc.) with a symbol rate of . The data is then demultiplexed by a serial to parallel converter resulting in a block of N complex symbols, .The parallel samples are then passed through an N point IFFT (in this case no oversampling is assumed) with a rectangular window of length N.Ts, resulting in complex samples .Assuming the incoming complex data is random it follows that the IFFT is a set of independent random complex sinusoids summed together. The samples, are then converted back into a serial data stream producing a baseband OFDM transmit symbol of length T=N.Ts [57]. A Cyclic Prefix (CP), which is a copy of the final part of the samples, is appended to the front of the serial data stream before RF up conversion and transmission. The CP combats the disrupting effects of the channel which introduce ISI. In the receiver the whole process is reversed to recover the transmitted data, the CP is removed prior to the FFT which reverses the effect of the IFFT [58]. The complex symbols at the output of the FFT, are then decoded and the original bit steam recovered. Thus, the IFFT and FFT blocks at the transmitter and at the receiver, respectively, are important components in an OFDM system. A lot of work has gone into the optimization of the FFT implementations and the design community has leveraged this trend to advantage leading to the popularity of OFDM based systems. The time-

Chapter Two

LTE and OFAM

frequency view of an OFDM signal is shown in Figure 2.5, where the important parameters like subcarrier spacing and OFDM symbol period are shown [59].

Figure 2.3 OFDM subcarrier spacing [56].

Input

Signal Mapper

IDFT OR IFFT

S/P

P/S

Add CP

D/A

Multipath Fading Ch. & noise

output

Signal demapper

Equalizer And P/S

DFT OR FFT

Figure 2.4 a block diagram of a basic OFDM system.

S/P

Remove CP

A/D

Chapter Two

LTE and OFAM

Figure 2.5 Time-Frequency view of OFDM signal [59]

2.6.1. Orthogonality of the subcarriers and OFDM: Two functions or signals are said to be orthogonal if they are mutually independent of each other. Orthogonality is a feature that lets multiple information signals to be transmitted skillfully over a common channel with the successful detection [24 and 76]. The subcarrier spacing is chosen so that the waveforms transmitted on different sub carriers are orthogonal in time, but overlap in frequency. The orthogonality is achieved by making the peak of each subcarrier signal coincide with the null of the other subcarrier signals resulting in a perfectly aligned and spaced subcarrier signal [77]. The principle of orthogonality state that if the time-averaged integral of the product of ( ) ( ) }, over a any two functions from a set of functions { ( ) ( ) joint existence time interval [ ] is equal to zero, irrespective of the limit of existence of the functions, then the functions are told to be orthogonal to each other within this time-interval [16]. Mathematically, it can be expressed as – ∫

( )

(2.1)

The orthogonality property of OFDM signals can be shown with the help of its spectrum. In the frequency domain every OFDM subcarrier has a ( ) frequency response, as shown in Figure 2.6 [10]. One of the key advantages of OFDM is its efficient use of the frequency band as the subcarriers are allowed to overlap each other in the frequency domain. The N equally spaced subcarriers will be orthogonal if the frequency separation between subcarriers is f = , where N.Ts is symbol duration, and rectangular windowing of the IFFT is performed. Under these conditions the subcarriers will have a waveform frequency response [78]. Simple rectangular pulse of the length is used and rectangular shape in time domain corresponds to a -square shaped spectrum in frequency domain as illustrated in Figure 2.6. The LTE sub-carrier spacing is set to Δf= 15 KHz [62].

Chapter Two

LTE and OFAM

Figure 2.6 Per-subcarrier pulse shape and spectrum of basic OFDM transmission [48] Figure 2.7 shows the frequency response of a 5 carrier system where it is seen that because of the orthogonal relationship the maximum of a particular sample corresponds to a null in all other carriers, therefore eliminating the effects of interference.

Figure 2.7: Frequency spectrum of 5 orthogonal subcarriers of an OFDM transmit signal [78]. The orthogonality among sub carriers can be viewed in time domain as shown in Figure 2.8. Each curve represents the time domain view of the wave for a subcarrier. As seen from Figure 2.3, in a single OFDM symbol duration, there are integer numbers of cycles of each of the subcarriers [62]

Chapter Two

LTE and OFAM

Figure 2.8: Time domain representation of the signal waveforms to show orthogonality among the subcarriers [62]

2.6.2. Guard Interval: Individual sub channels can be perfectly separated by the FFT at the receiver when there are no ISI and Inter-channel Interference (ICI) introduced by channel distortion. Practically these conditions cannot be acquired. Since the spectra of an OFDM signal is not precisely band limited, linear distortion like multipath fading caused sub channel to spread energy in the adjacent channels [79, 80]. Figure 2.9 illustrates the CP insertion technicality, the Guard Interval or CP is a periodic addition of the final part of an OFDM symbol that is added to the front of the symbol in the transmitter, and at the receiver the CP is removed before demodulation [81]. It serves as a recurrence of the end of the symbol, so allowing the linear convolution of a frequency selective multipath channel to be modeled as circular convolution which in turn might be transformed to the frequency domain utilizing a discrete Fourier transform (DFT). This process allows for simple frequency domain processing like channel estimation and equalization [82]. CP insertion, therefore, increases the size of the data symbol from to , being the duration of the guard-period containing the CP. The standard length of the guard-period in LTE is defined to be 4.69 Îźs, allowing the system to tolerate path variations up to 1.4 km (considering the standard LTE symbol length of 66.7 Îźs). When a cyclic extension longer than a channel impulse response is added, the negative effect of the previous symbol can be avoided by simply removing that extension. CP insertion implies the copying of the last part of the OFDM data symbol and attaching it to the timing at the beginning of the symbol, creating a break between signals (hence: guarding-period). The receiver can then sample the incoming waveform at optimum time, as time-dispersion problems (i.e. delays caused by reflections of the signal) up to the length of the guarding-period are ignored [45].

Chapter Two

LTE and OFAM

Figure 2.9 the CP insertion mechanism [83]

2.6.3 One-tap equalizer [10]: The tap-delay line model with path is considered for multipath fading channel. After Considering the effect of the multipath fading channel, the samples of The received signal can be expressed as: (

)

∑

( ) (

)

(

)

(2.2)

( ) is the impulse response of multipath fading channel with path gains where, { ( ) }, is the path delay of path, and ( ) is a zeromean, unit variance complex Gaussian noise. After discarding first G sample of the received signal and taking Z-point FFT, the output of FFT block is ( ) given as : (2.3) Where, the term is the channel response to the subcarrier frequency and is the Additive white Gaussian noise (AWGN) term in the frequency domain. To compensate the fading effect of the channel, one-tap equalizer is used and each element of the vector is multiplied by an equalized gain factor the output of equalizer may be written as – ̂

(2.4) is defined as –

Where, (|

(

))

(2.5)

Chapter Two

LTE and OFAM

2.7 OFDM based Multiple Access: Various multiple access schemes can be combined with OFDM transmission and they include orthogonal frequency division multiplexing-time division multiple access (OFDM-TDMA), OFDMA, and multicarrier code division multiple access (MCCDMA). In OFDM-TDMA, time-slots in multiples of OFDM symbols are used to separate the transmissions of multiple users as shown in figure. 2.10. This means that all the used subcarriers are allocated to one of the users for a finite number of OFDM symbol periods. The only difference from OFDM-TDMA is that the users capture the channel and use it for certain duration, i.e., the time dimension is used to separate the user signals [84]

Figure 2.10: Time â&#x20AC;&#x201C; Frequency view of an OFDM-TDMA Signal In OFDMA systems, both time and/or frequency resources are used to separate the multiple user signals. Groups of OFDM symbols and/or groups of subcarriers are the units used to separate the transmissions to/from multiple users. In figure 2.11, the time, frequency view of a typical OFDMA signal is shown in a case where there are 3 users. It can be seen from figure 2.11 that usersâ&#x20AC;&#x; signals are separated either in the time-domain by using different OFDM symbols and/or in the subcarrier domain. Thus, both the time and frequency resources are used to support multiuser transmissions. We shall discuss this technique in more detail in the subsequent sections and also compare it with OFDM-TDMA [85].

Chapter Two

LTE and OFAM

Figure 2.11: Time â&#x20AC;&#x201C; Frequency view of an OFDMA Signal [85]

2.8 Orthogonal Frequency Division Multiple Access: The approach used in LTEâ&#x20AC;&#x;s access techniques consists of using OFDMA for the downlink (DL) and SC-FDMA for the uplink (UL). The main reason that justifies different access techniques for the UL and DL is the fact that SC-FDMA optimizes range and power consumption at the UE, while OFDMA minimizes receiver complexity and enables frequency domain scheduling with flexibility in resource allocation. OFDMA is a multi-carrier transmission scheme in opposition to SC-FDMA. Both allow multiple user access, depending on the available bandwidth, by dynamically allocating each user to a specific time-frequency resource, depending on which duplexing is deployed. OFDM requires a large dynamic range due to PAPR [86 and 87]. The main difference between an OFDM system and an OFDMA one is represented in Figure 2.12. The different colors represent different users using resources. In OFDM, users are assigned to resources in the time domain only, while in OFDMA, users can be assigned also in the frequency domain, optimizing resource usage. In OFDMA systems, the multiple user signals are separated in the time and/or frequency domains. OFDMA has been developed with multi-user operation as its purpose, allowing a flexible assignment of bandwidth to users according to their needs. Typically, a burst in an OFDMA system will consists of several OFDM symbols. The subcarriers and the OFDM symbol period are the finest allocation units in the frequency and time domain, respectively. Hence, multiple users are allocated different slots in the time and frequency domain, i.e., different groups of subcarriers and/ or OFDM symbols are used for transmitting the signals to/from multiple users. For instance, we illustrate an example in figure 2.13 wherein the subcarriers in an OFDM symbol are represented by arrows and the lines shown at different times represent the different OFDM symbols. We have considered 3 users and we have shown how resources can be allocated by using the different subcarriers and OFDM symbols [88 and 89]. 20

Chapter Two

LTE and OFAM

Figure 2.12 Difference between OFDM and OFDMA resource by user allocation [86].

Figure 2.13: Example allocation of resources to users in an OFDMA system [85].

Figure 2.14 is a detailed block diagram of OFDMA. The LTE PHY (Physical Layer) specification has been designed to adapt bandwidths from 1.25 MHz to 20 MHz OFDM was selected as the main modulation scheme due to its robustness with a severe multipath fading. Downlink multiplexing is achieved through the OFDMA. OFDM is the modulation scheme for the DL. The primary subcarrier spacing is 15 kHz, with lower subcarrier spacing of 7.5 kHz available for some MB-SFN (Multicast-broadcast single-frequency network) scenarios. OFDM modulation parameters summarizes in Table 2-1 [90]

Chapter Two

LTE and OFAM

Table 2-1 Downlink OFDM Modulation Parameters [90] Transmission 1.25 MHz 2.5 5 MHz 10 MHz 15 MHz BW MHz Sub-frame duration Sub-carrier spacing Sampling frequency

FFT size No. occupied subcarrier

20 MHz

0.5 ms 15 kHz 192 MHz (1/2 x 3.84 MHz)

128 of 76

3.84 MHz

7.68 MHz (2 x 3.84 MHz)

15.36 23.04 30.72 MHz MHz MHz (6 (8 x 3.84 (4 x x 3.84 MHz) 3.84 MHz) MHz)

256 151

512 301

1024 601

1536 901

2048 1201

Chapter Two

LTE and OFAM

Figure 2.14 Complete block diagram of an OFDMA transmitter and receiver [91] 23

Chapter Two

LTE and OFAM

2.9 SC-FDMA: In cellular systems, the wireless communication service in a certain geographical area is supplied by multiple base stations. The downlink transmissions in cellular systems are one-to-many, whilst the uplink transmissions are many-to-one. A one-to-many service means that a base station transmits concurrent signals to multiple users‟ equipment‟s in its coverage area. This demands that the base station has very high transmission power ability; as a result of the transmission power is involved for transmissions to multiple users‟ equipment‟s [92]. On the other hand, in the uplink, a single user‟s equipment has all its transmission power available for its uplink transmissions to the base station. In the uplink, the design of an effective multiple access and multiplexing scheme is more challenging than on the downlink because of the many-to-one nature of the uplink transmissions. Another consequential requisite for uplink transmissions is the low signal peakiness by means of the limited transmission power at the user‟s equipment [92]. One of the main parameters that affect all mobile UE devices is their battery life. It is therefore necessary to ensure an economic and efficient power use in the transmission and reception of signals. With the RF power amplifier (i.e enhancer of mixed signals) and the transmitter being the parts with the highest energy consumption within the mobile UE; it is essential to establish a transmission model with near constant operating power level [45]. The downlink physical layer of LTE is depending on OFDMA. Thus, in spite of its many advantages, OFDMA has specific drawbacks like high sensitivity to frequency offset (Doppler spread by cause of mobility and Arising from the instability of electronics) and PAPR. PAPR occurs due to the random constructive addition of subcarriers and results in spectral spreading of the signal which leads to adjacent channel interference. It is a problem that could be insurmountable with high compression point power amplifiers and amplifier linearization techniques. While these approaches may be utilized on the base station, they become costly on the UE [93 and 94]. In LTE, a new concept is used for the access technique of the uplink, called SCFDMA. Its characteristics combine lower PAPR of single-carrier systems because there is only a single carrier unlike N carriers. (Which allows maintaining a lower operating power level than OFDMA) with immunity to multipath interference, as well as flexible subcarrier frequency allocation (as a crucial part of OFDM) [45]. Figure 2.15 shows the concepts of OFDMA and SC-FDMA.

Chapter Two

LTE and OFAM

Figure 2.15 frequency domain description of downlink and uplink LTE access technologies

SC-FDMA differs from OFDMA in one additional transmission step, caused by the single-path transmission of single-carrier systems. That transmission step, called resource element mapping (and its counterpart, resource element selection), shifts all symbols obtained through the FFT to the desired center frequency and passes them on to the IFFT for further conversion Figure 2.16. Since the power of the modulation signals used in this process is constant (QPSK (Quadrature Phase Shift Keying), 16QAM and 64QAM) and the result of the resource element mapping step is a waveform similar to the original, on another center frequency; the required result of a constant-power signal is achieved [45]. For practicality, SC-FDMA is implemented in LTE utilizing a Discrete Fourier Transform Spread OFDM transmission (DFTS-OFDM) which is repeatedly referred to as a frequency-domain generalization of SC-FDMA. The DFT is used to multiplex uplink transmissions in definite frequency allocation blocks within the general system bandwidth in accordance with eNodeB scheduler instructions. The bandwidth of the single carrier is specified based on the desired data rate by the user. Data remains serial and not parallelized as done on the downlink with OFDMA (i.e. one information bit is being transmitted at a time). This results in similar link performance parameters for the uplink and downlink. Nevertheless, there would be comparatively high ISI for the uplink because of the single carrier modulation. Thus, the eNodeB receiver requires a low-complexity equalizer to rectify for the distorting impacts of the radio channel. SC-FDMA is not as sensitive to Doppler Effect and frequency instability the as OFDM by cause of its single carrier nature [93].

Chapter Two

LTE and OFAM

Figure 2.16 Block diagram of an SC-FDMA transmitter and receiver [37]

Chapter Three

Peak-to-Average Power Ratio Reduction Chapter Three

Peak-to-Average Power Ratio Reduction:

High PAPR of transmitted signals is one of the major issues of the OFDM system. A large dynamic range of input data symbols is the main cause of getting high PAPR. An OFDM signal consists of independent data symbols modulated on N orthogonal subcarriers, and when these signals are added to the same phase, higher peak amplitude is observed. The value of this peak may be times of the average amplitude [10].

3.1 Definitions of PAPR: For a continuous time baseband OFDM signal, the PAPR of any signal is defined as the proportion of the maximum instantaneous power of the signal and its average power. If x (t) is a transmitted baseband OFDM signal, then PAPR is defined as: ,

, ( )-

( )

(3.1)

Where, is the average power of x (t) and can be computed in frequency domain because IFFT is a unitary transformation is useful duration of an OFDM symbol [95]. For a discrete OFDM signal, the PAPR is computed from time oversampled OFDM signal as: , ( )-

The

[ [

( ) ( )

, ( )- at (dB) =

]

(3.2)

] [ [

( ) ( )

]

(3.3)

Where, , - denotes the expectation operator and is the total number of subcarriers. The PAPR of pass band OFDM signal is approximately twice that of baseband PAPR [95]. The above power characteristics can also be described in terms of their magnitudes (not power) by defining the crest factor (CF), which is defined as the ratio between maximum amplitude of OFDM signal ( ) and root-mean-square (RMS) of the waveform. The CF is defined as: | ( )| ,|| ( )| |-

â&#x2C6;&#x161;

(3.4)

In most cases, the peak value of signal ( ) is equals to a maximum value of its envelope | ( )| However, it can be seen from Figure 3.1 that the appearance of peak amplitude is very rare, thus it does not make sense to use max | ( )| to represent the 27

Chapter Three

Peak-to-Average Power Ratio Reduction

peak value in real application. Therefore, the PAPR performance of OFDM signals is commonly measured by certain characterization constants which relate to probability [96].

Figure 3.1: High PAPR when sub-carriers are modulated by same symbols [96]

3.2 PAPR of OFDM signal [62]: The discrete time baseband OFDM signal is defined in (6). The PAPR of the discrete time OFDM signal determines the complexity of the digital circuitry in terms of the number of bits necessary to achieve the desired signal to quantization noise ratio during signal digitization and recovery. To better approximate the PAPR of a continuous time OFDM signal, the discrete time OFDM signal is to be obtained by L times oversampling. The oversampled discrete time OFDM signal can be obtained by performing LN point IFFT on the data block with (L-1) N zero padding as follows: , ( )-

(

)

√

, 0≤ n ≤NL-1

(3.5)

PAPR of the oversampled OFDM signal of becoming , ( )-

, ,

( )

(3.6)

where, E[. ] denotes the expectation operator and N is total number of sub-carriers. The PAPR of passband OFDM signal is approximately twice that of baseband PAPR. Complementary Cumulative Distribution Function (CCDF) for an OFDM signal can be written as: P (PAPR > PAP

(

)

(3.7)

Where PAP is the clipping level. This equation can be read as the probability that the PAPR of a symbol block exceeds some clip level PAP .

Chapter Three

Peak-to-Average Power Ratio Reduction

3.3 Oversampling discrete OFDM symbols to find true (continuous) peaks: The PAPR for the discrete-time baseband signal x [n] may not be the same as that of the continuous-time baseband signal ( ) In fact, the PAPR for , - is lower than that for ( ), simply because , - may not have all the peaks of ( ) In practice, the PAPR for the continuous-time baseband signal can be measured only after implementing the actual hardware, including digital-to-analog convertor (DAC). In other words, measurement of the PAPR of the continuous-time baseband signal is not straightforward. Therefore, there must be some means of estimating the PAPR from the discrete-time signal , -. Fortunately, it is known that , - can show almost the same PAPR as ( ) if it is L-times interpolated (oversampled) as shown in Figure 3.2 where L ≥ 4 [97 and 98].

Figure 3.2 Block diagram of L time‟s interpolator [83]

3.4 Distribution of PAPR: To design and develop an effective PAPR reduction technique, it is very important to accurately identify the distribution of PAPR in OFDM systems. The distribution of PAPR plays an important role in the design of the whole OFDM system. The distribution of PAPR can be used in determining the proper output back-off of the HPA to minimize the total degradation. It can be used directly to calculate the BER and to estimate the achievable information rates [10]. For the OFDM system, if we assume that the input data stream is statistically independent and identically distributed (i.e.) then the real and imaginary parts of x[n] are uncorrelated and orthogonal. From central limit theorem, it follows that, for large values of N, the real and imaginary parts of x[n] are independent and identically distributed (i.e.) Gaussian random variables, each with zero mean and variance ,| , - | -

(3.8)

The probability distribution of complex OFDM signals with large N is a complex Gaussian distribution given by following relation: * , -+

√

, -

(3.9)

Where Pr{.} denotes the probability distribution function. Where, is the variance of , -.The amplitude of OFDM signal has a Rayleigh distribution and its probability density function (PDF) is given by: * , -+

| , -|

(3.10) 29

Chapter Three

Peak-to-Average Power Ratio Reduction

The histogram plots for the real part, imaginary part and the absolute value of a time domain OFDM signal are shown in Figure 3.3(a), (b) and (c) respectively. The plots shown in Figures 3.3(a) and (b) are obtained after performing the computer simulations of an OFDM system having N=256 QPSK modulated subcarriers as shown in Fig. 2.4. The signal obtained from IFFT block of Figure 2.4 is complex OFDM signal. After that real, imaginary and absolute values of OFDM signal (x[n]) are calculated and their histograms are plotted [62]. The power of OFDM signal has chi-square distribution. The distribution of PAPR is often expressed on the one hand Complementary Cumulative Distribution Function (CCDF). In probability theory and statistics, the CCDF describes the probability that a real-valued random variable X with a given probability distribution will be found at a value greater than or equal to x [99 and 10]. The Cumulative Distribution Function (CDF) of the PAPR of the amplitude of a signal sample is given by ( )

( )

(3.11)

The CCDF of the PAPR of the data block is desired in our case is to compare outputs of different reduction techniques. This is given by: (

)

(

)

(3.12)

( ) ( Where,

(3.13) (

)

(3.14)

is the given reference level.

Figure 3.3 (a) 30

Chapter Three

Peak-to-Average Power Ratio Reduction

Figure 3.3 (b)

Figure 3.3 (c) Figure 3.3: Histogram of (a) Real part of OFDM signal amplitude (b) Imaginary part of OFDM signal amplitude (c) OFDM signal magnitude [63]. 31

Chapter Three

Peak-to-Average Power Ratio Reduction

3.5 Identification of the Problem: Multi-carrier phenomena is considered to be one of the major development in wireless communication and among them OFDM is becoming the important standard. However, high PAPR is the major drawback of OFDM, which results in lower power efficiency hence impedes in implementing OFDM. To overcome the low power efficiency requires not only large back off and large dynamic range DAC but also highly efficient HPA and linear converters. These demands result in costly hardware and complex systems. Therefore to lessen the difficulty of complex hardware design it has become imperative to employ efficient PAPR reduction techniques [100 and 101]. The drawback of a large dynamic range is that it places pressure on the design of components such as the word length of the IFFT/FFT pair, mixer stages, and most importantly the HPA, which must be designed to handle irregularly occurring large peaks, decreases the SQNR (Signal-to-Quantization Noise Ratio) of ADC (Analog-toDigital Converter) and DAC, The PAPR problem is more important in the uplink since the efficiency of power amplifier is critical due to the limited battery power in a mobile terminal. Failure to design components with a sufficiently large linear range result in saturation of the HPA [98, 78]. Saturation creates both in band distortion, increasing the BER and out of band distortion, or spectral splatter, which causes Adjacent Channel Interference (ACI). One obvious solution is to design the components to operate within large linear regions, however this is impractical as the components will be operating inefficiently and the cost becomes prohibitively high. This is especially apparent in the HPA where much of the cost and ~50% of the size of a transmitter lies which will be explained in next sections [98, 78].

3.5.1 Nonlinear HPA and DAC: HPA are used in the transmitter of communication systems for sufficient transmission power. To achieve maximum output power efficiency they have to be operated at or near the saturation region. [100] If the data on the subcarriers add up in a constructive manner at the transmitter, the resulting signal could exhibit large PAPR. As a result, the composite transmit signal could be severely clipped by the DAC and power amplifiers for their bounded dynamic range as described in Figure 3.4. In this case, the reconstructed output Ě&#x201A;( ) can possess a significant amount of distortion. It can be reduce the PAPR of an OFDM signal by modifying the signal characteristics in time-domain or frequency domain clipping of the composite OFDM signal causes several undesirable outcomes, such as signal distortion and spectral regrowth. For instance, clipping causes in band noise that results in a degradation of the BER performance .Moreover, higher-order harmonics that spill over into OOB spectrum can also result from signal clipping. Although filtering after the HPA can be employed to remove this spectral leakage, it is very power-inefficient, so it is an undesirable solution. Therefore, the dynamic range of DAC should be large enough to accommodate the largest peaks of signals or high PAPR values [102]. A high-precision DAC support high PAPR with acceptable amount of quantization noise, but could be very costly to a certain sampling rate of the system. On the other hand, a low-precision DAC would be cheaper, but the quantization noise will be significant, which reduces the signal SNR (Signal to Noise Ratio) when the dynamic range of DAC is increased to support high PAPR. Otherwise, the DAC will saturate and clipping will occur [48, and 103]. 32

Chapter Three

Peak-to-Average Power Ratio Reduction

Figure 3.4 An example illustrating effect of clipping. The dynamic range of the power amplifiers should also be large enough to accommodate large PAPR values. Otherwise, the power amplifiers may saturate and clipping might occur. The component cost of the DAC and power amplifiers increase with the increase in the dynamic range.

Chapter Three

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It is worth mentioning that the clipping of high signal peaks rarely happens, resulting in a comparatively low incidence clipping noise. In this manner, the impact of clipping at the transmitter on the error performance of the OFDM system liable to be subjected frequency selective fading is minimal [102]. If an HPA with limited linear range is utilized for amplification, it may operate near saturation and can cause OOB radiations and in-band distortion. The OOB distortion/noise is a major concern, especially in wireless communications, where large differences in signal strength from a mobile transmitter impose stringent requirements on ACI [104] Figure 3.5 demonstrates a classic input-output characteristic of a power amplifier. For prevent or limit signal distortion input signals must be preserved below the Non-linear area. The result is that the amplifier is not completely used [105] IBO = 10 OBO = 10

( (

)

(3.15)

)

(3.16)

IBO (Input Back-Off) or OBO (Output Back-Off) High PAPR results in a wide variety of OFDM signal amplitudes which due to nonlinear characteristics of HPA findings in inter-modulation among the various sub carriers and leading to an increment in BER. To realize a low BER and minimal signal distortion, HPA must be a large dynamic range and work in the linear amplifier region. But, these types of HPA are expensive and smaller power efficient. The power efficiency in wireless communication is very important for achieving efficient area coverage and small size terminals. Thus, the power efficient process of non-linear HPA is so important. Accordingly, it is best to target the reduction of PAPR the OFDM signals before transmitting the signal into nonlinear DAC and HPA [100].

Figure 3.5 Typical input-output characteristics of a power amplifier showing the Relation between Output Back-Off (OBO) and Input Back-Off (IBO) [98].

Chapter Three

Peak-to-Average Power Ratio Reduction

3.5.2 Power Saving [100]: A high dynamic range HPA has low power efficiency. The power could save by reducing PAPR. This power saving that is implemented in such a way to provide a direct correlation with the desired average output power. On the assumption a linear model of HPA, the power efficiency is: (3.17) (3.18) The η= HPA efficiency . = the average of the output power. . = A fixed amount of power regardless of their input power. For example: an OFDM signal with 256 sub carriers that demand an IBO equal to the PAPR at the probability level lower than 0.01%, i.e. (25.235).This makes η = 0.5/25.235≈1.98% The PAPR of OFDM systems has to reduce for avoiding this level of power inefficiency.

3.6 Factors influencing the PAPR: 3.6.1 The number of sub carriers: In Multi-Carrier Systems the complex base band signal for one symbol in an OFDM system can be expressed as follows: ( )

√

∑

(3.19)

Where N is the modulating symbol and is the number of sub carriers. For moderately large numbers of m-PSK (multiple phase-shift keying) sub carriers the quadrature components of x (t) each tends towards a Gaussian distribution (giving the sum of their power amplitude a Rayleigh distribution). Consequently, whilst the peak value possible is N times the individual sub carrier peak, the probability of any value close to that peak occurring is very low. For example, with only 24 sub carriers, the probability of the PAPR exceeding 4dB is and of exceeding 8dB is only [99].

3.6.2 The order of Modulation: High data bandwidth efficiency (in terms of b/s/Hz) this can be achieved by utilizing higher order modulations based, for instance, on QAM. When using a higher-order modulation such as QAM type, the PAPR of the summed OFDM signal is increased by the PAPR of the QAM constellation utilized. Nevertheless, the probability of these higher peaks happening is accordingly less. Furthermore, since among benefits of OFDM is one that sub carriers could have their modulation independently varied to adapt to channel conditions, the joined PAPR in any system utilizing this technique might are hard to predict and control. PAPR for an unfiltered base band signal is listed in the following Table 3.1. [100]. 35

Chapter Three

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Table 3.1 PAPR for picked modulation formats Modulation PAPR 256-QAM 4.23dB 64-QAM 3.68dB 256-QAM (modified) 2.85dB 16-QAM 2.55dB m-PSK (reference) 0 dB

3.6.3 Constellation shape: The last entry in Table 3.1 is for a constellation obtained by modifying 256- QAM to reduce PAPR. This modified constellation shape is shown in figure 3.6. However, there is an additional processor load associated with encoding and decoding this constellation.

Figure 3.6 256-QAM constellations: (a) regular and (b) modified mapping to reduce PAPR

3.6.4 Pulse Shaping: In terrestrial communications, it is popular to use pulse shaping to the base band signal, to decrease the bandwidth of the transmitted spectrum, but this causes overshoot and can increase the PAPR of the modulating signal by 4-5 dB [100].

3.7 The gauge for judgment of the PAPR reduction in OFDM systems [106, 107, 108]: Every method used to reduce the PAPR has some drawbacks and merits. There is always a trade-off between PAPR reduction and some other factors like bandwidth, computational complexity, average power etc. An ideal PAPR reduction technique should have following characteristics: 1) High potential to limit the PAPR: It is a key factor to consider in the selection of technology to reduce the PAPR with few adverse side effects like in-band distortion and OOB radiation. 2) Low average power: even though it can reduce PAPR through the average power of the original signals increase, it needs a bigger linear operation region in HPA and which led in the deterioration of BER performance.

Chapter Three

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3) Low implementation complexity: mainly, complexity techniques viewing better capability of PAPR reduction. Nevertheless, practically, both time and hardware requisites for the PAPR reduction must be minimal. 4) No bandwidth expansion: The bandwidth is an infrequent resource in systems. The bandwidth expansion has directly resulted in the data code rate loss because of side information (like the complementary bits in Complement Block Coding (CBC) and phase factors in PTS). Furthermore, when the side information is received in error unless some methods of protection like channel coding employed. For that reason, when channel coding is utilized, the loss in data rate is incremented further due to side information. Then, the loss in bandwidth because of side information must be avoided or at least be preserved minimal. 5) No BER performance degradation: The objective of the PAPR reduction is for the best system performance, including BER than that of the original OFDM system. For that reason, all the methods, which have an incrementation in BER at the receiver, must be paid more attention in practice. Additionally, if the side information is received in error at the receiver, which may also result in entire wrong data frame and thus the BER performance is reduced. 6) Without the additional power required: The design of a wireless system must always take into account the efficiency of power. If an operation of the technique which reduces the PAPR require more extra power, it deteriorates the BER performance when the transmitted signals are normalized back to the original power signal [109]. 7) No spectral spillage: Any PAPR reduction techniques cannot devastate OFDM fascinating technical features like immunity to the multipath fading. Thus, the spectral spillage must be avoided in the PAPR reduction. 8) Other factors: It must be driven greater concentration on the effect of the nonlinear devices utilized in signal processing loop in the transmitter like DACs, mixers and HPAs since the PAPR reduction fundamentally avoid nonlinear distortion as a result of these memories-less devices introducing into the communication channels. At the same time, the expense of these nonlinear devices is too the important factor to design the PAPR reduction scheme.

3.8 Fitness function-based approach for determining an appropriate Algorithm [110]: In order to determine an appropriate PAPR reduction algorithm for a given system, it is desirable to consider all above-listed requirements. The number and nature of these requirements may vary depending upon the system (or user) under consideration. For a given scenario and requirements, we propose to use the fitness value or appropriateness value of the algorithm, which is defined as the weighted sum of the relative changes in the above-listed factors. The appropriateness value provides a single metric for determining the appropriateness of a PAPR reduction algorithm. Suppose X1 be the relative degradation in BER performance at certain SNR level, for given channel conditions, AWGN or multipath, given by: X1 = â&#x2C6;&#x2019;10 ( ) (3.20) Let X2 be the relative increase in system complexity given by: X2 = â&#x2C6;&#x2019;10

(

)

(3.21)

Chapter Three

Peak-to-Average Power Ratio Reduction

Let X3 be the relative PAPR reduction given by: X3 = −10

(

)

(3.22)

Let X4 be the relative cost savings given by: X4 = −10

(

)

(3.23)

Let X5 be the relative increase in transmit power given by: X5 = −10

(

)

(3.24)

Let X6 be the relative increase in spectral spillage given by: X6 = −10

)

(3.25)

Let X7 be the relative reduction in goodput5 given by: X7 = −10

(

)

(3.26)

The aggregate fitness value of the PAPR reduction algorithm can be computed as the weighted sum of these factors, where the weights correspond to their relative importance levels. These weights can be determined as per the system or user requirements. Therefore, the fitness value of the algorithm is given by: ∑

(3.27)

Where ∑

(3.28)

Based on these fitness values, an appropriate algorithm can be chosen in order to achieve large reduction in PAPR values as well as satisfy other system requirements.

Chapter Four

PAPR Reduction Techniques Chapter Four PAPR Reduction Techniques

4.1There are three different way to divide the PAPR: 4.1.1The first way is [110]: PAPR reduction techniques can be categorized into deterministic and probabilistic approaches, as shown in Figure 4.1. Deterministic approaches guarantee that the PAPR of an OFDM signal does not exceed a predefined threshold, whereas the probabilistic approaches minimize the probability that the PAPR of an OFDM signal exceeds a predefined threshold. These categories will be discussed in the following sections 1) Deterministic Approach Deterministic PAPR reduction approaches can be classified into techniques that perform either time-domain based clipping or frequency-domain based coding. The simplest approach for PAPR reduction is to deliberately clip the amplitude of the signal to a predefined value before amplification [111]. However, the technique suffers from various drawbacks, such as signal distortion and spectral regrowth. Therefore, clipping alone is not a suitable option for PAPR reduction. Modified clipping techniques exist, which fall under the probabilistic approach explained in the next section. Coding techniques are applied to OFDM signals in order to map symbols to codes with smaller PAPR values [112] . Each symbol has a choice of two or more codes, where the code yielding the lowest PAPR is selected. However, this technique works well only when the number of subcarriers is small. With the increased number of subcarriers, the search space for finding codes with minimum PAPR increases exponentially and large lookup tables are needed for encoding and decoding. 2) Probabilistic Approach Probabilistic approaches attempt to minimize the number of occurrences of OFDM symbols with PAPR values exceeding a predefined threshold, while simultaneously minimizing the signal distortion and spectral growth. Probabilistic approaches can be classified according to whether time domain processing or frequency domain processing is involved: ď&#x201A;ˇ time Domain-Based Processing: Time domain-based processing approaches focus on manipulating the power of the signal in the time domain. This approach can be further classified into blind and nonblind techniques. Blind techniques imply that the receiver is oblivious to the changes made at the transmitter side, whereas non-blind techniques imply that the receiver requires a priori knowledge about the modifications made at the transmitter side for correctly demodulating the received signals. Thus, non-blind techniques require additional side information in order to operate, whereas blind techniques might degrade the error performance of the system since the receiver is transparent to the changes made at the transmitter side.

Chapter Four

PAPR Reduction Techniques

The simplest blind technique for PAPR reduction is to clip the amplitude of the signal to a predefined value and filter the signal to suppress the out-of-band interference [113,114, 115 ] . The clipping process might result in spectral regrowth, whereas filtering the signal might result in some peak regrowth. Therefore, clipping may not be an effective technique when reducing the PAPR of the OFDM signals as long as the transmitted OFDM signal is strictly band-limited. Even though numerous algorithms based on amplitude clipping and filtering have been proposed in the literature, it has been shown that clipping does not improve the reduction of total degradation [116]. Instead, an unclipped system outperforms a clipped system because of the inter-carrier interference (ICI) caused by clipping, and offsets the gain of the PAPR reduction [116]. Another technique called peak windowing can also reduce the PAPR, where large signal peaks are multiplied with a certain narrowband window such as Gaussian, Cosine, Kaiser, and Hamming windows [117]. Among the non-blind techniques, several companding4 techniques for compressing the large peaks of an OFDM signal in time domain, including Îź-law companding , and exponential companding , have been proposed in literature. By compressing the large peaks of an OFDM signal by companding, the dynamic range of the D/A converters are reduced. However, the receiver needs to expand the compressed signal for correct demodulation. ď&#x201A;ˇ Frequency Domain-Based Processing Frequency domain-based processing approaches focus on minimizing the correlation of the input signals since it is known that the PAPR of an OFDM signal is high when the input sequences are highly correlated. It has been shown that by altering the phase and/or power of the input sequence, it is possible to lower the correlation of the input sequence, thereby reducing the PAPR of an OFDM signal. However, some approaches also try to directly manipulate the correlation of the input signals. Frequency domain-based processing approaches can be further classified into blind and non-blind techniques. In blind phase adjustment-based techniques, the phase of the subcarriers are adjusted in order to reduce the coherence between the different subcarriers such that the PAPR value of the OFDM signal is reduced. The phase adjustments should be kept relatively small so as to minimize bit-error-rate (BER) performance degradation. For example, signal set expansion technique maps original signal set into an expanded signal set with two or more points, such as binary phase shift keying (BPSK) into quadrature phase shift keying (QPSK), which provides more freedom for phase selection and yields lower PAPR values for the OFDM signal [118]. Blind power-based techniques alter the power level of the subcarriers such that the PAPR of an OFDM signal is reduced. These techniques are suitable only for the MPSK-based OFDM system since the receiver is unaware of the information about the transmit power levels. For example, the input sequence envelope scaling technique adjusts the power of the subcarriers so that the power of the individual subcarriers becomes unequal yielding a minimized PAPR value [119]. Since the phase information of the original signal is unchanged, the receiver can decode the received signal without any side information. In blind power and phase-based techniques, both the phase and the power of the subcarriers are altered such that the PAPR of an OFDM signal is reduced. If the total transmit power needs to be kept constant, these techniques are suitable only for low order modulation techniques since the error robustness of the higher modulation techniques degrades rapidly with the blind phase and power alterations at the 40

Chapter Four

PAPR Reduction Techniques

transmitter. When the order of the modulation techniques in-creases, the complexity (and limitations) of the algorithm increases as well as transmit power level increases. For example, the active constellation extension (ACE) [120,121] and dynamic constellation shaping techniques allow changing the power and phase of some data symbols without affecting the error probability of the other data symbols. Non-blind power-based techniques, as well as power and phase-based techniques, would be suitable for the higher modulation schemes such as MQAM. Non-blind phase adjustment-based techniques update phases of the input sequence such that the PAPR of an OFDM signal is reduced. The information about the phase updates is transmitted to the receiver for correct demodulation. Several modified algorithms are proposed in literature, which avoid the requirement of explicit side information. For example, selective mapping (SLM)[9], partial transmit sequences (PTS) [122], random phase updating [123] techniques add random phase factors to each subcarriers in order to reduce PAPR with the information about the phase factors transmitted to the receiver. The blind techniques reduce the PAPR values at the cost of slight increase in the bit error rate of the system or increased transmit power level since the adjustments would result into increased noise level at the receiver, whereas the nonblind techniques reduce the PAPR values at the cost of a reduced information rate since the information about the adjustments made at the transmitter need to be transmitted to the receiver for the correct demodulation. A low autocorrelation coefficient of a signal is a sufficient condition for low PAPR. However this is not a necessary condition [124][125]. Non-blind autocorrelation minimization techniques attempt to minimize the autocorrelation of the input sequence `and the information about the changes is transmitted to the receiver for correct demodulation. For example, the selective scrambling [126] and interleaving techniques [127] attempt to break the long correlation patterns of the input sequences to reduce the PAPR. However, the techniques perform well only when the OFDM signal has moderate PAPR values since interleaving alone is not effective to break the correlation pattern when the input sequence are highly correlated. Attempts have been made to develop OFDM signals with a constant envelope to yield unity PAPR values [128] . The constant envelope waveforms have a constant instantaneous power. Continuous phase modulation (CPM) is a class of signaling that has very low side lobe power while maintaining the constant envelope property. However, CPM increases the complexity of the receiver and has a poor performance over frequency selective channels.

Chapter Four

PAPR Reduction Techniques

Figure 4.1.the first way taxonomy of PAPR Reduction techniques

Chapter Four

PAPR Reduction Techniques

4.1.2 The second way is : a) Distortion Based Techniques [11]-[8]-[4] b) Scrambling Techniques [17]-[16]-[8] As shown in figure 4.2 a. DISTORTION BASED TECHNIQUES The schemes that introduce spectral re-growth belong to this category. Distortion based techniques are the most straightforward PAPR reduction methods. Furthermore, these techniques distort the spectrum, this spectrum distortion or â&#x20AC;&#x153;spectral re-growthâ&#x20AC;? can be corrected to a certain extent by using filtering operation [62 ,129]. These methods reduce the PAPR by distorting the OFDM signal non-linearly. The methods like clipping and filtering, peak windowing, and non-linear companding are the example of these techniques. These techniques are applied after the generation of OFDM signal (after the IFFT) [130]. The distortion category attempts to reduce PAPR by manipulation of signal before amplification. Clipping of signal prior to amplification is a simplest method but it causes increase in both out-of-band (OOB) as well as in-band interference thus compromises upon performance of system. Amongst this category better techniques include companding, peak windowing, peak power suppression, peak cancellation, weighted multicarrier transmission etc. Any technique which is used to reduce PAPR should not only have high spectral efficiency but must be compatibility with the existing modulation schemes and at the same time must not be computational complex [100]. b. Scrambling techniques : Signal scrambling techniques are all variations on how to scramble the codes to decrease the PAPR. Coding techniques can be used for signal scrambling. Golay complementary sequences, Shapiro-Rudin sequences, M sequences, Barker codes can be used efficiently to reduce the PAPR. However with the increase in the number of carriers the overhead associated with exhaustive search of the best code would increase exponentially. More practical solutions of the signal scrambling techniques are block coding, Selective Level Mapping (SLM) and Partial Transmit Sequences (PTS). Signal scrambling techniques with side information reduces the effective throughput since they introduce redundancy [131] [132].

Chapter Four

PAPR Reduction Techniques

Figure 4.2.the second way taxonomy of PAPR Reduction techniques

Chapter Four

PAPR Reduction Techniques

4.1.3 The third way is [98]: These methods are basically divided in five categories: (1) The clipping technique (2) Coding Methods, (3) Probabilistic (Scrambling) Techniques (4) Pre-distortion Methods. 1. The clipping technique employs clipping or nonlinear saturation around the peaks to reduce the PAPR. It is simple to implement, but it may cause in-band and out-ofband interferences while destroying the orthogonality among the subcarriers. This particular approach includes block-scaling technique, clipping and filtering technique, peak windowing technique, peak cancellation technique, Fourier projection technique, and decision-aided reconstruction technique [133] [134]. 2. The coding technique is to select such code words that minimize or reduce the PAPR. It causes no distortion and creates no out-of-band radiation, but it suffers from bandwidth efficiency as the code rate is reduced. It also suffers from complexity to find the best codes and to store large lookup tables for encoding and decoding, especially for a large number of subcarriers. Golay complementary sequence, Reed Muller code, M-sequence, or Hadamard code can be used in this approach [133][134]. 3. The probabilistic (scrambling) technique is to scramble an input data block of the OFDM symbols and transmit one of them with the minimum PAPR so that the probability of incurring high PAPR can be reduced. While it does not suffer from the out-of-band power, the spectral efficiency decreases and the complexity increases as the number of subcarriers increases. Furthermore, it cannot guarantee the PAPR belowa specified level. This approach includes SLM (Selective Mapping), PTS (Partial Transmit Sequence). 4. The pre-distortion methods are based on the re-orientation or spreading the energy of data symbol before taking IFFT. The pre-distortion schemes include DFT spreading, pulse shaping or precoding and constellation shaping. The methods like Tone Reservation (TR) and Tone Injection (TI) are the example of constellation shaping schemes [10]. The DFT-spreading technique is to spread the input signal with DFT, which can be subsequently taken into IFFT. This can reduce the PAPR of OFDM signal to the level of Single-carrier transmission. This technique is particularly useful for mobile terminals in uplink transmission. It is known as the Single Carrier-FDMA (SC-FDMA), which is adopted for uplink transmission in the 3GPP LTE standard [135]. 4.1.4 And finally there is Hybrid techniques: Besides these different PAPR reduction techniques, some hybrid methods are also available in the literature [136 ,137,138 ] . These methods combine two or more than two techniques for PAPR reduction like clipping with coding, SLM with coding, precoding with clipping, interleaving and companding , Selective Mapping and Binary Cyclic Codes, combining Hadamard Transform and Hann peak windowing etc. The hybrid methods are considered as better choice for PAPR reduction because it possess the advantages of both techniques used in hybridization with slight increases in complexity.

Chapter Four

PAPR Reduction Techniques

4.2 Clipping and Filtering : The clipping is the simplest method of PAPR reduction. Clipping limits the maximum amplitude of OFDM signal to a pre-specified level. The implementation of clipping is relatively easy. The simplest and most widely used technique of PAPR reduction is to basically clip the parts of the signals that are outside the allowed region .For example; using HPA with saturation level below the signal span will automatically cause the signal to be clipped. For amplitude clipping, that is [109]:

(4.1) Where A is preset clipping level and it is a positive real number Generally, clipping is performed at the transmitter. However, the receiver need to estimate the clipping that has occurred and to compensate the received OFDM symbol accordingly. Typically, at most one clipping occurs per OFDM symbol, and thus the receiver has to estimate two parameters: location and size of the clip. However, it is difficult to get this information. Therefore, clipping method introduces both in band distortion and out of band radiation into OFDM signals, which degrades the system performance including BER and spectral efficiency. Filtering can reduce out of band radiation after clipping although it cannot reduce in-band distortion. However, clipping may cause some peak regrowth so that the signal after clipping and filtering will exceed the clipping level at some points [108] [109]. It has following drawbacks [98] [139]: (a) It causes in-band signal distortion, resulting in BER performance degradation. (b) It also causes out-of-band radiation, which imposes out-of-band interference signals to adjacent channels. The out-of-band radiation can be reduced by filtering, but the filtering may affect high-frequency components of in-band signal (aliasing) when the clipping is performed with the Nyquist sampling rate. (c) Filtering after clipping can reduce out-of-band radiation at the cost of peak regrowth. The signal after filtering operation may exceed the clipping level specified for the clipping operation. To reduce overall peak re-growth, a repeated clipping and filtering can be used to obtain a desirable PAPR at the cost of increase computational complexity . To reduce peak regrowth, a repeated clipping-and-filtering operation can be used to obtain a desirable PAPR at a cost of computational complexity increase. As improved clipping methods, peak windowing schemes attempt to minimize the out of band radiation by using narrowband windows such as Gaussian window to attenuate peak signals [140] Some of clipping techniques: 1. Repeated Clipping [13] The clipping technique is the simpler one which is used to cut the signal peak up to desired threshold level. But repeated clipping and filtering technique proved to be worthy one as it gives better result compared to earlier one. In this technique the peak regrowth which is generated in filtering can be minimized. So the repeated clip and filter process reduces these regrowth's in OFDM system 2. Reconstruction of Lost Clipped Signal 46

Chapter Four

PAPR Reduction Techniques

To remove the peak regrowth of signal oversampled sequence clipping is used which can reconstruct the clipped samples and mitigate the clipping distortion in presence of channel noise at the cost of bandwidth expansion. It is observed that by increasing small bandwidth , the performance of OFDM system can be improved . PAPR is the biggest problem in OFDM system. Many techniques are proposed for it. Clipping and filtering technique is considered to be the simplest one [114][106]. 3. Iterative Clipping & Filtering Technique This technique is used to eliminate the peak regrowth due to CF technique. In each iteration peak regrowth decreases significantly. The process of iteration undergoes FFT/IFFT and one extra IFFT is required for conversion into time domain in OFDM [115][106]. 4. Recursive Clipping and Filtering with Bounded Distortion (rcfbd) In RCF the signal is clipped by repeating process many times before feeding to power amplifier. When the process of repetition exhibit on the signal the out of band spectral density and the probability of the occurance of PAPR decreases but error rate increases due to increase in number of repetitions. The bit error rate increases due to increase in inband distortion. So to remove this increased error rate another improved technique is proposed called recursive clipping and filtering with bounded distortion (RCFBD) to achieve PAPR reduction. The idea of this technique is same as oversampled digital clipping in time domain and removing out of band components in frequency domains are used. But additional barrier on in band distortion of each subcarrier is applied during the recursive process. In this way PAPR can be reduced without producing any effect on the error rate [114][ 106]. RCFBD minimize PAPR and keeps the control on the distortion of data carried by each subcarrier. So by using this technique side information can be eliminated and receiver part becomes less complex and BER performance can be increased more. It is also more robust against AWGN noise [113].

4.3 Peak Windowing Method: It is an improved clipping method. The basic aim of peak windowing is to reduce the out-of-band radiation by using narrow band windows such as Gaussian window to attenuate peak signals. As a matter of fact, any window which is narrow in time domain and having good spectral properties can be used [10]. In 2008, an advance peak windowing method has been given by S. Cha which overcomes the drawback of normal peak windowing method. It effectively suppresses the peak signals to the desired threshold level in case when the successive peaks occur within a half of the window length [10]. The peak windowing method has been suggested by Van Nee and Wild [117]. This method, proposes that it is possible to remove large peaks at the cost of a slight amount of self-interference when large peaks arise infrequently. Peak windowing reduces PAPRs at the cost of increasing the BER and out-of-band radiation. Clipping is a one kind of simple introduces PAPR reduction technique which is selfinterference. The technique of peak windowing offers better PAPR reduction with better spectral properties. (Peak Windowing technique provides better PAPR reduction with better spectral properties than clipping) [141][142].

Chapter Four

PAPR Reduction Techniques

In peak windowing method we multiply large signal peak with a specific window, for example; Gaussian shaped window, cosine, Kaiser and Hamming window. In view of the fact that the OFDM signal is multiplied with several of these windows, consequential spectrum is a convolution of the original OFDM spectrum with the spectrum of the applied window. Thus, the window should be as narrow band as possible, conversely the window should not be too long in the time domain because various signal samples are affected, which results an increase in bit error rate (BER). Windowing method, PAPRs can be obtained to 4dB which from the number of independent subcarriers. The loss in signal-to-noise ratio (SNR) due to the signal distortion is limited to about 0.3dB. A back off relative to maximum output power of about 5.5dB is needed in spectra distortion at least 30dB below the in-band spectral density [141][142]. The PAPR reduction performance as well as spectral efficiency of peak windowing technique is better as compared to clipping. The major advantage of peak windowing is that PAPR reduction is achieved with minimal complexity for any number of sub carriers. The disadvantages include an increase in out-of-band interference and BER [100][143].

4.4 Envelope Scaling: The Envelope Scaling technique has been proposed by Foomooljareon and Fernando. They anticipated a new algorithm to reduce PAPR by scaling the input envelope for some subcarriers before they are sent to IFFT. They used 256 subcarriers with QPSK modulation technique, so that envelopes of all the subcarrie4rs are equal. The key idea of this scheme is that the input envelope in some sub carrier is scaled to achieve the smallest amount of PAPR at the output of the IFFT. Thus, the receiver of the system doesnâ&#x20AC;&#x;t need any side information for decoding the receiver sequence. This scheme is appropriate for QPSK modulation; the envelopes of all subcarriers are equal. Results show that PAPR can be reduced significantly at around 4 dB [144]. In Envelope Scaling, the input envelopes of sub carriers are scaled prior to IFFT. The base for this scheme is the facts that with PSK modulation all the sub carriers input envelops are equal. Hence input envelop of some sub carriers is scaled in such a way that minimum PAPR is achieved at IFFT output. The input which yields minimum PAPR is fed into the system. The phase information of the input sequence is similar to original however envelops are not the same. Hence decoding of sequence can be done by receiver without any requirement for side information .The major drawback of this method is that it can only be used when OFDM is employing PSK modulation. On the other hand if we use this method when QAM modulation is implemented by OFDM, then there is severe degradation in BER performance results [100] [145] .

4.5 Peak Reduction Carrier: Peak Reduction Carrier technique is proposed by Tan and Wassell. The technique is to use the data bearing peak reduction carriers (PRCs) to reduce the effective PAPR in the OFDM system. It includes the use of a higher order modulation scheme to represent a lower order modulation symbol. Hence the phase and amplitude of these carriers remains inside the constellation area which represents the data symbols being transmitted. This method is suitable for PSK modulation; where the envelopes of all subcarriers are the same. When the QAM modulation scheme will be implemented in the OFDM system, the carrier envelope scaling will result in the serious BER 48

Chapter Four

PAPR Reduction Techniques

degradation. To limit the BER degradation, amount of the side information would also be excessive when the number of subcarriers is large [141] Amongst drawbacks of PRCs, one is that the overall data transmission efficiency of the system is compromised if we try to achieve maximum PAPR reduction efficiency. At the same time the BER performance is also affected adversely because of employing constellation of higher order for carrying symbols of lower order results in higher probability of error [100]

4.6 Companding Technique: Non-linear companding is an especial clipping technique which offers good PAPR reduction with better BER performance, low implementation complexity, and no bandwidth expansion [109] [145]. The difference between clipping and companding is that the clipping process deliberately clips the large amplitude signals; therefore the signal cannot be recovered exactly. On the other hand, the companding transform compand the original signals using strict monotone increasing function; therefore the companded signals can be recovered correctly through the corresponding inversion of companding transform at the receiver. Clipping does not affect small amplitude signal, whereas companding enlarge the small signals while compressing the large amplitude signals. A lot of companding techniques are available. The basic concept of most of the companding techniques is to transform the Rayleigh distributed OFDM signal into a uniformly distributed signal [10]. It was based on the speech processing algorithm Îź-law and it has shown better performance than that of clipping method . The Îź-law companding transform mainly focuses on enlarging small amplitude signals while keeping peak signals unchanged, and thus it increase the average power of the transmitted signals and possibly results in exceeding the saturation region of HPA to make the system performance worse [140]. In order to overcome the problem of Îź-law companding (increasing average power) and to have efficient PAPR reduction, some more Companding Transform have been suggested [146,147,148,149,150, and 151] .

Figure 4.3 Block diagram of Companding of OFDM system

Chapter Four

PAPR Reduction Techniques

4.6.1 Square-Rooting Companding Technique ( SQRT) for PAPR Reduction in OFDM Systems: The block diagram of a typical OFDM system using the original SQRT technique for PAPR reduction is shown in figure 4.4. By using the SQRT technique, the original OFDM output signals is processed by (3.21) before they are converted into analog waveforms and amplified by the power amplifier â&#x2C6;&#x161;| |

(4.2)

is the new OFDM signal, and is the phase of During the entire signal processing, the phases of the OFDM output signals are kept unchanged while only the amplitudes are treated and changed [152]. For the complex Gaussian distributed signals, such as OFDM output signals, SQRT process changes the Rayleigh distribution of these signals into a Gaussian-like, close to Gaussian, distribution [16,152]; while the Chi-square distribution is converted, according to the analysis of these signals given in the previous section, to Rayleigh distribution. The latter is because the Rayleigh distribution in such signals represents voltage, while the Chi_square distribution represents the power of the same signals. However, not only the statistical distribution is changed by the SQRT process, but the values of the mean and variance of the processed OFDM output signals are also changed, and subsequently the values of the average power and peak power of these signals are altered also. To understand the effect of SQRT process on the power values of OFDM output signals, we assume normalized average power ( )

Figure 4.4 Block diagram of an OFDM system using SQRT technique When the average power is normalized, the value of the peak power is diminished by N because for the same PAPR. This assumption is applicable for all OFDM symbols as the average power is constant and equal to ( ) Hence, the PAPR can be analyzed according to (3.21) through the peak power only. The new value of normalized peak power is always greater than one because is constantly greater than in all OFDM symbols. Therefore, the SQRT process always causes a reduction in the value of the peak power of the normalized OFDM symbols, and as a result the PAPR is reduced in all sizes of OFDM blocks, N. 50

Chapter Four

PAPR Reduction Techniques

In [16, 152] the SQRT process is applied on the signals of all OFDM output symbols; therefore, the PAPR reduced without the need to send side information. The SQRT process changes the distribution of the power signals to Rayleigh distribution and reduces the value of average power from N to N1/2. The variance of the Rayleigh distribution equals ( ) [152] which is approximately equal to half the value of variance of the Gaussian distributed signals. The SQRT process in the SQRT OFDM system performs this statistical transformation, and therefore results in a constant degradation in the BER rate equal to 3 dB because of decreasing of variance to the half of that of the conventional OFDM system ( )

4.6.2 Exponential Companding Algorithm: A nonlinear companding algorithm, called “exponential companding”, developed to reduce the high (PAPR) of (OFDM) signals. Exponential companding technique adjusts both large and small signals and can keep the average power at the same level. By transforming the original OFDM signals into uniformly distributed signals, the exponential companding schemes can effectively reduce PAPR for different modulation formats and sub-carrier sizes [15]. Let | | be the power of the - .The amplitude of companded signal, have a uniform distribution in the interval , exponent is called the degree of a specific exponential companding scheme the CDF of | | is simply ( ) The amplitude of the | | of companded signal has the following CDF |

)

*| |

(4.3)

(4.4)

(4.5) (4.6)

The inverse function of |

)

| | (x)

√

(4.7)

On the other hand, given that ( )is a strictly monotonic\ increasing function, we have, |

*| |

)

(4.8) * (| |) |

( )+

(4.9)

( )) ,

(√ )

(4.10)

Considering the phase of input signals, the companding function ( ) is given by: ( )

( )

| |

(4.11)

Chapter Four

PAPR Reduction Techniques

( )√ 0

(

(4.12)

Where ( ) is sign function. „d‟ is the degree of companding scheme, is the variance of input signal applied for companding. The positive constant determines the average power of output signals. In order to keep the input and output signals at the same average power level, we let

| ]

(

√[

(

(4.13) |

)]

)

At the receiver side, the inverse function operation ( )

( )√

(

( ) of is used in the de-companding

)

(4.14) Figure 4.5 shows the exponential companding function ( ) with degree as a parameter. The companded signals have uniformly distributed amplitudes and powers, respectively for the cases and . When , the ( ) can compress large input signals and expand small signals simultaneously. While the -law companding scheme can only enlarge small signals and does not change the signal peaks, which leads to a higher average power level of output signals. As seen, the differences between exponential companding functions are ignorable when [15] [153].

Figure 4.5 The exponential companding function h(x). 52

Chapter Four

PAPR Reduction Techniques

4.6.3 Trapezoidal power companding: Is a nonlinear companding technique called “trapezoidal power companding” to reduce the high PAPR in a complex OFDM by transforming the original signals into new signals whose power is trapezoidally distributed. A flexible parameter is used to determine the shape of the trapezium so that the trapezoidal power companding scheme is able to meet the requirements for various conditions. Given an expected PAPR value, the scheme provides a closed-form solution that guarantees the actual PAPR the same as the expected [29]. A flexible trapezoidal design was introduced in [29], [154], transforming the amplitude of the signals into a distribution of various trapezoidal shapes. Since that scheme is based on the assumption that all signals are purely real or imaginary, consequently, when the design is used in a complex system, the theoretically estimated PAPR would be different than the actual value. This companding scheme has three desired properties mentioned above. It converts the power distribution of the original signals (as opposed to the amplitude used in [29]) into a trapezoidal distribution while keeping the average output power the same as the original signals. A parameter is used to determine the slope of the hypotenuse so that the trapezium could have a different shape The companding function ( ) is given by:

( )

| |

√.

√

(

)

| |

(4.15) | |

√ ( |

)

}

Where k is the slope of the trapezium. Is the maximum power The decompanding function at the receiver can be given as:

( )

| |

|√

(

|√

(

| | ( | | | |

)| (4.16)

}

Notice that when , the power distribution is actually a rectangular distribution, which is the same as the case in exponential companding .Since sometimes a received signal is so distorted that the square root part in (3.35) would be an imaginary or complex number, we then take the absolute value of the square root parts to eliminate any further potential phase distortion. When the flexible trapezoidal companding curve is then the same as the EC curve [29].

4.6.4 Hyperbolic tangent ( The hyperbolic tangent ( ( )=

(

) companding [30]:

) companding function is defined by

)

(4.17)

Chapter Four Where and the envelope x.

PAPR Reduction Techniques

are positive numbers controlling the companding level applied to

4.6.5 Error Function ( The error function ( ( )=

(

Where

and

) Companding [30]:

) is defined by

)

(4.18)

are positive numbers controlling the level of companding

4.6.6 Logarithm Function (log) Companding [30]: The logarithm ( ( )= Where and [83, 84]

(

) companding function is defined by )

(4.19)

are two positive numbers controlling the amount of companding.

4.7 Coding techniques: Many early papers considered how standard coding techniques could be applied to OFDM. The basic premise of coding is to insert redundant bits into the data stream which can be used for error correction at the receiver. Their application to PAPR reduction is in creating sequences of bits which will exhibit low PAPR after the IFFT. There are 2 types of error detection and correction codes, block codes and convolutional codes. Most papers relate to the block coding family for PAPR reduction. During the encoding process k information bits are encoded into n code d bits, therefore (n-k) redundant non information bits are added to the k information bits [78].The block code is referred to as an (n,k) code, and the rate of the code as Rc=k/n. Figure 4.6 is a block diagram showing where coding for PAPR reduction is located in an OFDM transmitter.

4.6 Block diagram of OFDM transmitter showing PAPR coding Different codes exhibit different degrees of error correction ability. Another important property of codes is the weight of the code, which is the number of non-zero elements in the codeword. Types of block codes are Hamming, Golay, and Reed- Solomon, some of which are used for PAPR reduction .

Chapter Four

PAPR Reduction Techniques

The basic idea of all coding schemes for the reduction of PAPR is to reduce the occurrence probability of the same phase of N signals. The coding method selects such code words that minimize or reduce the PAPR. It causes no distortion and creates no out-of-band radiation, but it suffers from bandwidth efficiency as the code rate is reduced. It also suffers from complexity to find the best codes and to store large lookup tables for encoding and decoding, especially for a large number of subcarriers [10]. A simple block coding scheme was introduced by Jones et al.[155], and its basic idea is that mapping 3 bits data into 4 bits codeword by adding a Simple Odd Parity Code (SOBC) at the last bit across the channels. The main disadvantage of SOBC method is that it can reduce PAPR for a 4-bit codeword [109]. Later, in 1996 Wulich applied the Cyclic Coding (CC) to reduce the PAPR [156]. In 1998, Fragiacomo proposed an efficient Simple Block Code (SBC) to reduce the PAPR of OFDM signals [157]. However, it is concluded that SBC is not effective when the frame size is large. Subsequently, Complement Block Coding (CBC) and Modified Complement Block Coding (MCBC) schemes were proposed to reduce the PAPR without the restriction of frame size [158][159]. CBC and MCBC are more attractive due to their flexibility on choosing the coding rate, frame size and low implementation complexity. CBC and MCBC utilize the complementary bits that are added to the original information bits to reduce the probability of the peak signals occurrence. To make comparisons, some results of the PAPR reduction obtained with different coding schemes have been shown in Table 4.1, in which the number of subblock is 2 and the coding rate for MCBC. Table 4.1 PAPR Reduction comparison with different coding schemes

(

)

About 3-dB PAPR reduction can be obtained when coding rate by using CBC with long frame size. It is also shown that the PAPR reductions obtained with ( ) ( ) CBC when coding rate are almost the same as that when . In addition, when coding rate is 3/4, more than 3-dB more PAPR reduction can be obtained using MCBC than the other schemes with any frame size. The flexibility in coding rate choice and low complexity makes the proposed CBC and MCBC schemes attractive for OFDM systems with large frame sizes and high coding rates [109]. The [160][161][162] authors used the Golay complementary sequences to achieve the PAPR reduction, in which more than 3-dB PAPR reduction had been obtained. Codes 55

Chapter Four

PAPR Reduction Techniques

with error correcting capabilities has been proposed in [163] to achieve more lower PAPR for OFDM signals by determining the relationship of the cosets of Reed-Muller codes to Golay complementary sequences. While these block codes reduce PAPR, they also reduce the transmission rate, significantly for OFDM systems with large number of subcarriers. In fact, let C be a code defined over an equal energy constellation, R denotes the rate and L denotes the length of the C, respectively, then C has possible codewords. Therefore, it is possible to compute the codewords with large PAPR by trying all the codewords of C and computing the peaks of the corresponding signals at some selected time points [109]. However, it is little hope for computing the PAPR of an arbitrary code when L is large. Even if it is possible, the complexity is still too high. Based on this motivates, authors of [159] proposed a novel method of computation and reduction of the PAPR and it mainly introduced a specific phase shift to each coordinate of all possible codewords where phase shifts are independent of the codewords and known both to transceiver, then it can be freely obtained more 4.5-dB PAPR reduction by using the optimized phase shifts. From this viewpoint, we also consider the coding scheme of PAPR reduction as a special phase optimization. In summarization, the inherent error control capability and simplicity of implementation make coding method more promising for practical OFDM systems design. However, the main disadvantage of this method is the good performance of the PAPR reduction at the cost of coding rate loss. Coding techniques for PAPR reduction where redundant bits are added to the bit stream before the IFFT. Properly chosen, these codewords ensure that the PAPR after the IFFT is kept low. These codes can be combined with existing COFDM to reduce the redundancy and complexity inherent in coding. A disadvantage of coding is that the complexity becomes prohibitively high with an increase in the number of subcarriers (>32). Various codewords were presented such as cyclic codes, ShapiroRudin Sequences, Golay Complementary codes, and Reed-Muller codes. Golay codes and their subset, second order Reed Muller codes were found to have excellent PAPR properties restricting the PAPR to 3dB. This reduction could be traded off with reductions in complexity and the code length. Still complexity remains a restrictive issue in coding [78].

4.8 Selective Mapping (SLM): In SLM, the basic idea is to generate a set of OFDM signals, all of them representing the same data block, and then transmitting the one with the lowest PAPR [9][10]. The major drawback of SLM method is that it is more computationally complex because more than one IFFT blocks are used. It also decreases the data rate because the selected signal index, called side information, must also be transmitted to allow for the recovery of the original data block at the receiver side. The eventual loss of the side information during transmission significantly degrades the error performance of the system since the whole data block is lost in this case. Therefore, a lot of work has been suggested as a modified SLM to reduce the computational complexity [164] and to reduce or to remove the side information transmitted [125]. In SLM, the input data sequences are multiplied by each of the phase sequences to generate alternative input symbol sequences. Each of these alternative input data sequences are then applied to IFFT operation, and then the one with the lowest PAPR is selected for transmission [165]. A block diagram of SLM techniques is shown in 56

Chapter Four

PAPR Reduction Techniques

Figure 4.7. The input data is partitioned into a data block Y of length N. Then these data block is multiplied element by element with phase sequence ( ) (

( )

(4.20)

resulting into U modified data blocks

( )

(

( )

- where (4.21)

After that, the N-point IFFT of each data block signal is given as â&#x20AC;&#x201C; ( )

â&#x2C6;&#x2018;

( )

is taken, the resulting OFDM

(4.22)

Among the OFDM data blocks ( ) , only one with the lowest PAPR is selected for transmission and the corresponding selected phase factor also transmitted to receiver as side information. For implementation of SLM OFDM systems, the SLM technique needs U- IFFT operation and the number of required bits as side information is , - for each data block. Therefore, the ability of PAPR reduction in SLM depends on the number of phase factors and the design of the phase factors. The major drawback of SLM method is that it is more computationally complex and less bandwidth efficient (side information is required). Therefore, a lot of work has been suggested as a modified SLM to reduce the computational complexity and to reduce or to remove the side information transmitted [10].

Figure 4.7 Block diagram of selective mapping (SLM) technique for PAPR reduction

4.9 Partial Transmit Sequence (PTS) : In PTS, the original data block is divided into multiple non-overlapping sub-blocks. Then these sub-blocks are rotated with rotation factors which are statistically independent. After that, the signal with the lowest PAPR is chosen for transmission. 57

Chapter Four

PAPR Reduction Techniques

There are several ways for the partition of the data sequence into multiple sub-blocks, including adjacent partition, interleaved partition and pseudorandom partition [122]. Among them, pseudo-random partitioning has been found to be the best choice. Similar to SLM, the major drawback of PTS is also the computational complexity (search complexity for optimal phase factor, and more than one IFFT blocks) and low data rate (required side information). Several techniques have been proposed in the literature to reduce the search complexity and overhead (by reducing/avoiding the usage of side information) [166]. The complexity of PTS is less than SLM [167]. In PTS method, the original frequency-domain data sequence is divided into multiple disjoint sub-blocks, which are then weighted by a set of phase sequences to create a set of candidates Finally, the candidate with the lowest PAPR is chosen for transmission [122]. A block diagram of PTS techniques is shown in Figure 4.8 The input data block in Y is divided in to M disjoint sub-blocks, which are represented by the vectors { ( ) + The input data block Y can be ( ) written in terms of as ∑

( )

for

(4.23)

Where, ( ) with = or 0 ( ) After that, the sub-blocks are transformed into M, time-domain partial transmit sequences by taking the IFFT of length N. These partial transit sequences can be written as: ( )

[

( )

] for

(4.24)

These partial sequences ( ) are then independently rotated by phase factors * , for The rotated partial sequences are then optimally combined to obtain the OFDM signals with lowest PAPR[10] . The time domain signal after combining is given by ̃

∑

( )

(4.25)

There are two main issues of any PTS scheme: to reduce the computational complexity for searching the optimal phase factors and to reduce the overhead by minimizing the side information. Suppose that there are W phase angles to be allowed, thus can has the possibility of W different values. Therefore, there are alternative representations for an OFDM symbol. The search complexity increases exponentially with the number of sub-blocks M To reduce the search complexity and overhead (by reducing/avoiding the usage of side information)[166].These methods achieve significant reduction in search complexity with marginal PAPR performance degradation. In 2007, R. J. Baxley et.al [167] gave a useful comparison between PTS and SLM techniques. It has been shown that the PTS outperforms SLM in terms of PAPR reduction at the cost of increase side.

Chapter Four

PAPR Reduction Techniques

Figure 4.8 Block diagram of partial transmit sequence (PTS) technique for PAPR Reduction

4.10 Tone Reservation : In TR subcarriers, called Peak Reduction Tones (PRTâ&#x20AC;&#x;s) [168], are set aside for PAPR reduction as shown in the transceiver block diagram in Figure 4.9. Tone reservation implemented a projection onto convex sets (POCS) method. Later, Tellado and Cioffi [169] discussed the idea of tone reservation as a linear programming problem that has an exact solution (the POCS method is suboptimal). The linear programming solution can be reached with complexity O [N log N]. The idea behind tone reservation is to isolate energy used to cancel large peaks to a predefined set of tones. These tones do not bear any useful information and are orthogonal to the data bearing tones. This orthogonality makes recovering the data trivial [100]. The advantages of TR technique include: 1. No need for side information 2. Fewer complex-multiplications as only one time IFFT operation is needed. But multiple iteration operations are needed after IFFT operation. 3. No special receiver operation is needed While promising, tone reservation has several shortcomings. First the data rate is necessarily decreased because some tones are used strictly for PAR reduction. The second problem is the difficulty of selecting which tones to reserve. A random search over all the possible sets, B, would greatly increase the complexity of tone reservation. Often the tones have to be chosen contiguously because fades often affect contiguous sets of sub carriers. These contiguous sets of tones are known to have bad PAR reduction abilities. The third issue is a tradeoff between the quantities of reserved tones and the rise in average power due to tone reservation. More the tones that are reserved, lesser the power needs to be allocated for PAPR reduction. On other hand, more reserved tones mean more unused bandwidth that could be data bearing [100]. 59

Chapter Four

PAPR Reduction Techniques

Figure 4.9 Block diagram of a Tone Reservation (TR) OFDM transceiver.

4.11 Tone Injection: Motivated by the data rate loss of tone reservation, Tellado introduced a new technique named tone injection [170] as shown in figure 4.10. It reduces the PAPR without compromising the data rate. In this method the size of the basic constellation is increased. Hence mapping of original constellation points into numerous corresponding points in the new stretched out constellation becomes possible. The distance between these duplicate points can be calculated as dâ&#x2C6;&#x161; , where M= constellation size, and . There is no effect on BER and all we have to do is add a modulo-D subsequent to FFT in the receiver side. Since mapping of each information unit into numerous corresponding constellation points is done, it gives a margin of free will which can be used reduction of PAPR [100]

Figure 4.10 Block diagram of a Tone Injection (TI) OFDM transceiver

Chapter Four

PAPR Reduction Techniques

4.12 Interleaving [171][172]: This method is also termed as Adaptive Symbol Selection Method .Multiple OFDM symbols are created by bit interleaving of input sequences .The basic Idea is to use W interleaving ways and selecting one with the lowest PAPR. Figure 4.11 shows an interleaver, PAPR Reduction capability depends on the number of interleaver used .To recover the signals the receiver need to know the information about which interleaver is used.

Figure 4.11: Interleaving

4.13 Active Constellation Extension (ACE) [173][174]: This technique deals with extending the constellation points outside the signal constellation which is then used to cancel the time domain peaks .Figure 4.12 shows the points where these constellation points can be extended. Is technique has several advantages like no loss of data, no degradation in system performance, lower BER as compared to other techniques and bears no side information like SLM. Some variations of this method like clipping-based ACE and Adaptive ACE in which repeated CAF an in later an adaptive control has been used to optimize the performance. The drawback is that the technique is useful for larger constellation size modulations only.

Figure 4.12 Active Constellation Extension (a) for QPSK (b) for 16 QAM

Chapter Four

PAPR Reduction Techniques

4.14 Dummy Sequence Insertion (DSI)[100]: In Dummy sequence insertion (DSI) [175], before IFFT stage in input data a dummy sequence is added. The sequences which are used may be complementary, correlation or any other sequence. Since dummy sequence is not used as side information hence any transmission error does not increase BER. DSI technique is united with PAPR threshold method. After IFFT, if PAPR is below specific threshold then signal is transmitted but if it is more than this specific level then insertion of dummy sequence is done to achieve the required results. The block diagram of DSI system is shown in figure

Figure 4.13 Block diagram of DSI system The main advantage of this technique is that BER is not degraded due to transmission errors in the dummy sequence. So far amongst different sequences, use of complementary sequence produces better results.

Chapter Five

Simulation Results and Analysis Chapter Five Simulation Results and Analysis

One of the major drawbacks of OFDM system is high PAPR of transmitting signals, which causes an earnest degradation in performance when a non-linear HPA is utilized. Therefore, it is compulsory to utilize a congruous PAPR reduction scheme at the transmitter. In this chapter, the different methods of PAPR reduction are given with results and new types of PAPR proposed.

5.1 OFDM System model: The system model used in the work is shown in figure 5.1. The OFDM parameter used in the test is the LTE parameters as shown in table 5.1. The system was tested under Rayleigh selective fading channel with parameter given in table 5.2 [176] I/P

S / P

Signal Mapper

IDFT OR IFFT

P / S

Add CP

D / A

Multipath Fading Ch. & noise

O/P P

Signal demapper

Equalizer And P/S

DFT OR FFT

S / P

Remove CP

A / D

Figure 5.1 OFDM system model. Table 5.1 LTE parameter FFT size Spacing frequency BW CP No symbol Sampling frequency Modulated type

128 15 KHz 1.25MHz 32 1000 192MHz QPSK

Table 5.2 Average Power and Relative Delays with 6 delay taps [176] Tap no. Relative delay (ns) Average Power (dB) 1 0 0.189 2 0.2 0379 3 0.5 0.239 4 1.6 .095 5 2.3 .061 6 5 .037

Chapter Five

Simulation Results and Analysis

The PAPR was evaluated statistically by using the complementary cumulative distribution function (CCDF). The CCDF of PAPR, for the proposed PAPR reduction techniques OFDMA downlink signal, is used to express the probability of exceeding a given threshold PAPR0 (i.e., CCDF ( )). A simulation result was compared with each other. PAPR was measured for the transmitted OFDM signal using the equation: | |

(5.1)

| |

In each case, the BER was measured. Initially, it is necessary to know the performance of OFDM system without any PAPR reduction techniques in order to compare it with the PAPR reduction techniques to find out the amount of improvement in PAPR in each case of PAPR reduction techniques and their impact on the BER. Fig (5.2) shows the CCDF of PAPR and SNR at BER for OFDM system without any PAPR reduction techniques which is equal to (10.84 dB) with PAPR equal to (25.6015 dB) while shows the BER for OFDM system without any PAPR reduction techniques and SNR at BER is equal to (11.4314 dB). 0

CCDF (Pr[PAPR>PAPR0])

Orignal

-1

-2

-3

6 PAPR0 [dB]

Figure (5.2.a) Bit error probability curve for qpsk using OFDM

simulated

-1

BER

-2

-3

-4

SNR

Figure (5.2.b) a) is CCDF of PAPR for OFDM system without any PAPR reduction techniques b) is BER for OFDM system without any PAPR reduction techniques 64

Chapter Five

Simulation Results and Analysis

5.2 PAPR techniques used: 5.2.1 Repeated clipping and frequency domain filtering (RCF): In the clipping technique hard limiting is applied to the amplitude of the complex values of the IFFT output. The filtering technique is designed to alleviate or cancel OOB distortion dependent on oversampling value but cannot correct in-band distortion. [98] Iterative clipping and filtering fft/iffft đ?&#x2018;&#x17D;

đ?&#x2018;?

đ?&#x2018; Ă&#x2014; (đ??ź

)

Zeroes

0 0

N*đ??ź

Nonlinear

N*đ??ź

Point inverse DFT over

Processing

Point DFT over

Clipping

sampling

rate đ??ź

đ?&#x2018; Ă&#x2014; (đ??ź Zeroes

rate đ??ź

Ratio =

sampling

N*đ??ź )

0 0

Point inverse DFT over sampling

Add cp

rate đ??ź

đ?&#x2018;?đ?&#x2018;

đ?&#x2018;&#x17D;đ?&#x2018; Input data zero padded

Interpolated baseband signal

Clipped Interpolated baseband signal

Frequency domain filtering

Figure 5.3 shows the block diagram of the new PAPR reduction scheme [177]. The input vector is first converted from the frequency to the time domain wing an oversize IFFT. N is the number of subcarriers in each OFDM symbol. For an oversampling factor of, the input vector is extended by adding ( ) zeros; in the middle of the vector. This results in the trigonometric interpolation of the time domain signal [178]. Trigonometric interpolation gives perfect interpolation when the original signal consists of integral frequencies over the FFT window. This is the case for OFDM. The input of the Nyquist frequency, has been omitted, as the interpolation technique does not work for this value [178]. This is not a practical limitation as all applications of OFDM null this input and most do not use a number of adjacent subcarriers. The interpolated signal is then clipped. In this Technique hard-limiting is applied to the amplitude of the complex values of the IFFT output [12] After an IFFT, the original signal is clipped in the time domain. The clipping can be described as shown below: *

â&#x2C6;&#x161;

Where

,| | -

| |

| | | |

(5.2)

represents the output of the time domain signal, ,| | -

(5.3) | |

, Is the threshold clipping level, power.

; ,| | - Is the mean

Chapter Five

Simulation Results and Analysis

The clipping ratio is defined as the ratio of the clipping level to the mean power of the unclipped baseband signal. As shown in the equation (5.2), the discrete time domain signal is clipped in the amplitude. At every point where the complex time domain signal exceeded the clipping level, the amplitude was reduced to the clipping level while the phase of the complex signal was unchanged [179]. The clipping is followed by frequency domain filtering to reduce OOB power caused by clipping. The filter consists of two FFT operations [12]. The clipped time domain signal c is then converted back into the discrete frequency domain using an FFT ,The inband discrete frequency components of the clipped signal are passed unchanged to the inputs of the second IFFT while the OOB components,

are nulled [13 and 180] this

technique is repeated, depending on iteration number, usually choose between one and four. In this work has been selected four. Although frequency domain filtering is a common signal processing technique the form shown in figure 5.3 is unusual. In most filtering applications the filter is designed to meet particular specifications in the continuous frequency domain. In this application, the wanted signal is an OFDM signal, which is the sum of discrete frequency components in each symbol period. The filter must therefore have as little effect as possible on the in-band discrete frequency domain while attenuating as much as possible any OOB components. This is precisely what is achieved by the simple filter structure in Figure 5.3 because the filter operates on a symbol by symbol basis; there is no filtering across symbol boundaries and so no resultant ISI. The filtering does cause some peak regrowth. However, this is much less than for clipping before interpolation [12, and 18] The clipping noise is added at the transmitter rather than the receiver. In fading channels this means that in general the clipping noise will cause less degradation in bit error rate than noise added in the channel since the clipping noise fades along with the signal. However the second oversize IFFT could be replaced by any of the transform, up sampling and filtering arrangements commonly used in OFDM systems. So the technique can be implemented by replacing the IFFT block in an existing OFDM system with the new configuration [12]. The FFT/IFFT transform filter can be replaced by DCT/IDCT transform and this technique has been described in [28]. In paper [18] present a new PAPR reduction technique which exploits the use of unused carriers as well as the phase information of pilot signals in OFDM systems to reduce the PAPR while not degrading channel estimation or frequency offset. Compared to conventional techniques such as clipping and windowing, this technique introduces much less OOB distortions and provides a lower BER since there is no interference to the original data signals. There is also no requirement for side information to be transmitted to the receiver. To reduce PAPR at LTE downlink, the RCF is applied to OFDM signal for different CR and oversampling filter and notes their impact on PAPR and BER. The reason to choose this method is because the filter improves the BER if the oversampling is high and clipping improves PAPR (it's possible to improve the BER & PAPR together and this way we have explained previously). 66

Chapter Five

Simulation Results and Analysis

The OFDM system model with RCF as shown in figure 5.4. For this simulation I = (1, pilot, 1.125, 1.25, 1.5, 2, 3, 4) and CR = (4, 3, 2, 1.75, 1.5), in order to see the impact of CR on the (BER) and (PAPR), this technique is repeated, depending on iteration number ( four is used in this simulation) The transmitted signals pass through Rayleigh fading channel.

D / A

Add CP

P / S

RCF

IDFT OR IFFT

S / P

+pilot symbol

Signal mapper

I / P

Multipath Fading Ch. & noise

S / P

Remove CP

đ?&#x2018;&#x192;

DFT OR FFT

Remove +pilot symbol

One Tap Equalizer And P/S

Signal demapper

O / P

A / D

Figure 5.4 the OFDM system model with RCF. Figure 5.5 illustrate the effect of repetition clipping and filtering on PAPR where CR =3, I =2, where CCDF of PAPR for, one RCF = 7.7581, two RCF = 6.5462, three RCF = 5.8319, and four RCF = 5.401, Note that there is an improvement in CCDF of PAPR for one RCF (2.8935 dB), two RCF (4.1054 dB), three RCF (4.8197 dB), and four RCF (5.2506 dB). But the proportion of improvement, between (N) RCF and (N-1) RCF decrease as N increase.

Whenever a CR reduces the PAPR is improving and contrast SNR at BER is increased, The best value of PAPR is for CR =1.5, but the SNR at BER for this case is the worst, as shown in table CR have a positive relationship with PAPR and negative relationship with SNR at BER Whenever oversampling increased the SNR at BER is improving and contrast PAPR is increased and vice versa. The best value of PAPR is for I =1 this mean there is no filter, but the SNR at BER for this case is the worst, while The best value of SNR at BER is for I =4, but the PAPR for this case is the worst, as shown in table (A.1) I have a positive relationship with SNR at BER and negative relationship with PAPR

Chapter Five

Simulation Results and Analysis

CCDF (Pr[PAPR>PAPR0])

Orignal One clip and filter Two clip and filter Three clip and filter Four clip and filter -1

-2

-3

6 PAPR0 [dB]

Figure 5.5 CCDF of PAPR for OFDM system with repeated clipping and frequency domain filtering where CR =3, I =2 Figure 5.6 shows the following:  There is a clear improvement in the CR3 CCDF of PAPR reduction in rate SNR at BER is relatively small compared with the CR4. Briefly, that‟s mean the percentage of improvement in CCDF of PAPR More than the degradation in BER  For the CR2 the PAPR improved more than CR3 and CR4 but SNR at BER gets worse  the CR1.75 had a little improvement in CCDF of PAPR in comparison with the CR2) but SNR at BER degradation more than The amount of improvement  For the CR1.5 the PAPR improved PAPR in comparison with the CR1.75 only in a small proportion, while SNR at BER Substantially worse. Figure 5.7 shows the impact of the oversampling (CCDF of PAPR) and (BER), is conclusion through drawing and table following:  whenever increase the PAPR will increase too only in small percentages, for this figure PAPR for I4 Worsened by (1.7978 dB) compared with I1  whenever increase the CCDF of PAPR will increase too only in small percentages, for this figure CCDF of PAPR for I4 Worsened by (.9939 dB) compared with I1  whenever increase the SNR at BER will improved , for this for I4 the SNR at BER improved by(5.5382) compared with I1  for I2 the SNR at BER Improved by (2.8466) , CCDF PAPR Worsened by (.5831) and PAPR Worsened by (.6559 dB) compared with I1

Chapter Five

Simulation Results and Analysis

The Conclusion from table (A.1) Summarizes as follows:  CR 4 have the best SNR at BER and the worst PAPR compared with the rest of the CR  PAPR at CR 3 better than CR 4 by (2-3 dB improvement in PAPR) but SNR at BER at CR3 worse than CR 4 only by small percentage (less than 1 dB in all cases) PAPR at CR 2 better than CR3 by (2.5 - 3.4 dB improvement in PAPR) but SNR at BER at CR2 worse than CR = 3 by (2- 3.7 dB degradation in SNR at BER )  PAPR at CR 1.75 better than CR2 by (Maximum improvement is 1.0059) but SNR at BER at CR1.75 worse than CR2by (2.2 – 3.4 dB degradation in SNR at BER )  CR 1.5 have the best PAPR and the worst SNR at BER compared with the rest of the CR, the SNR should higher than 30 dB have the desired SNR at BER that‟s mean SNR at BER is deteriorating by a large margin The following conclusion when comparing the proposed method with an OFDM system without PAPR reduction method:  As (CR =4, 3 and I =4, 3,2, pilot, 1.5,1.25) and when (CR =2and I =4) and finally (CR =4 and I = 1.125), The best one improvement in PAPR and CCDF of PAPR is at I =3 and CR =2. The improvement in PAPR by = (14.9490 dB), CCDF of PAPR = (6.2850 dB), and the SNR at BER by = (1.0134 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at I = Pilot and CR =2. The improvement in PAPR by = (16.1583 dB), and CCDF of PAPR = (6.9604 dB), while the SNR at BER deteriorated by = (-1.5686 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at I = 1 and CR =2. The improvement in PAPR by = (17.3529 dB), and CCDF of PAPR = (7.7214 dB), while the SNR at BER deteriorated by = (-2.9442 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at I = 1 and CR =1.75. The improvement in PAPR by = (18.2213 dB), and CCDF of PAPR = (8.2460 dB), while the SNR at BER deteriorated by = ( -5.2886 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at I = 1 and . The improvement in PAPR by = (19.2177 dB), and CCDF of PAPR = (7.9400 dB), while the SNR at BER deteriorated by = ( -18.0686 dB).

Chapter Five

Simulation Results and Analysis

CCDF (Pr[PAPR>PAPR0])

CR CR CR CR CR

-1

=4 =3 =2 =1.75 =1.5

-2

-3

3 4 PAPR0 [dB]

Figure 5.6.a Bit error probability curve for qpsk using OFDM

CR =4 CR =3 CR =2 CR =1.75 CR =1.5

-1

BER

-2

-3

-4

15 SNR

Figure 5.6.b (a)CCDF of PAPR for OFDM system with RCF where I =2 (b) BER for OFDM system with RCF where I =2 70

Chapter Five

Simulation Results and Analysis

CCDF (Pr[PAPR>PAPR0])

I =1 I =pilot 1.125 I =1.25 I =1.5 I= 2 I =3 I =4

-1

-2

-3

3 PAPR0 [dB]

Figure 5.7.a Bit error probability curve for qpsk using OFDM

I =1 I =pilot 1.125 I =1.25 I =1.5 I= 2 I =3 I =4

-1

BER

-2

-3

-4

10 SNR

Figure 5.7.b Figure 5.7 (a) CCDF of PAPR for OFDM system with RCF where CR =3 (b) BER for OFDM system with RCF where CR =3. 71

Chapter Five

Simulation Results and Analysis

5.2.2 Repeated frequency domain filtering and clipping RFC: The proposed method is the same as previous method RCF, but with almost a simple change and it changes the location of the filter becomes before the clipping as shown in figure 5.8, the frequency domain filtering that depends on the interpolation As noted by previous results that improve the BER As noted by previous results but increases the PAPR. The basic idea of this method is proposed that this filter will improve the performance of the OFDM to improve the BER and then the clipping will improves PAPR method is the almost same as RCF, where have the same receiver and channel But there is a difference in One block in the transmitter. This block is RFC as shown in figure 5.9. Interpolated baseband signal followed by frequency domain filtering, the same filter which are explained in the case of RCF. The filtering signal is clipped in the time domain. The clipping block is described previously in the case of RCF. Iterative filtering fft/iffft and clipping đ?&#x2018;&#x17D; đ?&#x2018;?

N*đ??ź N*đ??ź đ?&#x2018; Ă&#x2014; (đ??ź

Zeroe

)

Point DFT over

Point inverse DFT over

0 0

đ?&#x2018; Ă&#x2014; (đ??ź

Zeroes

)

sampling

rate đ??ź

Nonlinear

Point inverse DFT over

Processin g

Ratio =

rate đ??ź

đ?&#x2018;?đ?&#x2018;

đ?&#x2018;&#x17D;đ?&#x2018; Input data zero padded

Interpolated baseband signal

Add cp

Clipping

samplin g rate đ??ź

sampling

N*đ??ź

Clipped Frequency domain filtering

the filtering signal

Figure 5.8 shows the block of the OFDM system model for this proposed

P / S

Add CP

RFC

IDFT OR IFFT

S / P

+pilot symbol

Signal mapper

I / P

D / A

Multipath Fading Ch. & noise

Figure 5.9 the OFDM system model with RFC. 72

S / P

Remove CP

đ?&#x2018;&#x192;

DFT OR FFT

Remove +pilot symbol

One Tap Equalizer And P/S

Signal demapper

O / P

A / D

Chapter Five

Simulation Results and Analysis

The following conclusion when comparing the proposed method with an OFDM system without PAPR reduction method as shown in table A.2:  At (CR =4,3 and I =4,3,2, pilot,1.5,1.25) , (CR =2and I =4,3) , ( CR =1.75 and I = 4) and finally (CR =4 and I = 1.125), The best one improvement in PAPR and CCDF of PAPR is at I =4 and CR =1.75. The improvement in PAPR by = (18.2789 dB), CCDF of PAPR = (8.0187 dB), and the SNR at BER( ) by = (0.6101 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at I = 3 and CR =1.75. The improvement in PAPR by = (18.0071 dB), and CCDF of PAPR = (8.0088 dB), while the SNR at BER( ) deteriorated by = (-0.2686 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at I =2 and CR =1.75. The improvement in PAPR by = (18.0153 dB), and CCDF of PAPR = (7.9920 dB), while the SNR at BER( ) deteriorated by = (-3.1811 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at I = 1.5 and CR =1.75. The improvement in PAPR by = (18.1813 dB), and CCDF of PAPR = (7.7593 dB), while the SNR at BER( ) deteriorated by = (-4.8773 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at I = 1.125 and CR =1.75. The improvement in PAPR by = (18.2306 dB), and CCDF of PAPR = (8.1500 dB), while the SNR at BER( ) deteriorated by = (-5.6826 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at I = 4 and CR =1.5. The improvement in PAPR by = (19.2106 dB), and CCDF of PAPR = (8.4242 dB), while the SNR at BER( ) deteriorated by = ( -16.7886 dB).

The following conclusion when comparing the proposed method with RCF: 1. CCDF of PAPR was improved in all cases except when (I = pilot and CR=4 by (-0.3570)). The improvement ratio was increased with the decrease of CR and the increase of I. The biggest improvement is in the case (I = 3 and CR =4 (2.9062)) 2. PAPR was improved in all cases except when (I = 1.125 and CR=2 by (0.0866)) and (I = 1.5 and CR=4 by (-0.0015)) the improvement ratio was increased with the decrease of CR and the increase of I. The biggest improvement is in the case( I = 3 and CR =4 (1.5600)) 3. SNR at BER( ) a) SNR at BER( ) was improved for (I =3 and I =4 in all cases of CR ) b) For I =2 SNR at BER( ) was improved in all cases except when (CR=2) deteriorated by (-0.1548) c) For I =1.5 SNR at BER( )was improved except when (CR=4) deteriorated by (-0.1236) d) For I =1.25 SNR at BER( ) was improved except when (CR=4) deteriorated by (-0.5700) and (CR=2) deteriorated by (-0.0375) e) For I =1.125 SNR at BER( ) was improved except when (CR=3) deteriorated by (-0.1700) and (CR=1.75) deteriorated by (-0.2390) f) For I =pilot SNR at BER( ) was improved except when (CR=4) deteriorated by (-0.2400) and (CR=3) deteriorated by (-0.1585) 73

Chapter Five

Simulation Results and Analysis

g) The best value of the improvement is where the (I =4 and CR =1. 75 by (2.1787)) 4. RFC is better than RCF because when I increase the SNR at BER( ) improve and PAPR almost preserves its value That was the conclusion from a comparison of figure 5.6 and figure 5.10 the following: 1. There is an obvious improvement in the CCDF of PAPR of the RFC In comparison with the RCF 2. There is an improvement in the SNR at BER( ) of the RFC In comparison with the RCF 3. The CCDF of PAPR of the RFC at CR=2 is better than the CCDF of PAPR of the RCF at CR=1.5, in addition to that the SNR at BER( ) of the RFC at CR=2 is better than the SNR at BER( ) of the RCF at CR=1.5 by ((17.71721 dB) 4. The amounts of improvement, are described in the table A.2 Figure 5.11 shows the impact of the oversampling (CCDF of PAPR) and (BER), is the conclusion through drawing and table following:  The PAPR for I(N) was increased only by A small amount compared with I1, for this figure PAPR was declined amount (0.0251 - 0.3086 dB).  The CCDF of PAPR for I(N) was increased only Avery small amount could be neglected in comparison with I1, for this figure PAPR was declined amount ( 0.0102 - 0.0837 dB).  SNR at BER( ) degraded whenever I increased between (0.8374 6.2451dB) RFC have the same complexity and cost RCF because RFC has not added a new function for RCF but the only change filter location.

Chapter Five

Simulation Results and Analysis

CCDF (Pr[PAPR>PAPR0])

CR =4 CR =3 CR =2 CR =1.75 CR =1.5

-1

-2

-3

3 4 PAPR0 [dB]

Figure 5.10.a Bit error probability curve for qpsk using OFDM

CR =4 CR =3 CR =2 CR =1.75 CR =1.5

-1

BER

-2

-3

-4

15 snr

Figure 5.10.b Figure 5.10 (a)CCDF of PAPR for OFDM system with RFC where I =2 (b) BER for OFDM system with RCF where I =2 75

Chapter Five

Simulation Results and Analysis

CCDF (Pr[PAPR>PAPR0])

I =pilot I =1.125 I =1.25 I =1.5 I= 2 I =3 I =4

-1

-2

-3

0.5

1.5

2.5 3 PAPR0 [dB]

3.5

4.5

Figure 5.11.a Bit error probability curve for qpsk using OFDM

I =pilot 1.125 I =1.25 I =1.5 I= 2 I =3 I =4

-1

BER

-2

-3

-4

SNR

Figure 5.11.b Figure 5.11 (a)CCDF of PAPR for OFDM system with RCF where CR =3 (b) BER for OFDM system with RCF where CR =3 76

Chapter Five

Simulation Results and Analysis

5.2.3 The OFDM System with discrete time companding: Compresses the signal at the input and expands the signal at output in order to keep the signal level above the noise level during processing. At the output, the original input signal is then restored by a simple attenuation. Companding increases the SNR when the input signal is low and therefore reduces the effect of a system‟s noise source.

5.2.3.1 A-law companding: In this companding method, the compressor characteristic is piecewise, made up of a linear segment for low level inputs and a logarithmic segment for high level inputs. Figure 5.12 shows the A-law compressor characteristics for different values of A. Corresponding to A=1, we observe that the characteristic is linear (no compression) which corresponds to a uniform quantization. A-law has mid riser at the origin. Hence it contains non-zero value. The practically used value of “A” is 87.6. The A-law companding is used for PCM telephone systems. The linear segment of the characteristic is for low level inputs whereas the logarithmic segment is for high level inputs. This technique can be used to reduce the PAPR which is the main disadvantage of OFDM [181, and 182]. | |

(

( )

)

[

{

0 (

| |

( ) | |

)

(5.4) | |

( )

}

Where     

x=input signal. y=output signal. ( ) =sign of the input (+ or -). |x|=absolute value (magnitude of x). A=87.6 (defined by CCITT (Consultative Committee for International Telephony and Telegraphy) ). This A-law companding technique is used in Europe, Asia, Russia, Africa, China, etc [183]. Initially, A companding as discussed used with OFDM. Figure 5.13 illustrates the effect of A parameter on the PAPR, CCDF of PAPR, and SNR at BER( ). When increasing the values of A parameter, the CCDF of PAPR improves. The relationship between A parameter and CCDF of PAPR is the inverse relationship. CCDF of PAPR (A =20) - CCDF of PAPR (A =120) = (1.15 dB) A is not linear companding , A possible divided into three areas. The first area is that when A increases lead to improvement in the CCDF of PAPR is relatively large compared with the second and third region (the example for this area is A (CCDF of PAPR (A =5) - CCDF of PAPR (A =20) = (2.955 dB) amount of improvement in the CCDF of PAPR ) In the second area, when A was increased the CCDF of PAPR was improved but a small quantity less than the first region example of this area when A (CCDF of PAPR (A =20) - CCDF of PAPR (A =120) = (1.15 dB) as is evident A increased by 77

Chapter Five

Simulation Results and Analysis

(100) and the improvement in CCDF of PAPR is (1.15 dB) while in the first area A increased by (15) but the improvement in CCDF of PAPR (2.955 dB)) In the third area, when A was increased the CCDF of PAPR was not affected even if improved but very small.

Normalized Output

Normalized input

Figure 5.12. A-law Compressor Characteristics [99].

22 20

SNR at (BER =10-4) CCDF of PAPR PAPR

18 16

[dB]

14 12 10 8 6 4 2

60 A

100

120

Figure 5.13 the relationship between A parameter and (PAPR, CCDF of PAPR and BER)

Chapter Five

Simulation Results and Analysis

When A parameter was increased the PAPR improved while the SNR at BER( ) deteriorated. The A parameter has a positive relationship BER with but an inverse relationship with the PAPR. PAPR (A =5) – PAPR (A = 20) = (4.466 dB) SNR at BER( ) (A= 5) – SNR at BER( ) (A=20) = (-4.4 dB) PAPR (A =20) – PAPR (A = 120) = (3.03 dB) SNR at BER( ) (A= 20) – SNR at BER( ) (A=120) = (-3.386 dB) In the first area A increased by (15) but the improvement in PAPR (4.466 dB) and the degradation in SNR at BER( ) (4.4 dB) while in the second area A increased by (100) and the improvement in PAPR (3.03 dB) and the degradation in SNR at BER( ) (-3.386 dB) . The first and the second area evident in the Figure 5.14 The following conclusion when comparing the proposed method with an OFDM system without PAPR reduction method:  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at A =5. The improvement in PAPR by = (6.6954 dB), and CCDF of PAPR = (4.200 dB), while the SNR at BER( ) deteriorated by = (-2.1686 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at A =10. The improvement in PAPR by = (10.9098 dB), and CCDF of PAPR = (6.1100 dB), while the SNR at BER( ) deteriorated by = (-4.6886 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at A =30. The improvement in PAPR by = (13.7470 dB), and CCDF of PAPR = (7.5200 dB), while the SNR at BER( ) deteriorated by = (-7.7686 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at A =100. The improvement in PAPR by = (14.2472 dB), and CCDF of PAPR = (8.2600 dB), while the SNR at BER( ) deteriorated by = (-10.1886 dB). Figure 5.14 shows the CCDF of PAPR and the BER of A companding for various A parameter. For more details see table A.4

Chapter Five

Simulation Results and Analysis

CCDF (Pr[PAPR>PAPR0])

original A =5 A =10 A =30 A =50 A =87.6

-1

-2

-3

6 PAPR0 [dB]

5.14.a Bit error probability curve for qpsk using OFDM

A =5 A =10 A =30 A =50 A =87.6

-1

BER

-2

-3

-4

15 SNR

5.14.b Figure 5.14 (a) CCDF of PAPR OFDM system A companding for various A parameter. (b) BER for OFDM system A companding for various A parameter. 80

Chapter Five

Simulation Results and Analysis

5.2.3.2 μ-law companding technique: In the μ-law companding, the compressor characteristic is piecewise, made up of a linear segment for low level inputs and a logarithmic segment for high level inputs. Figure 5.15 shows the μ-law compressor characteristics for different values of μ. Higher the value of μ more is the compression. Corresponding to μ=0, we observe that the characteristic is linear (no compression) which corresponds to a uniform quantization. μ-law has mid tread at the origin. Hence it contains a zero value. The practically used value of “μ” is 255 [183]. The signal by utilized μ -Law compression characteristic is defined as: ( )

| |

(

)

( )

(5.5)

Where V is the peak amplitude of the signal, and x is the instantaneous amplitude of the input signal. Decompression is simply the inverse of (5.5). Compression improves the quantization resolution of small amplitude signals at the cost of lowering the resolution of large signals. This also introduces quantization noise; however, the effect of the quantization noise due to reduction in resolution of the peaks is relatively small as the peaks occur less frequently. The compression algorithm as described by amplifying the signals of lower amplitude with the peaks remaining unchanged. [184], [185].

Normalized Output

Normalized input

Figure. 5.15 μ-law Compressor Characteristics [186] A-law and law coefficients are responsible for the compression ratio. Compression increases with increasing value of the coefficients. Originally A-law and -law companders were used for voice compression, as it can be seen, A-law and -law companders have logarithmic compressing profile. In fact they work as follows, instead of compressing the high peaks; companding schemes increase the value of small signals in a way, to bring them in the same level with the high peaks [186]. 81

Chapter Five

Simulation Results and Analysis

Thus original Gaussian distributed OFDM signal will be transformed to a signal with quasi uniform distribution. However, because of increased level of the small signals, average power of the signal will be increased. That means noise will be increased as well. This is disadvantage of A-law and -law companding schemes as compared with exponential companding, which is claimed to adjust both small and large signals without changing the average power of the signal [187]. Figure 5.16 illustrates the effect of parameter on the PAPR, CCDF of PAPR and SNR at BER( ). In General, when parameter was increased, the CCDF of PAPR was decreased except for some cases are as follows:  At the CCDF of PAPR Larger than the by (0.06 dB)  At the CCDF of PAPR Larger than the by (0.014 dB)  At the CCDF of PAPR Larger than the by (0.205 dB)  At the CCDF of PAPR Larger than the by (0.01 dB)  At the CCDF of PAPR Larger than the by (0.09 dB)  At the CCDF of PAPR Larger than the by ( 0.11 dB )  Even in exceptional cases, the amount of the decline is a few and not exceed (0.205 dB) The max CCDF of PAPR at ( ) = 6.416 dB while the min CCDF of PAPR at ( ) =2.17 dB The parameter has a positive relationship SNR at BER( ) with but an inverse relationship with the PAPR. When parameter was increased SNR at BER( ) deteriorated except at be better than the by (0.132 dB). In General, when parameter was increased, the PAPR was decreased except for some cases are as follows:  At the PAPR Larger than the by (0.7103 dB)  At the PAPR Larger than the by (2.3264 dB)  At the PAPR Larger than the by (4.6422 dB)  At the PAPR Larger than the by (0.2429 dB)  At the PAPR Larger than the by (0.3182 dB)  At the PAPR Larger than the by (0.436 dB)  At the PAPR Larger than the by (2.0735 dB) The max PAPR at ( ) = 17.4332 dB while the min PAPR at ( ) =10.8218 dB The max SNR at BER( ) at ( ) = 23.764 dB while the min SNR at BER( ) at ( ) =13.3363 dB The following conclusion when comparing the proposed method with an OFDM system without PAPR reduction method:  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at MU =10. The improvement in PAPR by = (9.0545 dB), and CCDF of PAPR = (5.0700 dB), while the SNR at BER( ) deteriorated by = (-3.2086 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at MU =20 . The improvement in PAPR by = (8.3442 dB), and CCDF of PAPR = (5.7620 dB), while the SNR at BER( ) deteriorated by = (-4.8186 dB). 82

Chapter Five 



Simulation Results and Analysis

For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at MU =100 . The improvement in PAPR by = (13.1873 dB), and CCDF of PAPR = (7.7200 dB), while the SNR at BER( ) deteriorated by = (-8.5686 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at MU =700 . The improvement in PAPR by = ( 14.7797 dB), and CCDF of PAPR = (8.6700 dB), while the SNR at BER( ) deteriorated by = (-12.0686 dB).

Fig (5.17) shows the CCDF of PAPR of companding and the BER of companding for various parameter and illustrates the former explanation.  PAPR improved by ( 8.1683 -14.7797 dB)  CCDF of PAPR improved by (4.4240 - 8.6700 dB)  The amount of SNR at BER( ) degradation is (1.9049 - 12.3326dB ) 22 20 18 16

[dB]

14 12 10 SNR at (BER =10-4) CCDF of PAPR PAPR

8 6 4 2

100

Figure 5.16 the relationship between BER)

150 MU

200

250

300

parameter and (PAPR, CCDF of PAPR and

Chapter Five

Simulation Results and Analysis

CCDF (Pr[PAPR>PAPR0])

orginal MU =5 MU =50 MU =100 MU =160 MU =200 MU = 255

-1

-2

-3

6 PAPR0 [dB]

Figure 5.17.a Bit error probability curve for qpsk using OFDM

MU =5 MU =50 MU =100 MU =160 MU =200 MU = 255

-1

BER

-2

-3

-4

15 SNR

5.17.b Figure 5.17 (a) CCDF of PAPR OFDM system companding for various parameters. (b) The BER of companding for various parameters. 84

Chapter Five

Simulation Results and Analysis

5.2.3.3 Rooting Companding Technique (RCT): The proposed Rooting companding has the same principle of SQRT. Rooting companding equation is given by: ( )

| |

( )

(5.6)

Where ( )= sign(x) sign(x) was used in RCT to maintain the phases of the OFDM signal Where the phases of the OFDM output signals are kept unchanged while only the amplitudes are treated and changed . The amount of change in amplitude depends on the value of R Rooting decompanding equation is given by: ( )

| |

( )

(5.7)

The following can be observed from table A.6 and figure 5.18  When y parameter decreases the PAPR and CCDF of PAPR also decrease while SNR at BER( ) increase  The best value for the PAPR is (2.8726) when R =0.1 while the worst value is (21.8631) when R=0.9  The best value for the CCDF of PAPR is (1.268) when R =0.1 while the worst value is (9.55) when R =0.9  The best value for the SNR at BER( ) is (11.6765) when R =0.9 while the worst value is (28.3) when R=0.1  PAPR improved by (3.7384 - 22.7289 dB )  CCDF of PAPR improved by (1.2900 -9.5720 dB )  The amount of SNR at BER( ) degradation is (0.2451 - 16.8686 dB )

Chapter Five

ď&#x201A;ˇ

Simulation Results and Analysis

The best one improvement in PAPR and CCDF of PAPR is at improvement in PAPR by = (19.6127 dB), and CCDF of PAPR = while the SNR at BER( ) deteriorated by = (-10.8186 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at improvement in PAPR by = (22.7289 dB), and CCDF of PAPR = while the SNR at BER( ) deteriorated by = (-16.8686 dB).

R =0.2. The (8.2655 dB),

R =0.1. The (9.5720 dB),

CCDF (Pr[PAPR>PAPR0])

original R = .9 R = .8 R =.7 R =.6 R = .5 R = .4 R =.3 R =.2 R = .1

-1

-2

-3

6 PAPR0 [dB]

5.18.a Bit error probability curve for qpsk using OFDM

R = .9 R = .8 R =.7 R =.6 R = .5 R = .4 R =.3 R =.2 R = .1

-1

BER

-2

-3

-4

15 SNR

5.18.b Figure 5.18 (a)CCDF of PAPR OFDM system RCT for various BER of RCT for various parameter 86

parameter. (b) The

Chapter Five

Simulation Results and Analysis

Figure 5.19 illustrates the effect of R parameter on the PAPR, CCDF of PAPR and SNR at BER( ). The y parameter has a positive relationship with PAPR and CCDF of PAPR but an inverse relationship with the SNR at BER( ).

30 SNR at (BER =10-4) CCDF of PAPR PAPR

[dB]

0 0.1

0.2

0.3

0.4

Figure 5.19 the relationship between SNR at BER( ))

0.5 R

0.6

0.7

0.8

0.9

parameter and (PAPR, CCDF of PAPR, and

5.2.3.4 New error function Companding (NERF) : The new type of companding was proposed depends on erf. The NERF companding equation is: ( )

(

| | √

)

( )

(5.8)

NERF De_companding: ( )

|√

| |

( )

(5.9)

When used this type of companding the PAPR was improved by (15.422 dB) and the CCDF of PAPR also was improved by (6.4045 dB) while the SNR at BER( ) was deteriorated by (2.2466 dB). The rate of improvement in the PAPR and CCDF of PAPR is greater than the rate of the decline in SNR at BER( ) as shown in figure 5.20 and table 4.3. Table 5.3 NERF performance NERF PAPR 10.1795

CCDF of PAPR 4.4355

SNR at BER( 13.678

)

Chapter Five

Simulation Results and Analysis

CCDF (Pr[PAPR>PAPR0])

Orignal NERF

-1

-2

-3

6 PAPR0 [dB]

5.20.a Bit error probability curve for qpsk using OFDM

NERF

-1

BER

-2

-3

-4

SNR

5.20.b Figure 5.20 (a) CCDF of PAPR OFDM system NERF companding (b) the BER of NERF companding. 88

Chapter Five

Simulation Results and Analysis

5.2.3.5 Absolute Exponential companding (AEXP) : The proposed AEXP equation is derived based on EXP companding and Trapezoidal power companding: ( )√ 0

( )

| |

(5.10 )

EXP companding since received signal with EXP companding is so distorted that the square root part in (5.10) would be an imaginary or complex number, we then take the absolute value of the square root parts to eliminate any further potential phase distortion. Where ( ) is sign function? The positive constant determines the average power output signals. In order to keep the input and output signals at the same average power level

[| | ]

(

* √[

(

(5.11 ) | |

)] +

)

At the receiver side, the inverse function operation

( )

( ) |√

(

| |

)

( ) of is used in the De_companding

(5.12)

The following can be observed from the table (A.7) and figure 5.21  When d parameter decreases the PAPR and CCDF of PAPR also decrease while SNR at BER( ) increase  Values less than d =.8 the deterioration in the SNR at BER( ) becomes large      

For The best value for the CCDF of PAPR is (2.92) when d =0.8 while the worst value is (5.1533) when d =2 The best value for the PAPR is (6.0806) when d =0.8 while the worst value is (13.0811) when d =2 The best value for the SNR at BER( ) is (14.73) when d =2 while the worst value is (24.833) when d =0.8 PAPR improved by (12.5205 - 19.5209 dB ) CCDF of PAPR improved by (5.6867 - 7.9136 dB ) The amount of SNR at BER( ) degradation is (3.2986 - 13.4016 dB )

Chapter Five

Simulation Results and Analysis

CCDF (Pr[PAPR>PAPR0])

orginal d=2 d = 1.8 d =1.6 d = 1.4 d =1.2 d=1 d = .8 d =.6 d = .4 d =.2

-1

-2

-3

6 PAPR0 [dB]

5.21.a Bit error probability curve for qpsk using OFDM

-1

BER

d=2 d = 1.8 d =1.6 d = 1.4 d =1.2 d=1 d = .8 d =.6 d = .4 d =.2

-2

-3

-4

15 SNR

5.21.b Figure 5.21 (a) CCDF of PAPR OFDM system AEXP companding for various parameters. (b) The BER of AEXP companding for various parameters. 90

Chapter Five

Simulation Results and Analysis

The following conclusion when comparing the proposed method with an OFDM system without PAPR reduction method:  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at d= 1.1. The improvement in PAPR by = (17.6492 dB), and CCDF of PAPR = (7.2405 dB), while the SNR at BER( ) deteriorated by = (-3.4186 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at d= 0.9. The improvement in PAPR by = (18.8515 dB), and CCDF of PAPR = (7.6480 dB), while the SNR at BER( ) deteriorated by = (-4.8686 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at d =0.8. The improvement in PAPR by = (19.5209 dB), and CCDF of PAPR = (7.9136 dB), while the SNR at BER( ) deteriorated by = (-13.4016 dB). Figure 5.22 illustrates the effect of d parameter on the PAPR, CCDF of PAPR and SNR at BER( ). The relationship between the d parameter and SNR at BER( ) is a direct correlation, while the relationship between the d parameter and PAPR, and CCDF of PAPR is the inverse relationship. 30 SNR at (BER =10-4) CCDF of PAPR PAPR

[dB]

0.2

0.4

0.6

0.8

Figure 5.22 the relationship between SNR at BER( ))

1 d

1.2

1.4

1.6

1.8

parameter and (PAPR, CCDF of PAPR, and

5.2.3.6 Cos companding: The new type of companding was proposed depends on cos . The proposed cos companding eauation is: ( )

( )√ 0

| |

(5.13 )

Chapter Five

Simulation Results and Analysis

At the receiver side, the inverse function operation, ( )

( )|

(

| |

( ) of is used in the de-companding

(5.14)

The positive constant determines the average power of output signals. In order to keep the input and output signals at the same average power level.

(

[| | ] [ √0

| |

)

(5.15 )

/1 ]

The following can be observed from table A.8 and the following figures  At y =2 the PAPR deteriorate incremented by (.7122 dB) as well as the CCDF of PAPR, deteriorate incremented by (.515 dB) until the SNR at BER( ) deteriorates by (3.9886 dB)  For , in this region whenever d decreased lead to improvement in (PAPR, CCDF of PAPR and SNR at BER( )) compared with values when y = 2.  Figure 5.23 shows the best CCDF of PAPR and PAPR at y =0.1. The amount of improvement in CCDF of PAPR by (9.9192 dB) and in PAPR (23.6085 dB) compared with OFDM system without companding . Whereas the BER deteriorates considerably in this case.  Figure 5.23 show the best value for the SNR at BER( ) in cos companding is when y = 1. Where it has less value deterioration in the SNR at BER( ) by (0.2717 dB) while PAPR improved by (9.9547 dB), as well as it CCDF of PAPR improved by (3.8892 dB),compared with OFDM system without companding.  For this area is better than region in terms of PAPR and CCDF of PAPR and almost have the same SNR at BER( ) as shown in figure 5.24. So were selected d values less or equal to one. The relationship between the y parameter in cos companding and PAPR is a direct correlation, as shown in figure 5.25. Whenever y increased the PAPR and CCDF of PAPR also increased. But it's different from SNR at BER( ). y =1 is the point of separation and switching between two contradictory in relation to the SNR at BER( ). For whenever y decreased the SNR at BER( ) degradation increases simply means the relationship is an inverse relationship between y and SNR at BER( ) when . While for is quite unlike the previous case. Whenever y decreased the SNR at BER( ) degradation also decreased.

Chapter Five

Simulation Results and Analysis

CCDF (Pr[PAPR>PAPR0])

original y=1 y = .9 y = .8 y = .7 y = .6 y = .5 y = .4 y = .3 y = .2 y = .1

-1

-2

-3

6 PAPR0 [dB]

5.23.a Bit error probability curve for qpsk using OFDM

-1

BER

y y y y y y y y y y

-2

-3

-4

= = = = = = = = = = 5

1 .9 .8 .7 .6 .5 .4 .3 .2 .1 10

15 SNR

5.23.b Figure 5.23 (a) CCDF of PAPR OFDM system cos companding for various parameter. (b) The BER of cos companding for various parameter 93

Chapter Five

Simulation Results and Analysis

The following conclusion when comparing the proposed method with an OFDM system without PAPR reduction method:  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at y =0.8. The improvement in PAPR by = (12.4811 dB), and CCDF of PAPR = (5.0440 dB), while the SNR at BER( ) deteriorated by = (-1.2652 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at y=0.6. The improvement in PAPR by = (15.3614 dB), and CCDF of PAPR = (6.2151 dB), while the SNR at BER( ) deteriorated by = (-2.8639 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at y=0.5. The improvement in PAPR by = (16.8440 dB), and CCDF of PAPR = (6.8657 dB), while the SNR at BER( ) deteriorated by = (-4.3334 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at y =0.4. The improvement in PAPR by = (18.3948 dB), and CCDF of PAPR = (7.4947 dB), while the SNR at BER( ) deteriorated by = (-6.3224 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at y=0.3. The improvement in PAPR by = (20.0315 dB), and CCDF of PAPR = (8.2500 dB), while the SNR at BER( ) deteriorated by = (-17.8522 dB).

30 2.1> y >1 1.1>y>0

PAPR

0 10

SNR at (BER =10 4)

Figure 5.24 the effect of y parameter of cos companding on PAPR and SNR at (BER = )

Chapter Five

Simulation Results and Analysis

30 SNR at (BER =10-4) CCDF of PAPR PAPR

[dB]

0 0.2

0.4

0.6

0.8

1.2

1.4

1.6

1.8

Figure 5.25 illustrates the effect of y parameter in cos companding on the PAPR, CCDF of PAPR and SNR at BER( )

5.2.3.7 tangent Rooting Companding (tanhR): The proposed companding depend on tanh and the companding equation will be as follows: ( )

((| | Ă&#x2014; ) )

( )

(5.16 )

Where k is positive numbers controlling the companding level applied to the envelope x, | | and sign(x) was used to maintain the phases of the OFDM signal. Decompanding equation will be as follows: ( )

| |

( )) |

( )

(5.17)

Figure 5.26 shows the CCDF of PAPR and BER for OFDM system with tanhR companding at k =10 and y change from 0.1 to 1 by 0.1 every time. The CCDF of PAPR was improved by (6.6795 - 23.9603 dB) and the PAPR was improved by (6.6999 - 10.1412 dB) while the SNR at BER( ) was deteriorated by (3.0726 18.5686 dB) compared with an OFDM system without companding. Whenever y decreased the PAPR and CCDF of PAPR was improved while increasing the SNR at BER( ) values

Chapter Five

Simulation Results and Analysis

CCDF (Pr[PAPR>PAPR0])

original y =1 y =.9 y =.8 y =.7 y =.6 y =.5 y =.4 y =.3 y =.2 y =.1

-1

-2

-3

6 PAPR0 [dB]

5.26.a Bit error probability curve for qpsk using OFDM

-1

BER

y y y y y y y y y y

-2

-3

=1 =.9 =.8 =.7 =.6 =.5 =.4 =.3 =.2 =.1

-4

15 SNR

5.26.b Figure 5.26 (a) CCDF of PAPR OFDM system tanhR companding for various parameter. (b) The BER of tanhR companding for various parameters 96

Chapter Five

Simulation Results and Analysis

Figure 5.27 and table A.10 shows the CCDF of PAPR and BER for OFDM system with tanhR companding at y=1 and with different k,  k = ( 5,10) The CCDF of PAPR was improved by ( 3.6235 - 6.6310 dB) and the PAPR was improved by (9.1388 - 16.6703 dB) while the SNR at BER ( ) was deteriorated by (0.4931 - 3.2172 dB) compared with an OFDM system without companding.  k = ( 15 ,20) . The SNR at BER ( ) was deteriorated considerably up. where at k =15 the SNR at BER( ) reach to 30 dB And more than 30 dB at k =20. While the CCDF of PAPR was improved by (8.2413 - 9.0172 dB) and the PAPR was improved by ( 20.3701 - 22.2234 dB) compared with an OFDM system without companding. Whenever k was increased the PAPR and CCDF of PAPR was improved while increasing the BER values Fig (5.28) and table(A.10) shows the CCDF of PAPR and SNR at BER( ) for OFDM system with tanh companding at y=.8 and with different k .  k = ( 5,10) The CCDF of PAPR was improved by (1.3349- 0.4511 dB) and the PAPR was improved by (3.0266 - 0.7832 dB) While the SNR at BER ( ) was deteriorated by (0.8925 - 0.2835 dB) compared with an OFDM system with tanh companding at y=1  k = ( 5,10) The CCDF of PAPR was improved by (4.9584 - 7.0821 dB) and the PAPR was improved by (12.1654 - 17.4535 dB) While the SNR at BER ( ) was deteriorated by (1.3856 - 3.5007 dB) compared with an OFDM system without companding.  k =15. The SNR at BER ( ) was improved by (7.5344) and the CCDF of PAPR was improved by (0.0722 dB) While the PAPR was deteriorated by (0.0115 dB) but the deterioration ratio is less than the proportion of improvement. At k =20, The SNR at BER ( ) was improved ,the CCDF of PAPR was improved by (0.0184 dB) and the PAPR was improved by ( 0.2169 dB).  k = ( 15,20). where at k =15 the The SNR at BER ( ) was deteriorated by (11.0342 dB )And more than 30 dB at k =20. While the CCDF of PAPR was improved by (8.3135 - 8.9988 dB) and the PAPR was improved by (20.3586 - 22.0065 dB) compared with an OFDM system without companding.

Figure 5.29 and table A.9 shows the following:  For k =5, 10, whenever y was decreased the PAPR and CCDF of PAPR improved while deteriorating the SNR at BER( ).  For k =15, whenever y was decreased the PAPR and CCDF of PAPR improved, but the rate of improvement is less than at k = 5, 10. The best value of SNR at BER( ) at y =.5 while the worst at y =1, where up to 30 dB.  For k =20, The influence of y on PAPR and CCDF of PAPR very little, except when y = .2 which improve the SNR at BER( ), and CCDF PAPR of PAPR with a small amount

Chapter Five

Simulation Results and Analysis

CCDF (Pr[PAPR>PAPR0])

k= k= k= k=

5 10 15 20

-1

-2

-3

4 PAPR0 [dB]

Figure 5.27.a Bit error probability curve for qpsk using OFDM

k= k= k= k=

-1

BER

5 10 15 20

-2

-3

-4

15 snr

Figure 5.27.b Figure 5.27 (a) CCDF of PAPR OFDM system tanhR companding for various parameters at y=1. (b) The BER of tanhR companding for various parameters at y=1.

Chapter Five

Simulation Results and Analysis

CCDF (Pr[PAPR>PAPR0])

k= k= k= k=

5 10 15 20

-1

-2

-3

3 PAPR0 [dB]

Figure 5.28.a Bit error probability curve for qpsk using OFDM

k= k= k= k=

-1

BER

5 10 15 20

-2

-3

-4

15 SNR

Figure 5.28.b Figure 5.28 (a) CCDF of PAPR OFDM system tanhR companding for various parameters at y=.8. (b) The BER of tanhR companding for various parameters at y=.8.

Chapter Five

Simulation Results and Analysis

The following conclusion when comparing the proposed method with an OFDM system without PAPR reduction method:  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k =5 ,y =0.8 . The improvement in PAPR by = ( 11.7543 dB), and CCDF of PAPR = (4.7819 dB), while the BER deteriorated by = (-1.2398 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k =10 ,y =0.9 . The improvement in PAPR by = ( 17.0445 dB), and CCDF of PAPR = (6.8431 dB), while the SNR at BER( ) deteriorated by = (-3.4062 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k=10 ,y =0.6 . The improvement in PAPR by = ( 18.5665 dB), and CCDF of PAPR = (7.5973 dB), while the SNR at BER( ) deteriorated by = (-5.3686 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k =10 , y=0.5. The improvement in PAPR by = ( 19.0855 dB), and CCDF of PAPR = (7.9224 dB), while the SNR at BER( ) deteriorated by = (-6.4557 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k = 5, y=0.2. The improvement in PAPR by = ( 22.0569 dB), and CCDF of PAPR = (9.3125 dB), while the SNR at BER( ) deteriorated by = (-13.2917 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k =20 ,y =0.2. The improvement in PAPR by = ( 23.0305 dB), and CCDF of PAPR = (9.7085 dB), while the SNR at BER( ) deteriorated by = (-17.5078 dB).

[dB]

BER k = 5 BER k = 10 BER k = 15 BER k = 20 PAPR k = 5 PAPR k = 10 PAPR k = 15 PAPR k = 20

0 0.2

0.3

0.4

0.5

0.6 y

0.7

0.8

0.9

Figure 5.29 illustrates the effect of y and k parameter in tanhR companding on the PAPR and SNR at BER( ) 100

Chapter Five

Simulation Results and Analysis

5.2.3.8 Logarithmic Rooting Companding (logR): The logarithm ( ( )

) Rooting companding equation will be as follows:

((| | ×

)

( )

(5.18)

Decompanding equation will be as follows: ( )

| |

. /

) |

( )

(5.19)

Where is positive number controlling the amount of companding. We used to control the companding level applied to the envelope x, | | and sign(x) was used to maintain the phases of the OFDM signal. Figure 5.30 shows the CCDF of PAPR and BER for OFDM system with log companding at y =1 and k change from 10 to 100 by 10 every time. The CCDF of PAPR was improved by (15.5911 - 6.6150 dB) and the PAPR was improved by (8.8595 -15.5911 dB) while the SNR at BER( ) was deteriorated by (1.0686 -18.5686 dB) compared with an OFDM system without companding. Whenever k was increased the PAPR and CCDF of PAPR was decreased while increasing the BER values Figure 5.31 shows the CCDF of PAPR and BER for OFDM system with tanh companding at k =10 and y change from 0.1 to 1 by 0.1 every time. The CCDF of PAPR was improved by (3.5255 - 9.9080 dB) and the PAPR was improved by (8.8595 -23.4987 dB) while the SNR at BER( ) was deteriorated by (1.0686 18.1686 dB) compared with an OFDM system without companding. Whenever y decreased the PAPR and CCDF of PAPR was improved while increasing the BER values The following conclusion when comparing the proposed method (from the table (A.12)) with an OFDM system without PAPR reduction method:  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k =10, y =0.9. The improvement in PAPR by = ( 9.8230 dB), and CCDF of PAPR = (4.0570 dB), while the SNR at BER( ) deteriorated by = (-1.2806 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k=10 ,y =0.6. The improvement in PAPR by = ( 14.4744 dB), and CCDF of PAPR = (5.9700 dB), while the SNR at BER( ) deteriorated by = (-3.3207 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k =20 ,y =0.5 . The improvement in PAPR by = ( 16.6873 dB), and CCDF of PAPR = (6.9120 dB), while the SNR at BER( ) deteriorated by = (-5.0018 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k =10, y =0.3. The improvement in PAPR by = ( 19.6992 dB), and CCDF of PAPR = (8.2150 dB), while the SNR at BER( ) deteriorated by = (-8.5686 dB). 101

Chapter Five 



Simulation Results and Analysis

For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at . The improvement in PAPR by = ( 21.9193 dB), and CCDF of PAPR = (9.2140 dB), while the SNR at BER( ) deteriorated by = (-12.7266 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at . The improvement in PAPR by = ( 23.5788 dB), and CCDF of PAPR = (9.9600 dB), while the SNR at BER( ) deteriorated by = (-18.1686 dB). 0

CCDF (Pr[PAPR>PAPR0])

-1

original k = 10 k = 20 k = 30 k = 40 k = 50 k =60 k = 70 k = 80 k = 90 k = 100

-2

-3

6 PAPR0 [dB]

Figure 5.30.a Bit error probability curve for qpsk using OFDM

k k k k k k k k k k

-1

BER

-2

= 10 = 20 = 30 = 40 = 50 =60 = 70 = 80 = 90 = 100

-3

-4

15 SNR

Figure 5.30.b Figure 5.30 (a)CCDF of PAPR OFDM system logR companding for various parameter. (b) the BER of logR companding for various parameter. 102

Chapter Five

Simulation Results and Analysis

CCDF (Pr[PAPR>PAPR0])

original y=1 y = .9 y = .8 y = .7 y = .6 y = .5 y = .4 y = .3 y = .2 y = .1

-1

-2

-3

6 PAPR0 [dB]

Figure 5.31.a Bit error probability curve for qpsk using OFDM

-1

BER

y y y y y y y y y y

-2

-3

-4

= = = = = = = = = = 5

1 .9 .8 .7 .6 .5 .4 .3 .2 .1 10

15 SNR

Figure 5.31.b Figure 5.31 (a)CCDF of PAPR OFDM system logR companding for various parameter. (b) The BER of logR companding for various parameter. 103

Chapter Five

Simulation Results and Analysis

5.2.4 Pre-distortion methods: The pre-distortion methods are based on the reorientation or spreading the energy of data symbol before taking IFFT. The pre-distortion schemes include DFT spreading, pulse shaping or precoding and constellation shaping. The methods like Tone Reservation (TR) and Tone Injection (TI) are the example of constellation shaping schemes [188]

5.2.4.1 Pulse Shaping or Pre-coding: The pulse shaping or pre-coding technique is an efficient and flexible way for reducing the PAPR of OFDM signals. In this method, each data block is multiplied by a pre-coding matrix prior to OFDM modulation and transmission. This method is data-independent and, thus, avoids block based optimization. It also works with an arbitrary number of subcarriers and any type of baseband modulation used. In terms of BER performance, it takes advantage of the frequency variation of the fading multipath channel and improves the BER of OFDM signals in comparison to conventional OFDM (no pre-coding). The implementation complexity of the proposed technique is acceptable, since a predefined pre-coding matrix is used and thus, no handshake is needed between transmitter and receiver. Having the same pre-coding matrix for all OFDM blocks avoids all the processing needed in block-based optimization methods [189]. Precoded OFDMA consists of using a precoding matrix P that spreads the energy of symbols over the subcarriers allocated to the user. Uniform energy distribution is favored in practice. [190] The OFDM system with an orthogonal precoder is considered. In precoded OFDM system instead of sending uncoded symbols (one per subcarrier), the idea is to send different linear combinations of the information symbols on the subcarriers. This corresponds to signal space diversity. [191] Precoding based techniques are simple linear techniques. These techniques can reduce the PAPR up to the PAPR of single carrier systems (Slimane, 2007). WHT precoding based techniques, DCT precoding based techniques, DHT precoding based techniques are common examples of precoding based PAPR reduction techniques (Slimane, 2007; Min & Jeoti, 2007; Baig & Jeoti, 2010a, 2010b, 2010c) [14] Figure. 5.32 shows the block diagram of Precoding Based OFDM System. We implemented the Precoding matrix P of dimension N × N before the IFFT to reduce the PAPR. The Precoding matrix P can be written as:

[

(

)

(

)

(

)

(

)

(

(5.20) )]

)(

Where P is a Precoding Matrix of size N ×N is shown in equation (5.20). The complex baseband OFDM signal with N subcarriers can be written as: ( )

√

∑

t 104

(5.21)

Chapter Five

Simulation Results and Analysis

We can express modulated OFDM vector signal with N subcarriers as: *

(5.22)

[192], [193]

Add CP

S / P

IDFT OR IFFT

Signal mapper

I / P

P / S

Multipath Fading Ch. & noise

đ?&#x2018;&#x192;

S / P

Remove CP

DFT OR FFT

One Tap Equalizer And P/S

Signal demapper

O / P

Figure 5.32 Block diagram of Precoding based OFDM system

5.2.4.2 Discrete Hartley transform (DHT) : The DHT is a linear transform. In DHT, N real numbers are transformed into N real numbers a. According to [91], the N-point DHT can be defined as follows: â&#x2C6;&#x2018; Where

â&#x2C6;&#x2018;

( )

(5.23)

and

P is precoding matrix of size NĂ&#x2014;N shown, m and n are integers from DHT is also invertible transform which allows us to recover the from inverse can be obtained by simply multiplying DHT of by [194].

. The and

5.2.4.3 Walsh-Hadamard Transform (WHT): The Hadamard transform (also known as the Walshâ&#x20AC;&#x201C;Hadamard transform, Hadamardâ&#x20AC;&#x201C; Rademacherâ&#x20AC;&#x201C;Walsh transform, Walsh transform, or Walshâ&#x20AC;&#x201C;Fourier transform) is an example of a generalized class of Fourier transforms. It performs an orthogonal, symmetric, involutional, linear operation on real numbers (or complex numbers, although the Hadamard matrices themselves are purely real). The Hadamard transform can be regarded as being built out of size Ă&#x2014; Ă&#x2014; Ă&#x2014; Ă&#x2014; DFTs, and is in fact equivalent to a multidimensional DFT of size. It decomposes an arbitrary input vector into a superposition of Walsh functions [195]. 105

Chapter Five

Simulation Results and Analysis

The WHT is a non-sinusoidal and it is an orthogonal technique which decomposes a signal into set of basic functions. These functions are called Walsh functions, the hadamard transform scheme reduce the occurrence of the high peaks comparing the conventional OFDM system. The hadamard transform used because it reduce the autocorrelation of the input sequence to reduce the PAPR of OFDM signal. It also not requires to send side information to the receiver [36] WHT can be implemented by a butterfly structure as in FFT. This means that applying WHT does not require the extensive increase of system complexity. The kernel of WHT can be written as follows:, -

(5.24)

√

(5.25)

[

]

(5.26)

Where denotes the binary complement of [14] Note that Hadamard transform is an orthogonal linear transform and can be implemented by a butterfly structure as in FFT. This means that applying Hadamard transform does not require the extensive increase of system complexity. The received vector signal corrupted by noise vector n can be recovered to as [11]: * *

+ *

(5.27)

(5.28)

The FWHT for a signal x of length N is defined as: ∑

(

Where i = 0,1,..., N-1 and

)

(5.29) (

) are Walsh functions

5.2.4.4 Discrete Cosine Transform (DCT): DCT is a technique to transform a signal into frequency domain. DCT denotes a row of data in terms of the sum of cosine functions that oscillate at different frequencies. DCT is similar to the DFT, but the DCT only uses real number without imaginary component. The idea of using DCT in this study is to reduce the autocorrelation of the input row to reduce PAPR and it does not require the information transmitted to the receiver. The idea of using DCT is for reduce auto-correlation from input data to reduce PAPR problem [117]. DCT matrix P of size N-by-N can be created by using equation

106

Chapter Five

Simulation Results and Analysis

√ (

√

)

(5.30)

{

}

and DCT can be defined as:∑

, .

/ - k=0,1 … N-1

(5.31)

5.2.4.5 Discrete Sine Transform (DST) Precoding Technique: For an input signal ∑

, the discrete sine transform (

)(

can be defined as:

)- k=1, 2 … N-1

(5.32)

DST precoding matrix D can be generated as follows: √ (

√

)

(5.33)

{

}

The DST matrix must satisfy the following criteria: 1. Same magnitude for all the elements of the precoding matrix. 2. The magnitude must be equal to . √ 3. The DST precoding matrix must be non-singular matrix. These criteria ensure that every output symbol has the same amount of information of every input data; it preserves the power at the output and also ensures the recovery of the original data at the receiver. When DST precoding is applied to the complex input vector of size M, this input vector is transformed to a new vector of size L that can be written as follows: Y = D. X = ,

(5.34)

Where D is a DST precoder matrix of size N = L × L, generated by Eq. (5.33) and ∑

correspond to

(5.35) row and

column of DST precoder matrix [40].

5.2.4.6 The Discrete Fourier Transform (DFT) Precoding: The only difference between the DFT-spread OFDM and the conventional OFDM is the presence of a DFT and an IDFT block in the transmitter and receiver, respectively

107

Chapter Five

Simulation Results and Analysis

In the DFT-spread OFDM, the PAPR of the signal is fairly low as compared with the conventional OFDM because the DFT operation spreads data into subcarriers [197]. The DFT of a sequence of length N can be defined as ( )

∑

–

( )

(5.36)

The sinusoids of the DFT (or Inverse Discrete Fourier Transform (IDFT)) form an orthogonal basis set and a signal in vector space of the DFT (or IDFT) can be represented as a linear combination of the orthogonal sinusoids. Thus the IDFT at the transmitter maps an input signal into a set of orthogonal subcarriers. Similarly the transform DFT is used at the receiver to reverse the mapping of IDFT and signal from the subcarriers are combined to form an estimate of the source signal from the transmitter. Since the basis function of DFT is uncorrelated, the correlation performed in DFT for the given subcarrier only sees energy for that corresponding subcarrier. The energy from other subcarrier does not contribute because they are uncorrelated. This separation of the signal energy is the reason that OFDM subcarriers spectrum can overlap without causing interference. [198]

5.2.4.7 Simulation results and analysis of OFDM system with precoding: 5 types of Pre-coding are used in this section and then compare them with each other. The best type of reduced PAPR and BER is the DFT pre-coder. The best type of reduced PAPR and BER is the DFT pre-coder as shown in figure 5.34 and table (A.3) but suffer from link performance loss in a frequency-selective channel when highorder modulation techniques are used. The presence of carrier frequency offsets (CFOs) between the transmitter and the receiver results in a loss of orthogonality among subcarriers and an intercarrier interference (ICI). CFOs also introduce multiple access interference (MAI) and degrade the bit error rate (BER) performance in the DFT pre-coder system. [92] The following is the conclusion from the table (A.3) and figure 5.33  WHT pecoder was improved each of the PAPR by (2.7941 dB), CCDF of PAPR by (0.8684 dB) and SNR at BER( ) (0.01dB). But the amount of improvement in WHT pre-coding is the least in comparison with the rest kinds of pre-coding  DCT pecoder was improved each of the PAPR by (7.5208 dB), CCDF of PAPR by (3.109 dB) and SNR at BER( ) (0.012dB). DCT pre-coding results better than WHT pre-coding but worse than the rest  DST pecoder was improved each of the PAPR by (8.1669 dB), CCDF of PAPR by (3.25 dB) and SNR at BER( ) (0.012 dB). DST and DCT have the same SNR at BER( ) but DST better than DCT in PAPR and CCDF of PAPR  DHT pecoder was improved each of the PAPR by (18.6731 dB), CCDF of PAPR by (7.423 dB) and SNR at BER( ) (0.058dB). DHT pecoder results are better than other types of pre-coding except DFT pre-coder.  DFT pecoder was improved each of the PAPR by (25.6118 dB), CCDF of PAPR by (10.773 dB) and SNR at BER( ) (0.171dB). DFT pecoder results are the best compared with other types of pre-coding 108

Chapter Five

Simulation Results and Analysis

CCDF (Pr[PAPR>PAPR0])

orignal WHT DCT DST DHT DFT

-1

-2

-3

6 PAPR0 [dB]

5.33.a Bit error probability curve for qpsk using OFDM

orignal WHT DCT DST DHT DFT

-1

BER

-2

-3

-4

SNR

5.33.b Figure 5.33 (a)CCDF of PAPR for OFDM system with different type of pre-coding (b) BER for OFDM system with different type of pre-coding

109

Simulation Results and Analysis of Hybrid PAPR techniques Chapter six Simulation Results and Analysis of Hybrid PAPR techniques 6.1 Hybrid pre-coding with RCF: Proposed a method based on the integration of all of precoding with RCF as shown in figure 6.1. The results of this method better than the results of the RCF and precoding each alone, except in the case of DHT with RCF (I = 2, pilot) ,where the results of the DHT itself better than the pre-coding with RCF hybrid (DHT with RCF).The best result for the PAPR is when RCF (I =1) with (DHT). WHT, DCT, DST, and DHT pre-coders are used with RCF is used with the following specifications (I =1, pilot, and 2, CR =4, 3, 2) The OFDM system model with the proposed technique as shown in figure 6.1.

D / A

Add CP

P / S

RCF

IDFT OR IFFT

S / P

+pilot symbol

Signal mapper

I / P

Multipath Fading Ch. & noise

S / P

Remove CP

đ?&#x2018;&#x192;

DFT OR FFT

Remove +pilot symbol

One Tap Equalizer And P/S

Signal demapper

O / P

A / D

Figure 6.1 the OFDM system model with precoding + RCF. The following conclusion from table A.29 when comparing the proposed method with an OFDM system without PAPR reduction method: ď&#x201A;ˇ There are improved in PAPR, CCDF of PAPR and BER in many points that have been tested, but The best one improvement in PAPR and CCDF of PAPR is at I = pilot, CR =2,and DHT. The improvement in PAPR by = (17.2780 dB), CCDF of PAPR = (7.2062 dB), and the SNR at BER( ) by = (0.1105 dB). ď&#x201A;ˇ For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at I = 1, CR =1.5, and DHT. The improvement in PAPR by = (20.4339 dB), and CCDF of PAPR = (8.9446 dB), while the SNR at BER( ) deteriorated by = (-1.1636 dB). ď&#x201A;ˇ For SNR at BER( )

110

Simulation Results and Analysis of Hybrid PAPR techniques The best one improvement in PAPR and CCDF of PAPR is at I = 1, CR =1.3, and DHT. The improvement in PAPR by = (21.0373 dB), and CCDF of PAPR = (9.1129 dB), while the SNR at BER( ) deteriorated by = (-1.6285 dB). The following conclusion from table A.29 when comparing the proposed method with an OFDM system with RCF method:  The PAPR and the CCDF of PAPR were improved and the least amount of improvement was when I=1, CR = 4 and WHT .The PAPR improvement is equal to (0.1148 dB) and the CCDF of PAPR improvement is equal to (0.0031 dB), while the vast amount of improvement is where I=1, CR = 4 and DHT the PAPR improvement is equal to (7.1348 dB) and the CCDF of PAPR improvement is equal to (3.0141 dB )  The SNR at BER( ) at I =2, pilot, there is an improvement in some of the points and the degradation the other .The largest amount of improvement is when I=2, CR =1.5 and DHT is equal to (17.1418 dB).The largest amount of degradation is when I=2, CR =1.5 and WHT is equal to (-0.4 dB).  The SNR at BER( ) was improved at I =1.The least amount of improvement in SNR at BER( ) when DST and CR = 4 and is equal to (0.063 dB). The largest amount of improvement is when DHT and CR =1.5 is equal to (16.905 dB). The following conclusion from table A.29 when comparing the proposed method with an OFDM system with pre-coding method:  The PAPR was improved, except when (DHT, I =2, pilot, and CR = 4,3 ,2 ,1.5) PAPR was degraded and the maximum degradation is (-7.269 dB).the least amount of improvement was at (DHT, I =1, and CR = 2) and is equal to (0.0008 dB), while the vast amount of improvement is where (WHT, I =1, and CR = 1.1) and is equal to (18.3732 dB).  The CCDF of PAPR was improved, except when (DHT, I =2, pilot, and CR = 4,3 ,2 ,1.5) and (DHT, I =2, and CR = 1.5) CCDF of PAPR were degraded and the maximum degradation is (-2.9015 dB).the least amount of improvement was at (DHT, I = Pilot, and CR = 1.5) and is equal to (0.1727 dB), while the vast amount of improvement is where (WHT, I =1, and CR = 1.1) and is equal to (8.6977 dB).  The SNR at BER( ) was degraded, except when (DHT, I =2, pilot, and CR = 4,3 ,2 ) SNR at BER( ) were degraded and the maximum improvement is (2.7701 dB).The least amount of degradation in SNR at BER( ) when d=1.1 and DHT and is equal to (dB). The largest amount of degradation is when I =1, CR =1.1, 1.3 and WHT is equal to (-18.37 dB). Figures (6.2, 6.3, 6.4, and 6.5) shows the performance of the hybrid pre-coding with RCF (at I =1 for different CR). Hybrid pre-coding with RCF is the same as the RCF where whenever CR decreased the (PAPR and CCDF of PAPR) improved and the SNR at BER( ) degraded but the proposed better than RCF because the improvement (CCDF and PAPR of PAPR) is greater than the amount of degradation in the SNR at BER( ) if we compared both with the original signal and RCF to gather. The best species is (`DHT + RCF) comes after ((DCT + RCF and RCF + DST) together) and in the end come (WHT + RCF)

111

Simulation Results and Analysis of Hybrid PAPR techniques 0

CCDF (Pr[PAPR>PAPR0])

WHT+RCF (CR = 4) WHT+RCF(CR = 3) WHT+RCF(CR = 2) WHT+RCF(CR = 1.5) WHT+RCF(CR = 1.3) WHT+RCF(CR = 1.1)

-1

-2

-3

3 4 PAPR0 [dB]

Figure 6.2.a Bit error probability curve for qpsk using OFDM

WHT+RCF (CR = 4) WHT+RCF(CR = 3) WHT+RCF(CR = 2) WHT+RCF(CR = 1.5) WHT+RCF(CR = 1.3) WHT+RCF(CR = 1.1)

-1

BER

-2

-3

-4

15 SNR

Figure 6.2.b Figure 6.2 (a)CCDF of PAPR for OFDM system with WHT +RCF where I =1 for different CR (b) BER for OFDM system with WHT +RCF where I =1 for different CR 112

Simulation Results and Analysis of Hybrid PAPR techniques 0

CCDF (Pr[PAPR>PAPR0])

DCT+RCF (CR = 4) DCT+RCF(CR = 3) DCT+RCF(CR = 2) DCT+RCF(CR = 1.5) DCT+RCF(CR = 1.3) DCT+RCF(CR = 1.1)

-1

-2

-3

3 4 PAPR0 [dB]

Figure 6.3.a Bit error probability curve for qpsk using OFDM

DCT+RCF (CR = 4) DCT+RCF(CR = 3) DCT+RCF(CR = 2) DCT+RCF(CR = 1.5) DCT+RCF(CR = 1.3) DCT+RCF(CR = 1.1)

-1

BER

-2

-3

-4

15 SNR

Figure 6.3.b Figure 6.3 (a)CCDF of PAPR for OFDM system with DCT +RCF where I =1 for different CR (b) BER for OFDM system with DCT +RCF where I =1 for different CR 113

Simulation Results and Analysis of Hybrid PAPR techniques 0

CCDF (Pr[PAPR>PAPR0])

DST+RCF (CR = 4) DST+RCF(CR = 3) DST+RCF(CR = 2) DST+RCF(CR = 1.5) DST+RCF(CR = 1.3) DST+RCF(CR = 1.1)

-1

-2

-3

3 4 PAPR0 [dB]

Figure 6.4.a Bit error probability curve for qpsk using OFDM

DST+RCF (CR = 4) DST+RCF(CR = 3) DST+RCF(CR = 2 ) DST+RCF(CR = 1.5) DST+RCF(CR = 1.3) DST+RCF(CR = 1.1)

-1

BER

-2

-3

-4

15 SNR

Figure 6.4.b Figure 6.4 (a)CCDF of PAPR for OFDM system with DST +RCF where I =1 for different CR (b) BER for OFDM system with DST +RCF where I =1 for different CR 114

Simulation Results and Analysis of Hybrid PAPR techniques

CCDF (Pr[PAPR>PAPR0])

-1

-2

DHT+RCF(CR =4) DHT+RCF(CR =3) DHT+RCF(CR =2) DHT+RCF(CR =1.5) DHT+RCF(CR =1.3) DHT+RCF(CR =1.1)

-3

0.5

1.5 2 PAPR0 [dB]

2.5

3.5

Figure 6.5.a Bit error probability curve for qpsk using OFDM

DHT+RCF DHT+RCF DHT+RCF DHT+RCF DHT+RCF DHT+RCF

-1

BER

(CR (CR (CR (CR (CR (CR

= = = = = =

4) 3) 2) 1.5) 1.3) 1.1)

-2

-3

-4

SNR

Figure 6.5.b Figure 6.5 (a)CCDF of PAPR for OFDM system with DHT +RCF where I =1 for different CR (b) BER for OFDM system with DHT +RCF where I =1 for different CR Figure 6.6 shows the proposed method with I =2 and CR =4, Can note the following form the figure 6.6, primarily the proposed method on despite of the different type of pre-coding used but it has almost the same PAPR (0.1690 dB), CCDF of PAPR (0.1493 dB) and SNR at BER( ) (0.1111 dB). This means that it is not based on the type of pre-coding. 115

Simulation Results and Analysis of Hybrid PAPR techniques

CCDF (Pr[PAPR>PAPR0])

WHT + RCF DCT + RCF DST + RCF DHT+ RCF -1

-2

-3

3 4 PAPR0 [dB]

Figure 6.6.a Bit error probability curve for qpsk using OFDM

WHT + RCF DCT + RCF DST + RCF DHT + RCF

-1

BER

-2

-3

-4

5 SNR

Figure 6.6.b Figure 6.6 (a)CCDF of PAPR for OFDM system with different type of pre-coding +RCF where I =2, CR =4 (b) BER for OFDM with different type of pre-coding +RCF where I =2, CR =4 Figure 6.7 shows the proposed method with I =pilot and CR =4 , Can note the following form the figure 6.7 ,primarily the proposed method on despite of the different type of pre-coding used but it has almost the same PAPR (0.0378 dB), 116

Simulation Results and Analysis of Hybrid PAPR techniques CCDF of PAPR(0.0219 dB) and SNR at BER( is not based on the type of pre-coding.

) 0.3759 dB). This means that it

CCDF (Pr[PAPR>PAPR0])

DCT+RCF DST+RCF DHT +RCF

-1

-2

-3

3 4 PAPR0 [dB]

Figure 6.7.a Bit error probability curve for qpsk using OFDM

DCT +RCF DST +RCF DHT +RCF -1

BER

-2

-3

-4

5 snr

Figure 6.7.b Figure 6.7 (a)CCDF of PAPR for OFDM system with different type of pre-coding +RCF where I =pilot, CR =4 (b) BER for OFDM with different type of pre-coding +RCF where I =pilot, CR =4 117

Simulation Results and Analysis of Hybrid PAPR techniques Figure 6.8 shows the proposed method with I =1 and CR =1.5. Here depends on the type of the pre-coding, where different values of (PAPR, CCDF of PAPR and BER). For each type of pre-coding, and best as is evident is the DHT of figure 6.8.

CCDF (Pr[PAPR>PAPR0])

WHT + RCF DCT + RCF DST + RCF DHT + RCF -1

-2

-3

0.5

1.5

2.5

PAPR0 [dB]

Figure 6.8.a Bit error probability curve for qpsk using OFDM

WHT + RCF DCT + RCF DST + RCF DHT + RCF

-1

BER

-2

-3

-4

SNR

Figure 6.8.b Figure 6.8(a) CCDF of PAPR for OFDM system with different type of pre-coding +RCF where I =pilot, CR =4 (b) BER for OFDM with different type of pre-coding +RCF where I =1, CR =1.5 118

Simulation Results and Analysis of Hybrid PAPR techniques 6.2 Hybrids RCF with companding: The clipping is the easiest technique to reduce the power by setting a maximum level for the transmitted signal. In addition to these benefits in clipping, the use of frequency domain filtering, this improves the BER. On the other hand, the companding has also been considered a good technique, because it has the good PAPR reduction capability with no bandwidth expansion and low computational complexity. The other advantage of companding is that the signal can be recovered at the receiver through inverse companding transform [10] With the understanding on RCF and companding techniques, an idea emerged to combine the philosophy of companding and RCF. This hybrid technique shows good results because of first RCF reduce the PAPR and improves the BER constant and then companding more reduces the amount of the PAPR. The OFDM system model with the proposed technique is as shown in figure 6.9. RCF is used with the following specifications (I =2, CR =4, 3, 2) as for the companding has been using all kinds of previous companding.

Add CP

P / S

Companding

RCF

IDFT OR IFFT

S / P

+pilot symbol

Signal mapper

I / P

D / A

Multipath Fading Ch. & noise

Remove CP

S / P

De-Companding

DFT OR FFT

Remove +pilot symbol

One Tap Equalizer And P/S

Signal demapper

O / P

A / D

Figure 6.9 the OFDM system model with RCF with companding .

6.2.1 RCF + A companding: The following conclusion from table A.13 and figure 6.10 when comparing the proposed method with an OFDM system without PAPR reduction method:  At A =5 and CR =4 , There is an improvement in PAPR by = 17.1305 dB ,CCDF of PAPR = 6.9730 dB, and the BER by =0.3409 dB. This point was chosen because all the variables improved.  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at A =10 and CR =4 . The improvement in PAPR by = (18.8751 dB), and CCDF of PAPR = (7.7750 dB), while the SNR at BER( ) deteriorated by = (-1.5004 dB).  For SNR at BER( ) 119

Simulation Results and Analysis of Hybrid PAPR techniques



The best one improvement in PAPR and CCDF of PAPR is at A = 20 and CR =4 . The improvement in PAPR by = (20.0433 dB), and CCDF of PAPR = (8.3573 dB), while the SNR at BER( ) deteriorated by = (-3.5686 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at A =4 and CR =4. The improvement in PAPR by = (20.8021 dB), and CCDF of PAPR = (8.7150 dB), while the SNR at BER( ) deteriorated by = (-5.5686 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at A=200 and CR =4. The improvement in PAPR by = (21.9809 dB), and CCDF of PAPR = (9.2880 dB), while the SNR at BER( ) deteriorated by = (-8.4493 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at A =90 and CR =2. The improvement in PAPR by = (22.3041 dB), and CCDF of PAPR = (9.6580 dB), while the SNR at BER( ) deteriorated by = (-12.7686 dB).

The following conclusion from table A.13 and figure 6.10 when comparing the proposed method with an OFDM system with a companding PAPR reduction method:  The PAPR was improved and the least amount of improvement was when A = 90 and CR = 4 and is equal to (6.9071 dB), While the vast amount of improvement is where A = 5 and CR = 1.5 and is equal to (13.4875 dB).  Less the amount of improvement in CCDF of PAPR when A = 100 and CR = 4 and is equal to (0.815 dB), While the vast amount of improvement is where A = 5 and CR = 1.5 and is equal to (4.64 dB).  The SNR at BER( )was improved when CR =4,3 . The vast amount of improvement is where A = 10 and CR = 4 and is equal to (3.1882 dB), while Less the amount of improvement in SNR at BER( )when A = 30 and CR = 3 and is equal to (0.88 dB)  The SNR at BER( ) was degraded when CR = 2. The least amount of degradation in SNR at BER( ) when A =80 and CR = 2 and is equal to (1.2408 dB). The largest amount of degradation is when A= 90 and is equal to (3.0453 dB).  The SNR at BER( ) was degraded when CR = 1.5, when A =5 the amount of degradation is equal to (-16.4 dB). The following conclusion from table A.13 and figure 6.10 when comparing the proposed method with an OFDM system with RCF method:  The PAPR and the CCDF of PAPR were improved and the least amount of improvement was when A =5 and CR = 1.5 .The PAPR improvement is equal to (2.9643 dB) and the CCDF of PAPR improvement is equal to (1.6257 dB), while the vast amount of improvement is where A = 200 and CR = 4 and the PAPR improvement is equal to (10.8726 dB) and the CCDF of PAPR improvement is equal to (4.64 dB )  The SNR at BER( ) was degraded, except when A =5 and CR =1.5 the BER maintains its value. The least amount of degradation in SNR at BER( ) when A =5 and CR = 4 and is equal to (-2.3226 dB). The largest amount of degradation is when A= 90 and CR =2 is equal to (-12.072 dB).

120

Simulation Results and Analysis of Hybrid PAPR techniques 30 original RCF (CR=4) RCF (CR=3) RCF (CR=2) A RCF (CR=4) + A RCF (CR=3) + A RCF (CR=2) + A

PAPR

SNR at (BER =10- 4)

Figure 6.10.a 11 original RCF (CR=4) RCF (CR=3) RCF (CR=2) A RCF (CR=4) + A RCF (CR=3) + A RCF (CR=2) + A

10 9

CCDF of PAPR

8 7 6 5 4 3 2 1

SNR at (BER =10- 4)

Figure 6.10.b Figure 6.10(a) Shows the values of the PAPR and SNR at BER = for each of the RCF, Acompanding , and Hybird (RCF+A ) b) Shows the values of the CCDF of PAPR and SNR at BER = for each of the RCF, Acompanding , and Hybird (RCF+A )

6.2.2 RCF + : The following conclusion when from table A.14 and figure 6.11 comparing the proposed method with an OFDM system without PAPR reduction method:  At =5 and CR = 4, There is an improvement in PAPR by = (16.5081 dB), CCDF of PAPR = (6.7470 dB), and the SNR at BER( ) by = (0.8014 dB). This point was chosen because all the variables improved.  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at =10 and CR =3. The improvement in PAPR by = (18.8243 dB), and CCDF of PAPR = (7.8965 dB), while the SNR at BER( ) deteriorated by = (-1.4324 dB). 121

Simulation Results and Analysis of Hybrid PAPR techniques 



For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at =30 and CR =3. The improvement in PAPR by = (20.2789 dB), and CCDF of PAPR = (8.5620 dB), while the BER deteriorated by = (-3.5686 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at =70 and CR =3 . The improvement in PAPR by = (21.0934 dB), and CCDF of PAPR = (8.9200 dB), while the SNR at BER( ) deteriorated by = (-5.4399 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at = 220 and CR =3. The improvement in PAPR by = (22.2829 dB), and CCDF of PAPR = (9.2400 dB), while the SNR at BER( ) deteriorated by = (-7.8013 dB).

The following conclusion from table A.14 and figure 6.11 when comparing the proposed method with an OFDM system with companding PAPR reduction method:  The PAPR was improved and the least amount of improvement was when = 220 and CR = 4 and is equal to (7.1092 dB), While the vast amount of improvement is where = 80 and CR = 2 and is equal to (12.6705 dB).  Less the amount of improvement in CCDF of PAPR when = 220 and CR = 4 and is equal to (0.8325 dB), while the vast amount of improvement is where = 5 and CR = 1.5 and is equal to (4.1715 dB).  The SNR at BER( ) was improved when CR =4,3 . The vast amount of improvement is where = 50 and CR = 4 and is equal to (3.0554 dB), while Less the amount of improvement in SNR at BER( )when = 255 and CR = 3 and is equal to (1.468 dB)  The SNR at BER( ) was degraded when CR = 2. The least amount of degradation in SNR at BER( ) when =240 and is equal to (-0.985 dB). The largest amount of degradation is when = 20 and is equal to (-2.625 dB).  The SNR at BER( ) was degraded when CR = 1.5, when =5 the amount of degradation is equal to (-16.6637dB). The following conclusion from table A.14 and figure 6.11 when comparing the proposed method with an OFDM system with RCF method:  The PAPR was improved and the least amount of improvement was when = 5 and CR = 1.5 and is equal to (2.8281 dB), while the vast amount of improvement is where = 255 and CR = 4 and is equal to (10.4574 dB).  Less the amount of improvement in CCDF of PAPR when = 5 and CR = 2 and is equal to (1.4656 dB), while the vast amount of improvement is where = 255 and CR = 4 and is equal to (4.6677 dB).  The SNR at BER( ) was degraded, except when =5 and CR =1.5 the SNR at BER( ) maintains its value. The least amount of degradation in SNR at BER( ) when MU =5 and CR = 4 and is equal to (-1.8621 dB). The largest amount of degradation is when MU= 220 and CR =2 is equal to (-10.872 dB).

122

Simulation Results and Analysis of Hybrid PAPR techniques 30 original RCF (CR =4) RCF (CR =3) RCF (CR=2) MU RCF (CR =4) + MU RCF (CR =3) + MU RCF (CR=2) + MU

PAPR

SNR at (BER =10- 4)

Figure 6.11.a 11 original RCF (CR =4) RCF (CR =3) RCF (CR=2) MU RCF (CR =4) + MU RCF (CR =3) + MU RCF (CR=2) + MU

10 9

CCDF of PAPR

8 7 6 5 4 3 2 1

SNR at (BER =10- 4)

Figure 6.11.b Figure 6.11 ( a) Shows the values of the PAPR and SNR at BER = for each of the RCF, companding , and Hybird (RCF+ ). b) Shows the values of the CCDF of PAPR and SNR at BER = for each of the RCF, companding , and Hybird (RCF+ ).

6.2.3 RCF + RCT: The following conclusion from table A.15 and figure 6.12 when comparing the proposed method with an OFDM system without PAPR reduction method: ď&#x201A;ˇ At these values (R =0.9, 0.8, 0.7, 0.6 and CR = 4, 3) There are improved in PAPR, CCDF of PAPR and the BER dB). Point was chosen because all the variables improved. The best one improvement in PAPR and CCDF of PAPR is at R =0.6

123

Simulation Results and Analysis of Hybrid PAPR techniques



and CR =3. The improvement in PAPR by = (17.2514 dB), CCDF of PAPR = (7.0968 dB), and the SNR at BER( ) by = (0.5079 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at R =0.5 and CR =3. The improvement in PAPR by = (18.4159 dB), and CCDF of PAPR = (7.5330 dB), while the SNR at BER( ) deteriorated by = (-0.8186 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at R =0.4 and CR =3 . The improvement in PAPR by = (19.6235 dB), and CCDF of PAPR = (8.1034 dB), while the SNR at BER( ) deteriorated by = (-2.4115 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at R =0.3 and CR =3. The improvement in PAPR by = (20.9606 dB), and CCDF of PAPR = (8.7200 dB), while the SNR at BER( ) deteriorated by = (-5.0018 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at R= 0.2 and CR =3. The improvement in PAPR by = (22.3722 dB), and CCDF of PAPR = (9.3400 dB), while the SNR at BER( ) deteriorated by = (-8.5686 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at R =0.1 and CR =2. The improvement in PAPR by = (24.2450 dB), and CCDF of PAPR = (10.2400 dB), while the SNR at BER( ) deteriorated by = (-17.8776 dB).

The following conclusion from table A.15 and figure 6.12 when comparing the proposed method with an OFDM system with RCT PAPR reduction method:  The PAPR and the CCDF of PAPR were improved and the least amount of improvement was when R =0.1 and CR = 4 .The PAPR improvement is equal to (0.9227 dB) and the CCDF of PAPR improvement is equal to(0.363 dB), While the vast amount of improvement is where R =0.9 and CR = 1.5 and the PAPR improvement is equal to (13.9836 dB) and the CCDF of PAPR improvement is equal to (6.201dB )  The SNR at BER( ) was improved when CR =4,3 . The vast amount of improvement is where R =0.7 and CR = 4 and is equal to (3.0902 dB), while Less the amount of improvement in CCDF of PAPR when R =0.1 and CR = 3 and is equal to (1.8866 dB)  The SNR at BER( ) was degraded when CR = 2. The least amount of degradation in SNR at BER( ) when R =0.5 and is equal to (-0.7285 dB). The largest amount of degradation is when R =0.3 and is equal to (-1.475 dB).  The SNR at BER( ) was degraded when CR = 1.5, when R =0.9 the amount of degradation is equal to (-18.3235dB). The following conclusion from table A.15 and figure 6.12 when comparing the proposed method with an OFDM system with RCF method:  The PAPR was improved and the least amount of improvement was when R =0.9 and CR = 1.5 and is equal to (0.5034 dB), while the vast amount of improvement is where R =0.1and CR = 4 and is equal to (12.5433 dB).

124

Simulation Results and Analysis of Hybrid PAPR techniques  

Less the amount of improvement in CCDF of PAPR when R = 0.9 and CR =2 and is equal to (0.1506 dB), while the vast amount of improvement is where R= 0.1 and CR = 4 and is equal to (5.5127 dB). The SNR at BER( ) was degraded, except when R =0.1 and CR =1.5 the SNR at BER( ) maintains its value. The least amount of degradation in CCDF of PAPR when R =0.9 and CR = 4 and is equal to (-0.1291 dB). The largest amount of degradation is when R= 0.1 and CR =2 is equal to (-17.181 dB).

30 original RCF (CR=4) RCF (CR=3) RCF (CR=2) Roots RCF (CR=4) + Rooting RCF (CR=3) + Rooting RCF (CR=2) + Rooting

PAPR

SNR at (BER =10- 4)

figure 6.12.a 12 original RCF (CR=4) RCF (CR=3) RCF (CR=2) Roots RCF (CR=4) + Rooting RCF (CR=3) + Rooting RCF (CR=2) + Rooting

CCDF of PAPR

SNR at (BER =10 4)

Figure 6.12. b figure 6.12 (a) Shows the values of the PAPR and SNR at BER = for each of the RCF, RCT , and Hybird (RCF+RCT) b) Shows the values of the CCDF of PAPR and SNR at BER = for each of the RCF, RCT, and Hybird (RCF+ RCT). 125

Simulation Results and Analysis of Hybrid PAPR techniques 6.2.4 RCF + AEXP: The following conclusion from table A.16 and figure 6.13 when comparing the proposed method with an OFDM system without PAPR reduction method:  At these values ((d =2-1by .1 every time and CR = 4) and when (d =1.8, 1.5 and CR =3) There are improved in PAPR, CCDF of PAPR and SNR at BER( )). The best one improvement in PAPR and CCDF of PAPR is at d = 1 and CR =4. The improvement in PAPR by = (18.7316 dB) ,CCDF of PAPR = (7.7135 dB), and the SNR at BER( ) by = ( 0.2467 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at d =0.9 and CR =3. The improvement in PAPR by = (19.6985 dB), and CCDF of PAPR = (8.1400 dB), while the SNR at BER( ) deteriorated by = (-1.5186 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at d =0.7 and CR =3. The improvement in PAPR by = (20.7361 dB), and CCDF of PAPR = (8.5535 dB), while the SNR at BER( ) deteriorated by = (-3.3069 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at d =0.6 and CR =4. The improvement in PAPR by = (21.0273 dB), and CCDF of PAPR = (8.6875 dB), while the SNR at BER( ) deteriorated by = (-4.7686 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at d =0.5 and CR =4. The improvement in PAPR by = (21.6682 dB), and CCDF of PAPR = (8.9810 dB), while the SNR at BER( ) deteriorated by = (-17.7786 dB). The following conclusion from table A.16 and figure 6.13 when comparing the proposed method with an OFDM system with AEXP companding PAPR reduction method:  The PAPR was improved except when d =1.6 and CR =3 the PAPR was degraded by (-1.6855dB) .the least amount of improvement was when d =0.4 and CR = 4 and is equal to (0.1356 dB), while the vast amount of improvement is where d =2 and CR = 1.5 and is equal to (4.7537 dB).  Less the amount of improvement in CCDF of PAPR when d =0.4 and CR = 4 and is equal to (0.14 dB), while the vast amount of improvement is where d =2 and CR = 1.5 and is equal to (1.7799 dB).  The SNR at BER( ) was improved when CR =4,3 except when (d =0.4 and CR =4,3) and when (d =0.5 and CR =3) the SNR at BER( ) maintains its value. The vast amount of improvement is where d =0.7 and CR = 4 and is equal to (15.5893 dB), while Less the amount of improvement in SNR at BER( ) when d=0.1 and CR = 4 and is equal to (0.79 dB)  The SNR at BER( ) was degraded when CR = 2 except when (d =0.6, 0.5, 0.4) the SNR at BER( ) maintains its value. The least amount of degradation in SNR at BER( ) when d =0.7 and is equal to (-0.9 dB). The largest amount of degradation is when R =1.1 and is equal to (-7.65 dB).  The SNR at BER( ) was degraded when CR = 1.5, when d =2. the amount of degradation is equal to (-15.27 dB).

126

Simulation Results and Analysis of Hybrid PAPR techniques

30 original RCF (CR=4) RCF (CR=3) RCF (CR=2) AEXP RCF (CR=4) + AEXP RCF(CR=3) +AEXP RCF(CR=2) +AEXP

PAPR

SNR at (BER =10- 4)

Figure 6.13.a 11 original RCF (CR=4) RCF (CR=3) RCF (CR=2) AEXP RCF (CR=4) + AEXP RCF (CR=3) + AEXP RCF(CR=2) + AEXP

10 9 8

CCDF of PAPR



The following conclusion from table A.16 and figure 6.13 when comparing the proposed method with an OFDM system with RCF method:  The PAPR was improved and the least amount of improvement was when d =2 and CR = 1.5 and is equal to (0.0555 dB), while the vast amount of improvement is where d =0.4 and CR = 4 and is equal to (11.2249 dB).  Less the amount of improvement in CCDF of PAPR when d =2 and CR =2 and is equal to (0.3051 dB), while the vast amount of improvement is where d =0.4 and CR = 4 and is equal to (4.8177 dB). The SNR at BER( ) was degraded, except when d =2 and CR =1.5 the SNR at BER( ) maintains its value. The least amount of degradation in SNR at BER( ) when d =1.3 and CR = 4 and is equal to (-1.6321 dB). The largest amount of degradation is when d =0.4 and CR =4 is equal to (-21.2321 dB).

7 6 5 4 3 2 1

SNR at (BER =10- 4)

Figure 6.13.b Figure 6.13 (a) Shows the values of the PAPR and SNR at BER = RCF, AEXP companding , and Hybird (RCF+AEXP) b) Shows the values of the CCDF of PAPR and SNR at BER = RCF, AEXP companding , and Hybird (RCF+ AEXP) 127

for each of the for each of the

Simulation Results and Analysis of Hybrid PAPR techniques 6.2.5 RCF + cos : The following conclusion from table A.17 and figure 6.14 when comparing the proposed method with an OFDM system without PAPR reduction method:  At these values ((y =1,0.9, 0.8,0.7 and CR = 4,3) and when ( y =0.6 and CR =4) There are improved in PAPR, CCDF of PAPR and BER ). The best one improvement in PAPR and CCDF of PAPR is at y =.7 and CR =3. The improvement in PAPR by = (17.1463 dB),CCDF of PAPR = (7.0651 dB), and the BER by = ( 0.5250 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at y=0.5 and CR =4. The improvement in PAPR by = (18.2929 dB), and CCDF of PAPR = (7.5582 dB), while the SNR at BER( ) deteriorated by = (-1.1636 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at y = 0.6 and CR =2. The improvement in PAPR by = (19.7667 dB), and CCDF of PAPR = (8.1941 dB), while the SNR at BER( ) deteriorated by = (-3.4776 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at y = 0.4 and CR =2. The improvement in PAPR by = (21.3061 dB), and CCDF of PAPR = (8.8769 dB), while the SNR at BER( ) deteriorated by = (-5.3379 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at y = 0.3 and CR =3. The improvement in PAPR by = (21.4320 dB), and CCDF of PAPR = (8.9568 dB), while the SNR at BER( ) deteriorated by = (-5.7164 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at y = 0.2 and CR =3. The improvement in PAPR by = (22.6995 dB), and CCDF of PAPR = (9.5194 dB), while the SNR at BER( ) deteriorated by = (-15.6263 dB). The following conclusion from table A.17 and figure 6.14 when comparing the proposed method with an OFDM system with cos companding PAPR reduction method:  The PAPR was improved and the least amount of improvement was when y =0.3 and CR = 4 and is equal to (0.8658 dB), While the vast amount of improvement is where y =1 and CR = 1.5 and is equal to (8.2456 dB).  The CCDF of PAPR was improved and the least amount of improvement is where y =0.3 and CR = 4 and is equal to (0.4597 dB). , while the vast amount of improvement is where y =1 and CR = 1.5 and is equal to (3.4937 dB).  The SNR at BER( ) was improved when CR =4, 3. The vast amount of improvement when y=0.3 and CR = 4 and is equal to (4.4287 dB), while Less the amount of improvement in BER when y = 1 and CR = 3 and is equal to (2.061dB)  The SNR at BER( ) was degraded when CR = 2. The least amount of degradation in SNR at BER( ) when y =0.6 and is equal to (-0.3705 dB). The largest amount of degradation is when y =0.3 and is equal to (-8.3 dB).  The SNR at BER( ) was degraded when CR = 1.5, when y =1 the amount of degradation is equal to (-18.2000 dB).

128

Simulation Results and Analysis of Hybrid PAPR techniques The following conclusion from table A.17 and figure 6.14 when comparing the proposed method with an OFDM system with RCF method:  The PAPR was improved and the least amount of improvement was when y =1 and CR = 1.5 and is equal to (1.0679 dB), While the vast amount of improvement is where y =0.1 and CR = 4 and is equal to (12.7916 dB).  Less the amount of improvement in CCDF of PAPR when y=1 and CR = 1.5 and is equal to (0.2694 dB), while the vast amount of improvement is where y=0.1 and CR = 4 and is equal to (5.6361 dB).  The SNR at BER( ) was degraded, except when y =1 and CR =1.5 the BER maintains its value. The least amount of degradation in SNR at BER( ) when y =1 and CR = 4 and is equal to (-0.3076 dB). The largest amount of degradation is when y =0.2, 0.1 and CR =4 is equal to (21.2321 dB). 30 original RCF (CR=4) RCF (CR=3) RCF (CR=2) cos RCF (CR=4) + cos RCF (CR=3) +cos RCF (CR=2) + cos

PAPR

SNR at (BER =10- 4)

Figure 6.14.a 11 original RCF (CR=4) RCF (CR=3) RCF (CR=2) cos RCF (CR=4) + cos RCF (CR=3) + cos RCF (CR=2) + cos

10 9

CCDF of PAPR

8 7 6 5 4 3 2 1

SNR at (BER =10- 4)

Figure 6.14.b Figure 6.14 (a) Shows the values of the PAPR and SNR at BER = for each of the RCF, cos companding , and Hybird (RCF+cos) (b)Shows the values of the CCDF of PAPR and SNR at BER = for each of the RCF, cos companding , and Hybird (RCF+ cos) 129

Simulation Results and Analysis of Hybrid PAPR techniques 6.2.6 RCF + NERF : The following conclusion from table A.18 and figure 6.15 when comparing the proposed method with an OFDM system without PAPR reduction method: At these values (CR = 4, 3) and There are improved in PAPR, CCDF of PAPR and BER). The best one improvement in PAPR and CCDF of PAPR is at CR =3. The improvement in PAPR by = (17.0615 dB), CCDF of PAPR = (7.2730 dB), and the SNR at BER( ) by = (0.5314 dB). 26 original RCF (CR=4) RCF (CR=3) RCF (CR=2) RCF (CR=1.5) NERF RCF (CR=4) + RCF (CR=3) + RCF (CR=2) + RCF (CR=1.5)

24 22 20

PAPR

18 16 14

NERF NERF NERF + NERF

12 10 8 6

SNR at (BER =10- 4)

Figure 6.15.a 11 original RCF (CR=4) RCF (CR=3) RCF (CR=2) RCF (CR=1.5) NERF RCF (CR=4) + NERF RCF (CR=3) + NERF RCF (CR=2) + NERF RCF (CR=1.5) + NERF

10 9

CCDF of PAPR

8 7 6 5 4 3 2

SNR at (BER =10 4)

Figure 6.15.b Figure 6.15 (a) Shows the values of the PAPR and SNR at BER = for each of the RCF, cos companding , and Hybird (RCF+NERF) b) Shows the values of the CCDF of PAPR and SNR at BER = for each of the RCF, cos companding , and Hybird (RCF+ NERF) 130

Simulation Results and Analysis of Hybrid PAPR techniques 6.2.7 RCF + tanhR : The following conclusion from table A.19 when comparing the proposed method with an OFDM system without PAPR reduction method:  At these values (k =5,10,15 and y =1 ,0.8 for CR = 4,3) and when (k =20 and y =1 ,0.8 for CR =3) There are improved in PAPR, CCDF of PAPR and BER ). The best one improvement in PAPR and CCDF of PAPR is at k=20 y = .8 for CR =4 . The improvement in PAPR by = (18.6958 dB), CCDF of PAPR = (7.5530 dB), and the SNR at BER( ) by = ( 0.1527dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k=10, y=.5 and CR =4 . The improvement in PAPR by = (19.3352 dB), and CCDF of PAPR = (7.9060 dB), while the SNR at BER( ) deteriorated by = (-1.2824 dB). The best one improvement in CCDF of PAPR is at k=5 , y=.5 and CR =3 . The improvement in PAPR by = (19.3226 dB), and CCDF of PAPR = (7.9160 dB), while the SNR at BER( ) deteriorated by = (--1.4569dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k =20, y=0.5 and CR =3. The improvement in PAPR by = (21.1850 dB), and CCDF of PAPR = (8.7750 dB), while the SNR at BER( ) deteriorated by = (-3.4943 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k =30, y=0.5 and CR =3. The improvement in PAPR by = (21.9924 dB), and CCDF of PAPR = (9.0990 dB), while the SNR at BER( ) deteriorated by = (-5.1066 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k=40, y =0.5and CR =3. The improvement in PAPR by = (22.5983 dB), and CCDF of PAPR = (9.4120 dB), while the SNR at BER( ) deteriorated by = (-8.2502 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k= 40, y =0.2 and CR =3.The improvement in PAPR by = (23.9630 dB), and CCDF of PAPR = (10.0640 dB), while the SNR at BER( ) deteriorated by = (-13.1497 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k = 20, y=.2 and CR =2. The improvement in PAPR by = (24.0164 dB), and CCDF of PAPR = (10.1040 dB), while the SNR at BER( ) deteriorated by = (-18.1686 dB). The following conclusion from table A.19 when comparing the proposed method with an OFDM system with tanhR companding PAPR reduction method:  The PAPR was degraded at (k =15, 20, y =1,0.8 and CR = 4, 3, 2), (k= 10, y =1,0.8 and CR=4,3), (k =15 ,y =0.5 ,and CR =4) , and when (k =20 ,y =0.5 ,and CR =4,3). The least amount of degradation in PAPR when k=10, y=0.8 and CR = 3 and is equal to (-0.5458 dB). The largest amount of degradation is when k =15, y = 1 and CR =4 is equal to (-4.717 dB).  The CCDF of PAPR was degraded at [for CR = 4, 3, 2 at (k =20, y =1,0.8,0.5 ) and (k =15, y =1,0.8) ], [for CR =4,3 at (k =15 ,y =0.5 ),and((k= 10, y =1,0.8 and CR=4,3)], and when [k =10 ,y =0.5 ,and CR =4] The least amount of degradation in CCDF of PAPR when k=20, y=0.5 and CR = 2 and is equal to (-0.0085 dB). The largest amount of degradation is when k =20, y = 1 and CR =4 is equal to (2.0457 dB). 131

Simulation Results and Analysis of Hybrid PAPR techniques 



 

Except the points already mentioned, the PAPR was improved and the least amount of improvement was when k =20, y =0.5 and CR = 2 and is equal to (0.0043 dB), while the vast amount of improvement is where k =5, y =1 and CR = 1.5 and is equal to (8.0365dB). Except the points already mentioned, the CCDF of PAPR was improved and Less the amount of improvement in CCDF of PAPR when k =10, y =1 and CR = 2 and is equal to (0.3585 dB), While the vast amount of improvement is where k =5, y =1 and CR = 1.5 and is equal to (3.6052 dB). The SNR at BER( ) was degraded when CR =2 at (k =5, y =1,0.8,0.5,0.2 ), ( y =0.2, k =10,15,20), and when (k =10 ,y =0.5). The least amount of degradation in SNR at BER( ) when k=5, y=0.8 and is equal to (-0.11 dB). The largest amount of degradation is when k =10, y = 0.2 is equal to (-2.4641 dB). The SNR at BER( ) was degraded when CR = 1.5, when k =5, y =1 .The amount of degradation is equal to (-18.0900 dB). Except the points already mentioned, the SNR at BER( ) was improved and the vast amount of improvement is where k =15, y =1 and CR = 4 and is equal to (19.803 dB), while Less the amount of improvement in SNR at BER( ) when k =5, y=0.8 and CR = 2 and is equal to (1.1217 dB).

The following conclusion from table A.19 when comparing the proposed method with an OFDM system with RCF method:  The PAPR was improved and the least amount of improvement was when k=5 ,y =1 and CR = 1.5 and is equal to (0.0685 dB), While the vast amount of improvement is where k =40, y =0.2 and CR = 4 and is equal to (12.6433 dB).  The CCDF of PAPR was improved, except when k =5 ,y =1 and CR =2 the CCDF of PAPR was degraded by (-0.0264 dB). Less the amount of improvement in CCDF of PAPR when k =5, y= 1 and CR = 3 and is equal to (0.027 dB), While the vast amount of improvement is where k =40, y =0.2 and CR = 4 and is equal to (5.5377 dB).  The SNR at BER( ) was degraded, except when k =5 ,y =1 and CR =1.5 the SNR at BER( ) maintains its value. The least amount of degradation in SNR at BER( ) when k =5, y =1 and CR = 3 and is equal to (-0.1386 dB). The largest amount of degradation is when k =50, y =1, 0.8, 0.2 and CR =3 is equal to (20.5860 dB).

6.2.8 RCF +logR : The following conclusion from table A.20 when comparing the proposed method with an OFDM system without PAPR reduction method:  At these values (k =5,10,20 and y =1 ,0.8 for CR = 4,3) and when (k =30,40,50 ,70 and y =1 ,0.8 for CR =4) and finally, when (k =30,40 and y =1 for CR =3) There are improved in PAPR, CCDF of PAPR and BER ). The best one improvement in PAPR and CCDF of PAPR is at k=70, y = .8 for CR =4 . The improvement in PAPR by = (18.0511 dB) ,CCDF of PAPR = (7.3815 dB), and the SNR at BER( ) by = (0.2025 dB).  For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k=5, y =0.5 and CR =3. The improvement in PAPR by = (19.4862 dB), and CCDF of PAPR = (8.0155 dB), while the SNR at BER( ) deteriorated by = (-1.5345 dB). 132

Simulation Results and Analysis of Hybrid PAPR techniques 



For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k=90, y=0.5 and CR =3. The improvement in PAPR by = (20.9335 dB), and CCDF of PAPR = (8.6886 dB), while the SNR at BER( ) deteriorated by = (-3.1466 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k=50 ,y =0.5and CR =2. The improvement in PAPR by = (21.5265 dB), and CCDF of PAPR = (9.0164 dB), while the SNR at BER( ) deteriorated by = (-8.2869 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k=70 ,y =0.2and CR =3. The improvement in PAPR by = (23.3205dB), and CCDF of PAPR = (9.7780 dB), while the SNR at BER( ) deteriorated by = (-10.1498 dB). For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k=.9, y =0.2 and CR =2. The improvement in PAPR by = (23.7727 dB), and CCDF of PAPR = (10.0043 dB), while the SNR at BER( ) deteriorated by = (-16.1064 dB).

The following conclusion from table A.20 when comparing the proposed method with an OFDM system with logR companding PAPR reduction method:  The PAPR was improved and the least amount of improvement was when k =5, y =1 and CR = 1.5 and is equal to (0.4088 dB), While the vast amount of improvement is where k =90, y =0.2 and CR = 4 and is equal to (11.9078 dB).  The CCDF of PAPR was improved and the least amount of improvement was when k =5, y =1 and CR = 2 and is equal to (0.2136 dB), while the vast amount of improvement is where k =90, y =0.2 and CR = 4 and is equal to (5.2067 dB).  The SNR at BER( ) was degraded when k =5 ,y =1 and CR =1.5 the BER degraded by (-17.9300 dB).  The SNR at BER( ) was improved when CR =4, 3. The least amount of improvement in BER when k =5, y =1 and CR = 3 and is equal to (2.0783 dB). The largest amount of improvement is when k =90, y =1 and CR =4 is equal to (18.215 dB).  The SNR at BER( ) was degraded when CR = 2, except when (k =30,90 and y =1) and CR =1.5 the SNR at BER( ) was improved by (0.5122 - 0.825 dB). The least amount of degradation in SNR at BER( ) when k=10, y =1 and is equal to (-0.1639 dB). The largest amount of degradation is when k=90, y =0.8 and is equal to (-4.0838 dB). The following conclusion from table A.20 when comparing the proposed method with an OFDM system with RCF method:  The PAPR was improved and the least amount of improvement was when k=5, y =1 and CR = 1.5 and is equal to (0.4088 dB), while the vast amount of improvement is where k =90, y =0.2 and CR = 4 and is equal to (11.9078 dB).  Less the amount of improvement in CCDF of PAPR when k =5, y= 1 and CR = 2 and is equal to (0.2136 dB), While the vast amount of improvement is where k =90, y =0.2 and CR = 4 and is equal to (5.2067 dB).  The SNR at BER( ) was degraded, except when k =5 ,y =1 and CR =1.5 the SNR at BER( ) maintains its value. The least amount of degradation in SNR at BER( ) when k =5, y =1 and CR = 3 and is equal to (-0.1171 dB). The largest amount of degradation is when k =90, y =1 and CR =2 is equal to (-17.047 dB). 133

Simulation Results and Analysis of Hybrid PAPR techniques 6.3 Hybrid RFC with companding: The process of companding enlarges the amplitudes of the small signals, while the peaks remain unchanged. Therefore, the average power is increased and thus the Peak-to Average Power Ratio (PAPR) of the OFDM systems can be reduced, which in turn helps in increasing the efficiency of the power amplifiers and also reduces the complexity of the Analog-to-Digital Converter (ADC) and Digital-to-Analog Converter (DAC) [53]. As we demonstrated earlier RFC batter than RCF, because it reduces the impact of the filter on the PAPR and also when increasing I the BER improves. The previous method hybrid technique depend on RCF and companding give a good result and as we demonstrated earlier RFC batter than RCF, because it reduces the impact of the filter on the PAPR and also the BER improves. When increasing I. So we proposed a new hybrid method based on RFC and companding. The OFDM system model with RFC and companding was shown in the Figure 6.16. This hybrid technique shows good results better than the previous method, because of first RFC improve the PAPR and the BER more than RCF and then companding more reduces the amount of the PAPR. RFC is used with the following specifications (I =4, CR =4, 3, 2) as for the companding has been using all kinds of previous companding.

Add CP

P/ S

Companding

RFC

IDFT OR IFFT

S/ P

+pilot symbol

Signal mapper

I / P

D / A

Multipath Fading Ch. & noise

Remove CP

S/ P

De-Companding

DFT OR FFT

Remove +pilot symbol

Signal demapper

One Tap Equalizer And P/S

O / P

A / D

Figure 6.16 the OFDM system model with RFC + companding .

6.3.1 RFC + A companding: The following conclusion from table A.21 and figure 6.17 when comparing the proposed method with an OFDM system without PAPR reduction method: ď&#x201A;ˇ At these values (A =5, 10 for CR = 4, 3) and There are improved in PAPR, CCDF of PAPR and BER). The best improvement in PAPR and CCDF of PAPR is at A=10 for CR =3. The improvement in PAPR by = (19.8043 dB), CCDF of PAPR = (8.4633 dB), and the SNR at BER( ) by = ( 0.5300 dB). ď&#x201A;ˇ For SNR at BER( ) 134

Simulation Results and Analysis of Hybrid PAPR techniques The best improvement in PAPR and CCDF of PAPR is at A =20 and CR =3. The improvement in PAPR by = (20.6801 dB), and CCDF of PAPR = (8.8636 dB), while the SNR at BER( ) deteriorated by = (-1.5522 dB). The best improvement in CCDF of PAPR is at A =20 and CR =2. The improvement in PAPR by = (20.6641 dB), and CCDF of PAPR = (8.8670 dB), while the SNR at BER( ) deteriorated by = (-1.4438 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at A = 80 and CR =4 . The improvement in PAPR by = (21.5086 dB), and CCDF of PAPR = (9.0404 dB), while the SNR at BER( ) deteriorated by = (-3.5335 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at A = 100 and CR =2. The improvement in PAPR by = (21.9802 dB), and CCDF of PAPR = (9.4272 dB), while the SNR at BER( ) deteriorated by = (-4.4427 dB). The best improvement in CCDF of PAPR is at A = 120 and CR =3. The improvement in PAPR by = (21.9362 dB), and CCDF of PAPR = (9.4719 dB), while the SNR at BER( ) deteriorated by = (-5.3212 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at A=80 and CR =2. The improvement in PAPR by = (22.2111 dB), and CCDF of PAPR = (9.8161dB), while the SNR at BER( ) deteriorated by = (-8.2440 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at A =90 and CR =2. The improvement in PAPR by = (21.0327 dB), and CCDF of PAPR = (9.4036 dB), while the SNR at BER( ) deteriorated by = (-18.1686 dB). The following conclusion from table A.21 and figure 6.17 when comparing the proposed method with an OFDM system with A companding PAPR reduction method:  The PAPR was improved and the least amount of improvement was when A = 40 and CR = 4 and is equal to (7.1833 dB), While the vast amount of improvement is where A = 5 and CR = 1.5 and is equal to (14.3373 dB).  The least amount of improvement in CCDF of PAPR when A = 100 and CR = 4 and is equal to (0.8784 dB), while the vast amount of improvement is where A = 5 and CR = 1.5 and is equal to (5.2036 dB).  The SNR at BER( ) was improved when CR =4,3,2 . The vast amount of improvement is where A = 70 and CR = 4 and is equal to (6.3199 dB), while The least amount of improvement in SNR at BER( ) when A = 5 and CR = 2 and is equal to (0.5206 dB)  The SNR at BER( ) was degraded when CR = 1.5, when A =5. The amount of degradation is equal to (-16 dB). The following conclusion from table A.21 and figure 6.17 when comparing the proposed method with an OFDM system with RCF method:  The PAPR and the CCDF of PAPR were improved and the least amount of improvement was when A =5 and CR = 1.5 .The PAPR improvement is equal to (1.8221 dB) and the CCDF of PAPR improvement is equal to(0.9794 dB), while the vast amount of improvement is where A = 200 and CR = 4 and the PAPR improvement is equal to (10.6436 dB) and the CCDF of PAPR improvement is equal to (4.5529 dB ) 135

Simulation Results and Analysis of Hybrid PAPR techniques ď&#x201A;ˇ

The SNR at BER( ) was degraded. The least amount of degradation in SNR at BER( ) when A =5 and CR = 1.5 and is equal to (-1.38 dB). The largest amount of degradation is when A= 80 and CR =2 is equal to (-11.2484 dB).

30 original RFC (CR=4) RFC (CR=3) RFC (CR=2) A RFC (CR=4) + A RFC (CR=3) + A RFC (CR=2) + A

PAPR

SNR at (BER =10-4)

Figure 6.17.a 12 original RFC (CR=4) RFC (CR=3) RFC (CR=2) A RFC (CR=4) + A RFC (CR=3) + A RFC (CR=2) + A

CCDF of PAPR

SNR at (BER =10 4)

Figure 6.17.b Figure 6.17 (a) Shows the values of the PAPR and SNR at BER = for each of the RFC, companding , and Hybird (RFC+ ). b) Shows the values of the CCDF of PAPR and SNR at BER = for each of the RCF, companding , and Hybird (RFC+ ). 136

Simulation Results and Analysis of Hybrid PAPR techniques 6.3.2 RFC +

companding:

The following conclusion from table A.22 and figure 6.18 when comparing the proposed method with an OFDM system without PAPR reduction method:  At these values ( =5,10,20 for CR = 4,3) and when ( =30 ,40 for CR = 4) There are improved in PAPR, CCDF of PAPR and SNR at BER( )). The best improvement in PAPR is at =40 for CR =4. The improvement in PAPR by = (20.0157 dB) ,CCDF of PAPR = (8.3564 dB), and the SNR at BER( ) by = ( 0.0116dB). The best improvement in CCDF of PAPR is at =20 for CR =3 . The improvement in PAPR by = (19.9252 dB) ,CCDF of PAPR = (8.5044 dB), and the SNR at BER( ) by = ( 0.2052 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at = 40 and CR =3. The improvement in PAPR by = (20.6523 dB), and CCDF of PAPR = (8.8336 dB), while the SNR at BER( ) deteriorated by = (-1.2110 dB).  For SNR at BER( ) The best improvement in PAPR is at =180 and CR =4. The improvement in PAPR by = (21.4145 dB) , and CCDF of PAPR = (9.0247 dB), while the SNR at BER( ) deteriorated by = (-3.4087 dB). The best improvement in CCDF of PAPR is at =80 and CR =3. The improvement in PAPR by = (21.1482 dB) , and CCDF of PAPR = (9.0856 dB), while the SNR at BER( ) deteriorated by = (-2.7686 dB).  For SNR at BER( ) The best improvement in PAPR is at =240 and CR =4. The improvement in PAPR by = (21.605 dB), and CCDF of PAPR = (9.0964 dB), while the SNR at BER( ) deteriorated by = (-3.9316 dB). The best improvement in CCDF of PAPR is at =255and CR =4. The improvement in PAPR by = (21.1370 dB),and CCDF of PAPR = (9.1356 dB), while the SNR at BER( ) deteriorated by = (-3.8762 dB). A.22 and figure 6.18 when comparing the proposed method with an OFDM system with companding PAPR reduction method:  The PAPR was improved and the least amount of improvement was when = 120 and CR = 4 and is equal to (6.3834 dB), While the vast amount of improvement is where = 80 and CR = 3 and is equal to (12.0968 dB).  The least amount of improvement in CCDF of PAPR when = 220 and CR = 4 and is equal to (0.9042 dB), while the vast amount of improvement is where = 5 and CR = 3 and is equal to (3.222 dB).  The SNR at BER( ) was improved when CR =4,3. The vast amount of improvement is where = 50 and CR = 4 and is equal to (6.4429 dB), while The least amount of improvement in SNR at BER( ) when = 5 and CR = 3 and is equal to (4.8568 dB)  The SNR at BER( ) was improved when CR =2. The vast amount of improvement is where = 220 and is equal to (2.4273 dB), while The least amount of improvement in CCDF of PAPR when = 30 and is equal to (1.2065 dB)  The SNR at BER( ) was degraded when CR = 1.5, when =5 .the amount of degradation is equal to (-16.6637 dB). 137

Simulation Results and Analysis of Hybrid PAPR techniques The following conclusion from table A.22 and figure 6.18 when comparing the proposed method with an OFDM system with RFC method:  The PAPR was improved and the least amount of improvement was when = 5 and CR = 1.5 and is equal to (1.4566 dB), While the vast amount of improvement is where = 240 and CR = 4 and is equal to (10.1873 dB).  The least amount of improvement in CCDF of PAPR when = 5 and CR = 1.5 and is equal to (0.8031 dB), while the vast amount of improvement is where = 255 and CR = 4 and is equal to (4.3606 dB).  The SNR at BER( ) was degraded the least amount of degradation in SNR at BER( ) when =5 and CR = 1.5 and is equal to (-1.78 dB). The largest amount of degradation is when = 255 and CR =2 is equal to (-10.882 dB). 30 original RFC (CR=4) RFC (CR=3) RFC (CR=2) MU RFC (CR=4) + MU RFC (CR=3) + MU RFC (CR=2) + MU

PAPR

SNR at (BER =10-4)

Figure 6.18.a 12 original RFC (CR=4) RFC (CR=3) RFC (CR=2) MU RFC (CR=4) + MU RFC (CR=3) + MU RFC (CR=2) + MU

CCDF of PAPR

SNR at (BER =10- 4)

Figure 6.18.b Figure 6.18 (a) Shows the values of the PAPR and SNR at BER = for each of the RFC, companding , and Hybird (RFC+ ) b) Shows the values of the CCDF of

138

Simulation Results and Analysis of Hybrid PAPR techniques PAPR and SNR at BER = (RFC+ )

for each of the RFC,

companding , and Hybird

6.3.3 RFC + RCT: The following conclusion from table A.23 and figure 6.19 when comparing the proposed method with an OFDM system without PAPR reduction method:  At these values (R =0.9 ,0.8 ,0.7,0.6,0.5 and CR = 4,3,2) and when (R =0.4 and CR = 4,3) There are improved in PAPR, CCDF of PAPR and the SNR at BER( ) dB). These points were chosen because all the variables improved. The best improvement in PAPR and CCDF of PAPR is at R =0.5 and CR =2. The improvement in PAPR by = (20.5192 dB) ,CCDF of PAPR = (8.7312 dB), and the SNR at BER( ) by = ( 0.0156 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at R = 0.4 and CR = 2 . The improvement in PAPR by = (21.3277 dB), and CCDF of PAPR = (9.0651 dB), while the SNR at BER( ) deteriorated by = (-2.3554 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at R = 0.2 and CR =3 . The improvement in PAPR by = (22.4752 dB), and CCDF of PAPR = (9.4425 dB), while the SNR at BER( ) deteriorated by = (-5.4198 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at R= 0.2 and CR =2. The improvement in PAPR by = (23.2521 dB), and CCDF of PAPR = (9.8332 dB), while the SNR at BER( ) deteriorated by = (-7.4169 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at R= 0.1 and CR =3. The improvement in PAPR by = (23.9486 dB), and CCDF of PAPR = (10.1129 dB), while the SNR at BER( ) deteriorated by = (-12.0518 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at R = 0.1and CR =2 . The improvement in PAPR by = (24.3546 dB), and CCDF of PAPR = (10.3164 dB), while the SNR at BER( ) deteriorated by = (-14.1974 dB). The following conclusion from table A.23 and figure 6.19 when comparing the proposed method with an OFDM system with RCT companding PAPR reduction method:  The PAPR and the CCDF of PAPR were improved and the least amount of improvement was when R =0.1 and CR = 4 .The PAPR improvement is equal to (0.9577 dB) and the CCDF of PAPR improvement is equal to(0.3934 dB), While the vast amount of improvement is where R =0.9 and CR = 1.5 and the PAPR improvement is equal to (15.9263 dB) and the CCDF of PAPR improvement is equal to (7.3321 dB )  The SNR at BER( ) was improved when CR =4,3,2 . The vast amount of improvement is where R =0.6 and CR = 4 and is equal to (6.298 dB), while The least amount of improvement in SNR at BER( ) when R =0.4 and CR = 2 and is equal to (2.3582 dB)

139

Simulation Results and Analysis of Hybrid PAPR techniques 

The SNR at BER( ) was degraded when CR = 1.5, when R =0.9 the amount of degradation is equal to (-17.0065 dB). 30 original RFC (CR=4) RFC (CR=3) RFC (CR=2) Roots RFC (CR=4) + Rooting RFC (CR=3) + Rooting RFC (CR=2) + Rooting

PAPR

SNR at (BER =10- 4)

Figure 6.19.a 12 original RFC (CR=4) RFC (CR=3) RFC (CR=2) Roots RFC (CR=4) + Rooting RFC (CR=3) + Rooting RFC (CR=2) + Rooting

CCDF of PAPR

SNR at (BER =10- 4)

Figure 6.19.b figure 6.19 (a) Shows the values of the PAPR and SNR at BER = for each of the RFC, RCT, and Hybird (RFC+ RCT) b) Shows the values of the CCDF of PAPR and SNR at BER = for each of the RFC, RCT, and Hybird (RFC+ RCT). The following conclusion from table A.23 and figure 6.19 when comparing the proposed method with an OFDM system with RFC method:  The PAPR was improved and the least amount of improvement was when R =0.9 and CR = 1.5 and is equal to (0.4541 dB), while the vast amount of improvement is where R =0.1and CR = 4 and is equal to (12.2689 dB).  The least amount of improvement in CCDF of PAPR when R = 0.9 and CR =2 and is equal to (0.1621 dB), while the vast amount of improvement is where R= 0.1 and CR = 4 and is equal to (5.1904 dB).  The SNR at BER( ) was degraded the least amount of degradation in SNR at BER( ) when R =0.9 and CR = 4 and is equal to (-0.2943 dB). The largest amount of degradation is when R= 0.1 and CR =3 is equal to (-17.4447 dB). 140

Simulation Results and Analysis of Hybrid PAPR techniques 6.3.4 RFC + AEXP: The following conclusion from table A.24 and figure 6.20 when comparing the proposed method with an OFDM system without PAPR reduction method:  At these values ((d =2-0.7 by .1 every time and CR = 4,3) and when (d =2 and CR =2) and finally when(d =0.6 and CR =2)There are improved in PAPR, CCDF of PAPR and SNR at BER( )). The best improvement in PAPR and CCDF of PAPR is at d = 0.6 and CR =4. The improvement in PAPR by = (21.0509dB) ,CCDF of PAPR = (8.7178 dB), and the SNR at BER( ) by = ( 0.0116 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at d =0.6 and CR = 3. The improvement in PAPR by = (21.3545 dB) , and CCDF of PAPR = (8.8589 dB), while the SNR at BER( ) deteriorated by = (-1.0503 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at d= 0.5 and CR =4 . The improvement in PAPR by = (21.6833 dB), and CCDF of PAPR = (9.0020 dB), while the SNR at BER( ) deteriorated by= (-2.9912 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at d = 0.7 and CR =2 . The improvement in PAPR by = (21.7102 dB), and CCDF of PAPR = (9.0772 dB), while the SNR at BER( ) deteriorated by = (-5.6201 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at d =0.6 and CR =2. The improvement in PAPR by = (22.1581 dB), and CCDF of PAPR = (9.2287 dB), while the SNR at BER( ) deteriorated by = (-16.9286 dB). The following conclusion from table A.24 and figure 6.20 when comparing the proposed method with an OFDM system with AEXP companding PAPR reduction method:  The PAPR was improved the least amount of improvement was when d =0.4 and CR = 4 and is equal to (0.1528 dB), while the vast amount of improvement is where d =2 and CR = 1.5 and is equal to (5.8486 dB).  The least amount of improvement in CCDF of PAPR when d =1.2 and CR = 4 and is equal to (0.0659 dB), while the vast amount of improvement is where d =2 and CR = 1.5 and is equal to (2.6752 dB).  The SNR at BER( ) was improved when CR =4,3,2 except when (d =0.4 and CR =4, 3, 2) and when (d =0.5 and CR =2) the SNR at BER( ) maintains its value. The vast amount of improvement is where d =0.7 and CR = 4 and is equal to (19.5598 dB), while The least amount of improvement in CCDF of PAPR when d=1 and CR = 2 and is equal to (0.217 dB)  The SNR at BER( ) was degraded when CR = 1.5, when d =2 The amount of degradation is equal to (-15.27 dB). The following conclusion from table A.24 and figure 6.20 when comparing the proposed method with an OFDM system with RFC method:  The PAPR and the CCDF of PAPR were improved except when d =2 and CR =1.5,2 they were degraded. The least amount of improvement was when d =1.9 and CR = 2 and the PAPR improvement is equal to (0.2144 dB) and the CCDF of PAPR improvement is equal to (0.0784 dB), while the vast amount of 141

Simulation Results and Analysis of Hybrid PAPR techniques

ď&#x201A;ˇ

improvement is where d =0.4 and CR = 4 and the PAPR improvement is equal to (10.9327 dB) and the CCDF of PAPR improvement is equal to (4.5281 dB). The SNR at BER( ) was degraded the least amount of degradation in SNR at BER( ) when d =2 and CR = 4 and is equal to (-1.5141 dB). The largest amount of degradation is when d =0.4 and CR =4 is equal to (-24.3287 dB). 30 original RFC (CR=4) RFC (CR=3) RFC (CR=2) AEXP RFC (CR=4) + AEXP RFC (CR=3) + AEXP RFC (CR=2) + AEXP

PAPR

SNR at (BER =10 4)

Figure (6.20.a) 11 original RFC (CR=4) RFC (CR=3) RFC (CR=2) AEXP RFC (CR=4) + AEXP RFC (CR=3) + AEXP RFC (CR=2) + AEXP

10 9

CCDF of PAPR

8 7 6 5 4 3 2 1

SNR at (BER =10- 4)

Figure (6.20.b) Figure 6.20 (a) Shows the values of the PAPR and SNR at BER = RFC, AEXP companding , and Hybird (RFC+AEXP) b) Shows the values of the CCDF of PAPR and SNR at BER = RFC, AEXP companding , and Hybird (RFC+ AEXP).

142

for each of the for each of the

Simulation Results and Analysis of Hybrid PAPR techniques 6.3.5 RFC + cos : The following conclusion from table A.25 and figure 6.21 when comparing the proposed method with an OFDM system without PAPR reduction method:  At these values ((y =1,0.9, 0.8,0.7 and CR = 4,3,2) and when ( y =0.6,0.5 and CR =4,3 ) and finally at (y =.4 and CR =4) There are improved in PAPR, CCDF of PAPR and SNR at BER( )). The best improvement in PAPR and CCDF of PAPR is at y =0.7 and CR =2. The improvement in PAPR by = (19.9896 dB), CCDF of PAPR = (8.3417 dB), and the SNR at BER( ) by = (1.1469 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at y = 0.5 and CR =2 . The improvement in PAPR by = (21.2527 dB) , and CCDF of PAPR = (8.8872 dB), while the SNR at BER( ) deteriorated by = (-1.3793 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at y =0.4 and CR =2 . The improvement in PAPR by = (21.9786 dB), and CCDF of PAPR = (9.1968 dB), while the SNR at BER( ) deteriorated by = (-3.2063 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at y =0.3 and CR =2. The improvement in PAPR by = (22.7251 dB), and CCDF of PAPR = (9.5413 dB), while the SNR at BER( ) deteriorated by = (-5.8059 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at y=0.2 and CR =3. The improvement in PAPR by = (22.8710 dB), and CCDF of PAPR = (9.5895 dB), while the SNR at BER( ) deteriorated by = (-13.8501 dB). The following conclusion from table A.25 and figure 6.21 when comparing the proposed method with an OFDM system with cos companding PAPR reduction method:  The PAPR and the CCDF of PAPR were improved and the least amount of improvement was when y =0.3 and CR = 4.The PAPR improvement is equal to (1.0157 dB) and the CCDF of PAPR improvement is equal to(0.5335 dB), while the vast amount of improvement is where y =1 and CR = 1.5 and the PAPR improvement is equal to (10.1993 dB) and the CCDF of PAPR improvement is equal to (4.5714 dB )  The SNR at BER( ) was improved when CR =4, 3,2. The vast amount of improvement when y=0.3 and CR = 4 and is equal to (7.6706 dB), while The least amount of improvement in SNR at BER( ) when y = 0.8 and CR = 2 and is equal to (2.5218 dB)  The SNR at BER( ) was degraded when CR = 1.5, when y =1 the amount of degradation is equal to (-17.8 dB). The following conclusion from table A.25 and figure 6.21 when comparing the proposed method with an OFDM system with RFC method:  The PAPR and the CCDF of PAPR were improved and the least amount of improvement was when y =1 and CR = 1.5 .The PAPR improvement is equal to (1.0296 dB) and the CCDF of PAPR improvement is equal to(0.1372 dB), while the vast amount of improvement is where y =0.1 and CR = 4 and the PAPR improvement is equal to (12.5275 dB) and the CCDF of PAPR improvement is equal to (5.3006 dB ) 143

Simulation Results and Analysis of Hybrid PAPR techniques ď&#x201A;ˇ

The SNR at BER( ) was degraded, the least amount of degradation in SNR at BER( ) when y =1 and CR = 2 and is equal to (-0.0444 dB). The largest amount of degradation is when (y =0.2, 0.1 and CR =4)(y =0.1 and CR =3) is equal to (-23.9287 dB). 30 original RFC (CR=4) RFC (CR=3) RFC (CR=2) cos RFC (CR=4) + cos RFC (CR=3) + cos RFC (CR=2) + cos

PAPR

SNR at (BER =10 4)

Figure 6.21.a 12 original RFC (CR=4) RFC (CR=3) RFC (CR=2) cos RFC (CR=4) + cos RFC (CR=3) + cos RFC (CR=2) + cos

CCDF of PAPR

SNR at (BER =10 4)

Figure 6.21.b Figure 6.21 (a) Shows the values of the PAPR and SNR at BER = RCF, cos companding , and Hybird (RCF+cos) b) Shows the values of the CCDF of PAPR and SNR at BER = RCF, cos companding , and Hybird (RCF+ cos) 144

for each of the for each of the

Simulation Results and Analysis of Hybrid PAPR techniques 6.3.6 RFC + NERF : The following conclusion from table A.26 and figure 6.22 when comparing the proposed method with an OFDM system without PAPR reduction method: At these values ( CR = 4,3) and There are improved in PAPR, CCDF of PAPR and SNR at BER( )). The best improvement in PAPR and CCDF of PAPR is at CR =3. The improvement in PAPR by = (17.3493 dB) ,CCDF of PAPR = (7.3822 dB), and the SNR at BER( ) by = ( 3.6500 dB). 30 original RFC (CR=4) RFC (CR=3) RFC (CR=2) RFC (CR=1.5) NERF RFC (CR=4) + RFC (CR=3) + RFC (CR=2) + RFC (CR=1.5)

PAPR

NERF NERF NERF + NERF

SNR at (BER =10- 4)

Figure 6.22.a 11 original RFC (CR=4) RFC (CR=3) RFC (CR=2) RFC (CR=1.5) NERF RFC (CR=4) + NERF RFC (CR=3) +NERF RFC (CR=2) + NERF RFC (CR=1.5) + NERF

10 9

CCDF of PAPR

8 7 6 5 4 3 2 1

SNR at (BER =10- 4)

Figure 6.22.b Figure (a) Shows the values of the PAPR and SNR at BER = for each of the RCF, cos companding , and Hybird (RCF+NERF) (b)Shows 6.22 the values of the CCDF of PAPR and SNR at BER = for each of the RCF, cos companding , and Hybird (RCF+ NERF)

145

Simulation Results and Analysis of Hybrid PAPR techniques 6.3.7 RFC + tanhR : The following conclusion from table A.27 when comparing the proposed method with an OFDM system without PAPR reduction method:  There are improved in PAPR, CCDF of PAPR and SNR at BER( ) in many points that have been tested, but The best improvement in PAPR and CCDF of PAPR is at k=40, y = .5 for CR =4 . The improvement in PAPR by = (20.7866 dB),CCDF of PAPR = (8.5636 dB), and the SNR at BER( ) by = ( 0.1129 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at k =20, y =0.5 and CR =2 . The improvement in PAPR by = (21.4382 dB) , and CCDF of PAPR = (9.0560 dB), while the SNR at BER( ) deteriorated by = (-1.5155 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at k =30, y=0.5 and CR =2 . The improvement in PAPR by = (21.7998 dB), and CCDF of PAPR = (9.2006 dB), while the SNR at BER( ) deteriorated by = (-3.2532 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at k =40, y=0.5 and CR =2. The improvement in PAPR by = (22.1212 dB), and CCDF of PAPR = (9.3121 dB), while the SNR at BER( ) deteriorated by = (-4.5295 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at k=40, y=0.2 and CR =3. The improvement in PAPR by = (23.7408 dB), and CCDF of PAPR = (9.9982 dB), while the SNR at BER( ) deteriorated by = (-8.0074 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is k=40, y=0.2 and CR =2. The improvement in PAPR by = (24.1411 dB), and CCDF of PAPR = (10.2047 dB), while the SNR at BER( ) deteriorated by = (-13.0440 dB). The following conclusion from table A.27 when comparing the proposed method with an OFDM system with tanhR companding PAPR reduction method:  The PAPR was degraded at (k =20, y =1,0.8,0.5 and CR = 4, 3, 2), (k= 10, y =1,0.8 and CR=4,3), (k =15 ,y =0.5 ,and CR =4,3) , and when (k =10 ,y =0.5 ,and CR =4). The least amount of degradation in SNR at BER( ) when k=20, y=0.5 and CR = 2 and is equal to (-0.3872 dB). The largest amount of degradation is when k =20, y = 1 and CR =4 is equal to (-8.4594 dB).  The CCDF of PAPR was degraded at (k =15, 20, y =1,0.8,0.5 and CR = 4, 3), (k= 20, y =1,0.8 and CR=2), (k =10 ,y =1,0.8 ,and CR =4,3) , and when (k =10 ,y =0.5 ,and CR =4). The least amount of degradation in SNR at BER( ) when k=15, y=0.8 and CR = 2 and is equal to (-0.2065 dB). The largest amount of degradation is when k =20, y = 1 and CR =4 is equal to (-3.426 dB).  Except the points already mentioned, the PAPR was improved and the least amount of improvement was when k =20, y =0.2 and CR = 4 and is equal to (0.1722 dB), while the vast amount of improvement is where k =5, y =1 and CR = 1.5 and is equal to (10.4523 dB).  Except the points already mentioned, the CCDF of PAPR was improved and the least amount of improvement in CCDF of PAPR when k =10, y =1 and CR = 2 and is equal to (0.0055 dB), While the vast amount of improvement is where k =5, y =1 and CR = 1.5 and is equal to (4.9898 dB). 146

Simulation Results and Analysis of Hybrid PAPR techniques  

The SNR at BER( ) was degraded when CR = 1.5, when k =5, y =1 the amount of degradation is equal to (-16.346 dB). Except the points already mentioned, the SNR at BER( ) was improved and The vast amount of improvement is where k =20, y =1 and CR = 4 and is equal to (23.5747 dB), while the least amount of improvement in SNR at BER( ) when k =5, y=0.2 and CR = 2 and is equal to (3.0185 dB).

The following conclusion from table A.27 when comparing the proposed method with an OFDM system with RFC method:  The PAPR was improved except when k =5, y =1 and CR =2 the SNR at BER( ) degraded by (-0.1918 dB), the least amount of improvement was when k=5, y =1 and CR = 1.5 and is equal to (0.1303 dB), While the vast amount of improvement is where k =40, y =0.2 and CR = 4 and is equal to (12.0608 dB).  The CCDF of PAPR was improved , except when k =5,y =1 and CR =1.5,2,3,4 the CCDF of PAPR was degraded. The least amount of improvement in CCDF of PAPR when k =10, y= 1 and CR = 2 and is equal to (0.0255 dB), while the vast amount of improvement is where k =40, y =0.2 and CR = 4 and is equal to (5.0792 dB).  The SNR at BER( ) was degraded, except when k =10, y =1 and CR =3 the SNR at BER( ) was improved by (0.018 dB). The least amount of degradation in SNR at BER( ) when k =10, y =1 and CR = 4 and is equal to (-0.1085 dB). The largest amount of degradation is when k =30, y =0.2 and CR =2 is equal to (16.7147 dB).

6.3.8 RFC +logR : The following conclusion from table A.28 when comparing the proposed method with an OFDM system without PAPR reduction method:  There are improved in PAPR, CCDF of PAPR and SNR at BER( ) in many points that have been tested, but the best improvement in PAPR and CCDF of PAPR is at k=90, y = .5 for CR =3 . The improvement in PAPR by = (20.6844 dB) ,CCDF of PAPR = (8.6603 dB), and the SNR at BER( ) by = ( 0.2112 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at k=5, y=0.5 and CR =2 . The improvement in PAPR by = (20.9759 dB) , and CCDF of PAPR = (8.9296 dB), while the SNR at BER( ) deteriorated by = (-1.3390 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at k=50 ,y =0.5 and CR =2 . The improvement in PAPR by = (21.6271 dB), and CCDF of PAPR = (9.1756 dB), while the SNR at BER( ) deteriorated by = (-2.6441 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at k =90 ,y= 0.5 and CR =2. The improvement in PAPR by = (21.8732 dB), and CCDF of PAPR = (9.2881 dB), while the SNR at BER( ) deteriorated by = (-4.1962 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at k=90, y=0.2 and CR =. The improvement in PAPR by = (23.3566 dB), and CCDF of PAPR = (9.8323 dB), while the SNR at BER( ) deteriorated by = (-6.8513 dB).  For SNR at BER( ) 147

Simulation Results and Analysis of Hybrid PAPR techniques The best improvement in PAPR and CCDF of PAPR is at = and CR =. The improvement in PAPR by = (23.8801 dB), and CCDF of PAPR = (10.0978 dB), while the SNR at BER( ) deteriorated by = (-11.4916 dB). The following conclusion from table A.28 when comparing the proposed method with an OFDM system with logR companding PAPR reduction method:  The PAPR was improved and the least amount of improvement was when k =90 ,y =1 and CR = 4 and is equal to (0.4691 dB), while the vast amount of improvement is where k =10, y =1 and CR = 2 and is equal to (8.8239 dB).  The CCDF of PAPR was improved and the least amount of improvement was when k =70 , y =1 and CR = 4 and is equal to (0.2321 dB), while the vast amount of improvement is where k =10, y =1 and CR = 2 and is equal to (4.1088 dB).  The SNR at BER( ) was improved . The least amount of improvement in SNR at BER( ) when k =90 ,y =0.2 and CR = 2 and is equal to (1.437 dB). The largest amount of improvement is when k =90, y =1 and CR =4 is equal to (22.5168 dB). The following conclusion from table A.28 when comparing the proposed method with an OFDM system with RFC method:  The PAPR was improved and the least amount of improvement was when k=5 ,y =1 and CR = 1.5 and is equal to (0.0422 dB), while the vast amount of improvement is where k =90, y =0.2 and CR = 4 and is equal to (11.5599 dB).  The CCDF of PAPR was improved, except when k =5 ,y =1 and CR =2 the CCDF of PAPR was degraded by (-0.0018 dB). The least amount of improvement in CCDF of PAPR when k =5, y= 1 and CR = 3 and is equal to (0.0727 dB), while the vast amount of improvement is where k =90, y =0.2 and CR = 4 and is equal to (4.8674 dB).  The SNR at BER( ) was degraded, except when k =5 ,y =1 and CR =1.5, 2. the BER was improved by (8.7803 - 0.1415 dB). The least amount of degradation in SNR at BER( ) when k =5, y =1 and CR = 4 and is equal to (-0.0429 dB). The largest amount of degradation is when k =90, y =0.2 and CR =2 is equal to (14.496 dB).

6.4 Pre-coding + companding: The companding scheme can be implemented with low complexity, without any iterative computations, comparing with coding, partial transmit and selective mapping schemes, in which either delay due to coding or extra overheads to synchronize transmitter and receiver are required. On the other hand, the pre-coding has also been considered as a best among all these techniques, because it improves the PAPR without increasing much complexity and destroying the orthogonality between subcarriers. The pre-coding also improves the BER in comparison to the normal OFDM system because of diversity gain obtained due to the spreading of the data symbol on more than one subcarrier. The OFDM system model with the proposed technique is as shown in figure 6.23. WHT, DCT, DST, and DHT pre-coders are used as for the companding has been using all kinds of previous companding. The results of the proposed method are good and the best result for the PAPR is when (DHT + tanhR).

148

Simulation Results and Analysis of Hybrid PAPR techniques

Add CP

P / S

Companding

S / P

IDFT OR IFFT

Signal mapper

I / P

D / A

Multipath Fading Ch. & noise

S / P

Remove CP

đ?&#x2018;&#x192;

De-Companding

DFT OR FFT

One Tap Equalizer And P/S

Signal demapper

O / P

A / D

Figure 6.23 the OFDM system model with precoding + companding .

6.4.1 Pre-coding + A companding: The following conclusion from table A.30 and figure 6.24 when comparing the proposed method with an OFDM system without PAPR reduction method: ď&#x201A;ˇ For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at A =5 and DHT . The improvement in PAPR by = (20.9180 dB), and CCDF of PAPR = (8.4193 dB), while the SNR at BER( ) deteriorated by = (-1.0169 dB). ď&#x201A;ˇ For SNR at BER( ) The best improvement in PAPR is at A = 20 and DHT. The improvement in PAPR by = (21.5586 dB), and CCDF of PAPR = (8.6612 dB), while the SNR at BER( ) deteriorated by = (-2.8809 dB). The best improvement in CCDF of PAPR is at A = 15 and DHT. The improvement in PAPR by = (21.4655 dB), and CCDF of PAPR = (8.6243 dB), while the SNR at BER( ) deteriorated by = (-2.3884 dB). ď&#x201A;ˇ For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at A =120 and DHT. The improvement in PAPR by = (21.9516 dB), and CCDF of PAPR = (8.7124 dB), while the SNR at BER( ) deteriorated by = (-5.0546 dB). ď&#x201A;ˇ For SNR at BER( ) The best improvement in CCDF of PAPR is at A =120 and DST. The improvement in PAPR by = (17.4182 dB), and CCDF of PAPR = (8.9910 dB), while the SNR at BER( ) deteriorated by = (-10.7326 dB). The following conclusion from table A.30 and figure 6.24 when comparing the proposed method with an OFDM system with A companding PAPR reduction method: ď&#x201A;ˇ The PAPR was improved except when (WHT and A =30, 87.6 ,100,120) the PAPR was degraded and maximum degraded at A =30 by (-0.9647). The least amount of improvement was when A = 40 and WHT and is equal to (0.0499 dB), 149

Simulation Results and Analysis of Hybrid PAPR techniques

 



While the vast amount of improvement is where A = 5 and DHT and is equal to (14.2226 dB). The least amount of improvement in CCDF of PAPR when A = 30 and WHT and is equal to (0.1178 dB), while the vast amount of improvement is where A = 5 and DHT and is equal to (4.2193 dB). The SNR at BER( ) was degraded at DST and WHT. The least amount of degradation in The SNR at BER( ) when A =70 and WHT and is equal to (0.036 dB). The largest amount of degradation is when A =50 and WHT and is equal to (-1.5167 dB). The SNR at BER( ) was improved at DHT. The least amount of improvement in The SNR at BER( ) when A =5 and is equal to (1.1517 dB). The largest amount of improvement is when A= 120 is equal to (5.377 dB). The SNR at BER( ) was improved at DCT and A= 30, 40, 70, 87.6, 100,120. The least amount of improvement in The SNR at BER( ) when A =40 and is equal to (0.0804 dB). The largest amount of improvement is when A= 70 is equal to (0.4065 dB). The SNR at BER( ) was degraded at DCT and A= 5, 10, 15, 20, 35, 50. The least amount of degradation in the SNR at BER( ) when A =15 and DHT and is equal to (-0.0769 dB). The largest amount of degradation is when A= 5 and WHT is equal to (-0.5053 dB).

The following conclusion from table A.30 and figure 6.24 when comparing the proposed method with an OFDM system with pre-coding method:  The PAPR and the CCDF of PAPR were improved and the least amount of improvement was when A =5 and DHT.The PAPR improvement is equal to (2.2752 dB) and the CCDF of PAPR improvement is equal to ( 0.9293 dB), while the vast amount of improvement is where A = 120 and WHT and the PAPR improvement is equal to (11.2105 dB) and the CCDF of PAPR improvement is equal to (7.6154 dB)  The SNR at BER( ) was degraded. The least amount of degradation in the SNR at BER( ) when A =5 and DHT and is equal to (-0.8663 dB). The largest amount of degradation is when A= 90 and WHT is equal to (-10.9249 dB).

150

Simulation Results and Analysis of Hybrid PAPR techniques 30 original WHT DCT DST DHT A WHT + A DCT+ A DST + A DHT + A

PAPR

0 10

SNR at (BER =10-4)

Figure 6.24.a 11 original WHT DCT DST DHT A WHT + A DCT+ A DST + A DHT + A

10 9

CCDF of PAPR

8 7 6 5 4 3 2 1 10

SNR at (BER =10-4)

Figure 6.24.b Figure 6.24 (a) Shows the values of the PAPR and SNR at BER = for each of the precodings, companding , and Hybird (precodings + ). b) Shows the values of the CCDF of PAPR and SNR at BER = for each of the precodings, companding , and Hybird (precodings + ).

151

Simulation Results and Analysis of Hybrid PAPR techniques 6.4.2 Pre-coding + : The following conclusion from table A.31 and figure 6.25 when comparing the proposed method with an OFDM system without PAPR reduction method:  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at MU =10 and DHT. The improvement in PAPR by = (20.9980 dB), and CCDF of PAPR = (8.4117 dB), while the SNR at BER( ) deteriorated by = (-1.3565 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at MU =50 and DHT. The improvement in PAPR by = (21.5810 dB), and CCDF of PAPR = (8.5662 dB), while the SNR at BER( ) deteriorated by = (-3.4700 dB).  For SNR at BER( ) The best improvement in PAPR is at MU =320 and DHT. The improvement in PAPR by = (21.9326 dB), and CCDF of PAPR = (8.7312 dB), while the SNR at BER( ) deteriorated by = (-5.5054 dB). The best improvement in CCDF of PAPR is at MU =320 and DHT. The improvement in PAPR by = (21.9143 dB), and CCDF of PAPR = (8.7404 dB), while the SNR at BER( ) deteriorated by = (-5.2295 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at MU = 1000 and DHT. The improvement in PAPR by = (22.0735 dB), and CCDF of PAPR = (8.7910 dB), while the SNR at BER( ) deteriorated by = (-6.1544 dB).  For SNR at BER( ) The best improvement in CCDF of PAPR is at MU = 1000 and DCT. The improvement in PAPR by = (18.6642 dB), and CCDF of PAPR = (9.3314 dB), while the SNR at BER( ) deteriorated by = (-12.2452 dB). The following conclusion from table A.31 and figure 6.25 when comparing the proposed method with an OFDM system with A companding PAPR reduction method:  The PAPR was improved except when (WHT and MU =60, 160,180) the PAPR was degraded and the maximum degraded at MU=160 by (-0.7606 dB). The least amount of improvement was when MU = 220 and WHT and is equal to (0.0305 dB), while the vast amount of improvement is where MU = 20 and DHT and is equal to (12.9585 dB).  The least amount of improvement in CCDF of PAPR when MU = 160 and WHT and is equal to (0.0919 dB), while the vast amount of improvement is where MU = 5 and DHT and is equal to (3.8402 dB).  The SNR at BER( ) was degraded at DST and WHT except at (MU =700 and WHT the SNR at BER( ) was improved by (0.1826 dB) . The least amount of degradation in The SNR at BER( ) when MU =100 and WHT and is equal to (-0.071dB). The largest amount of degradation is when MU =30 and DST and is equal to (-0.8137 dB).  The SNR at BER( ) was improved at DHT. The least amount of improvement in The SNR at BER( ) when MU =5 and is equal to (1.0094 dB). The largest amount of improvement is when MU= 700 is equal to (6.2032 dB).  The SNR at BER( ) at DCT there is an improvement in some of the points and the degradation the other .The largest amount of improvement is when MU =120 is equal to (0.4501 dB).The largest amount of degradation is when MU =10 152

Simulation Results and Analysis of Hybrid PAPR techniques is equal to (-0.3151 dB). Is clearly the amount of improvement and degradation less than 0.5 in all cases.

original WHT DCT DST DHT MU WHT + MU DCT+ MU DST + A DHT + MU

PAPR

0 10

SNR at (BER =10- 4)

Figure 6.25.a 11 original WHT DCT DST DHT MU WHT + MU DCT+ MU DST + MU DHT + MU

10 9

CCDF of PAPR

8 7 6 5 4 3 2 1 10

SNR at (BER =10-4)

Figure 6.25.b Figure 6.25 (a) Shows the values of the PAPR and SNR at BER = for each of the precodings, companding , and Hybird (precodings + ). b) Shows the values of the CCDF of PAPR and SNR at BER = for each of the precodings, companding , and Hybird (precodings + ). 153

Simulation Results and Analysis of Hybrid PAPR techniques The following conclusion from table A.31 and figure 6.25 when comparing the proposed method with an OFDM system with pre-coding method:  The PAPR and the CCDF of PAPR were improved and the least amount of improvement was when MU =5 and DHT .The PAPR improvement is equal to (1.9728dB) and the CCDF of PAPR improvement is equal to (0.7742 dB), While the vast amount of improvement is where MU = 1000 and WHT and the PAPR improvement is equal to (13.56 dB) and the CCDF of PAPR improvement is equal to (8.0035 dB ).  The SNR at BER( ) was degraded. The least amount of degradation in SNR at BER( ) when MU =5 and DHT and is equal to (-0.7449 dB). The largest amount of degradation is when MU =1000 and WHT is equal to (-12.5842 dB).

6.4.3 Pre-coding + RCT: The following conclusion from table A.32 and figure 6.26 when comparing the proposed method with an OFDM system without PAPR reduction method:  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at R =0.6 and DHT . The improvement in PAPR by = (20.1602 dB) , and CCDF of PAPR = (8.1603 dB), while the SNR at BER( ) deteriorated by = (-1.2760 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at R =0.4 and DHT . The improvement in PAPR by = (20.9808 dB), and CCDF of PAPR = (8.3274 dB), while the SNR at BER( ) deteriorated by = (-2.6866 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at R =0.3 and DHT. The improvement in PAPR by = (21.3993 dB), and CCDF of PAPR = (8.5410 dB), while the SNR at BER( ) deteriorated by = (-3.5957 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at R= 0.2 and DHT. The improvement in PAPR by = (21.8366 dB), and CCDF of PAPR = (8.5468 dB), while the SNR at BER( ) deteriorated by = (-5.9509 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at R = 0.1 and DHT. The improvement in PAPR by = (22.2932 dB), and CCDF of PAPR = (8.7149 dB), while the SNR at BER( ) deteriorated by = (-9.7284 dB). The best improvement in PAPR and CCDF of PAPR is at R = 0.2 and DST. The improvement in PAPR by = (21.2222 dB), and CCDF of PAPR = (8.9443 dB), while the SNR at BER( ) deteriorated by = (-10.7563 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at R =0.1 and DST. The improvement in PAPR by = (23.3532 dB), and CCDF of PAPR = (9.8705 dB), while the SNR at BER( ) deteriorated by = (-17.0023 dB).

154

Simulation Results and Analysis of Hybrid PAPR techniques 30 original WHT DCT DST DHT Rooting WHT + Rooting DCT+ Rooting DST + Rooting DHT + Rooting

PAPR

0 10

SNR at (BER =10 4)

Figure 6.26.a 12 original WHT DCT DST DHT Rooting WHT + Rooting DCT+ Rooting DST + Rooting DHT + Rooting

CCDF of PAPR

0 10

SNR at (BER =10 4)

Figure 6.26.b Figure 6.26 (a) Shows the values of the PAPR and SNR at BER = for each of the precodings, RCT, and Hybird (precodings +RCT). b) Shows the values of the CCDF of PAPR and SNR at BER = for each of the precodings, RCT, and Hybird (precodings +RCT ). 155

Simulation Results and Analysis of Hybrid PAPR techniques The following conclusion from table A.32 and figure 6.26 when comparing the proposed method with an OFDM system with RCT PAPR reduction method:  The PAPR and the CCDF of PAPR were improved .The least amount of improvement was when R =0.1 and WHT.The PAPR improvement is equal to (0.218 dB) and the CCDF of PAPR improvement is equal to (0.0754 dB), while the vast amount of improvement is where R =0.9 and DHT and the PAPR improvement is equal to (15.2604 dB) and the CCDF of PAPR improvement is equal to (6.3052 dB )  The SNR at BER( ) was improved at DHT. The least amount of improvement in The SNR at BER( ) when R = 0.8 and is equal to (0.1258 dB). The largest amount of improvement is when R=0.1 is equal to (7.1402 dB).  The SNR at BER( ) at DCT, DST and WHT there is an improvement in some of the points and the degradation the other .The largest amount of improvement is when R = 0.2 and DCT is equal to (0.4058 dB).The largest amount of degradation is when R=0.4 and WHT is equal to (-0.4257 dB). Is clearly the amount of improvement and degradation less than 0.5 in all cases. The following conclusion from table A.33 and figure 6.26 when comparing the proposed method with an OFDM system with pre-coding method:  The PAPR and the CCDF of PAPR were improved and the least amount of improvement was when R =0.9 and DHT.The PAPR improvement is equal to (0.356 dB) and the CCDF of PAPR improvement is equal to (0.1052 dB), While the vast amount of improvement is where R =0.1 and WHT and the PAPR improvement is equal to (20.1831 dB) and the CCDF of PAPR improvement is equal to (8.712 dB )  The SNR at BER( ) was degraded, except when R=0.9 and DHT the SNR at BER( ) was improved by (0.0718 dB).The least amount of degradation in SNR at BER( ) when R=0.9 and DCT and is equal to (-0.0147 dB). The largest amount of degradation is when R =0.1 and WHT is equal to (-17.0412 dB).

6.4.4 Pre-coding + AEXP: The following conclusion from table A.33 and figure 6.27 when comparing the proposed method with an OFDM system without PAPR reduction method:  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at d =0.8 and DHT . The improvement in PAPR by = (20.7461 dB), and CCDF of PAPR = (8.5315 dB), while the SNR at BER( ) deteriorated by = (-1.0379 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at d =0.4 and DHT. The improvement in PAPR by = (21.7546 dB) , and CCDF of PAPR = (8.7026 dB), while the SNR at BER( ) deteriorated by (-3.1563 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at d =0.3 and DHT. The improvement in PAPR by = (22.0123 dB) , and CCDF of PAPR = (8.7677 dB), while the SNR at BER( ) deteriorated by (-4.2985dB).  For SNR at BER( )

156

Simulation Results and Analysis of Hybrid PAPR techniques The best improvement in PAPR and CCDF of PAPR is at d =0.2 and DHT . The improvement in PAPR by = (22.2953 dB), and CCDF of PAPR = (8.8718 dB), while the SNR at BER( ) deteriorated by = (-18.5686 dB). The following conclusion from table A.33 and figure 6.27 when comparing the proposed method with an OFDM system with AEXP companding PAPR reduction method:  The PAPR was improved, except when (DHT and d =0.1, 0.2, 0.3, 0.4) and (WHT and d =0.1) PAPR was degraded and the maximum degradation is(-2.1186 dB).the least amount of improvement was when d = 0.2 and WHT and is equal to (0.0148 dB), while the vast amount of improvement is where d = 1.9 and DHT and is equal to (5.5788 dB).  The CCDF of PAPR was improved, except when (DHT and d =0.1, 0.2, 0.3, 0.4, 0.5) and (WHT and d =0.1, 0.2, 0.3) PAPR were degraded and the maximum degradation is (-1.5086 dB). The least amount of improvement in CCDF of PAPR when d =0.7 and WHT and is equal to (0.017 dB), while the vast amount of improvement is where d =0.9 and DHT and is equal to (2.085 dB).  The SNR at BER( ) was improved at DHT. The least amount of improvement in The SNR at BER( ) when d =2 and is equal to (1.5749 dB). The largest amount of improvement is when d =0.3 is equal to (14.2701 dB).  The SNR at BER( ) at DCT, DST and WHT there is an improvement in some of the points and the degradation the other .The largest amount of improvement is when d = 0.7 and WHT is equal to (11.8003 dB).The largest amount of degradation is when d = 1 and DST is equal to (-14.3dB). The following conclusion from table A.33 and figure 6.27 when comparing the proposed method with an OFDM system with pre-coding method:  The PAPR was improved, except when DHT and d =2, 1.9, 1.8, 1.7 PAPR was degraded and the maximum degradation is(-0.6937 dB).the least amount of improvement was when d = 1.6 and DHT and is equal to (0.0648 dB), while the vast amount of improvement is where d = 0.1 and WHT and is equal to (21.8494 dB).  The CCDF of PAPR was improved. The least amount of improvement in CCDF of PAPR when d =2 and DHT and is equal to (0.1988 dB), while the vast amount of improvement is where d =0.1 and WHT and is equal to (9.3313 dB).  The SNR at BER( ) was degraded. The least amount of degradation in SNR at BER( ) when d=1.1 and DHT and is equal to (-0.3896 dB). The largest amount of degradation is when d =0.2 and DHT is equal to (-18.418 dB).

157

Simulation Results and Analysis of Hybrid PAPR techniques 30 original WHT DCT DST DHT AEXP WHT + AEXP DCT+ AEXP DST +AEXP DHT +AEXP

PAPR

0 10

SNR at (BER =10 4)

Figure 6.27.a 12 original WHT DCT DST DHT AEXP WHT + AEXP DCT+ AEXP DST +AEXP DHT +AEXP

CCDF of PAPR

0 10

SNR at (BER =10 4)

figure 6.27.b Figure 6.27 (a) Shows the values of the PAPR and SNR at BER = for each of the precodings, AEXP companding , and Hybird (precodings +AEXP ). b) Shows the values of the CCDF of PAPR and SNR at BER = for each of the precodings,AEXP companding , and Hybird (precodings +AEXP ).

158

Simulation Results and Analysis of Hybrid PAPR techniques 6.4.5 Pre-coding + cos : The following conclusion from table A.34 and figure 6.28 when comparing the proposed method with an OFDM system without PAPR reduction method:  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at y=0.6 and DHT . The improvement in PAPR by = (19.8641 dB), and CCDF of PAPR = (8.1707 dB), while the SNR at BER( ) deteriorated by = (-1.2639 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at y = 0.4 and DHT . The improvement in PAPR by = (20.8133 dB) , and CCDF of PAPR = (8.3525 dB), while the SNR at BER( ) deteriorated by = (-2.7903 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at y = 0.3 and DHT. The improvement in PAPR by = (21.2690 dB), and CCDF of PAPR = (8.5732 dB), while the SNR at BER( ) deteriorated by = (-4.1634 dB).  For SNR at BER( ) The best improvement in CCDF of PAPR is at y = 0.3 and DHT. The improvement in PAPR by = (20.5789 dB), and CCDF of PAPR = (8.6358 dB), while the SNR at BER( ) deteriorated by = (-8.8379 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at y = 0.2 and DST. The improvement in PAPR by = (22.1250 dB), and CCDF of PAPR = (9.2996 dB), while the SNR at BER( ) deteriorated by = (-18.0413 dB). The following conclusion from table A.34 and figure 6.28 when comparing the proposed method with an OFDM system with cos companding PAPR reduction method:  The PAPR was improved .The least amount of improvement was when y = 0.3 and WHT and is equal to (0.0641 dB), while the vast amount of improvement is where y = 1 and DHT and is equal to (7.7327 dB).  The CCDF of PAPR was improved except at y =1 and WHT the CCDF of PAPR was degraded by (-0.0179 dB). The least amount of improvement in CCDF of PAPR when y =0.4 and WHT is equal to (0.0715 dB), while the vast amount of improvement is where y =0.1 and DHT and is equal to (3.7092 dB).  The SNR at BER( ) was improved at DHT. The least amount of improvement in The SNR at BER( ) when y = 1 and is equal to (0.0513 dB). The largest amount of improvement is when y =0.3 is equal to (5.7052 dB).  The SNR at BER( ) at DCT, DST and WHT there is an improvement in some of the points and the degradation the other .The largest amount of improvement is when y = 0.3 and DST is equal to (1.0307 dB).The largest amount of degradation is when y =0.3 and DCT is equal to (-1.6335 dB). The following conclusion from table A.34 and figure 6.28 when comparing the proposed method with an OFDM system with pre-coding method:  The PAPR was improved, except when DHT and y =1, 9, the PAPR was degraded and the maximum degradation is (-0.8692 dB). the least amount of improvement was when y = 0.8 and DHT and is equal to (0.0275 dB), while the vast amount of improvement is where y = 0.1 and WHT and is equal to (20.8281 dB). 159

Simulation Results and Analysis of Hybrid PAPR techniques  

The CCDF of PAPR was improved. The least amount of improvement in CCDF of PAPR when y =1 and DHT is equal to (0.2092 dB), while the vast amount of improvement is where y =0.1 and WHT and is equal to (8.9201 dB). The SNR at BER( ) was degraded. The least amount of degradation in SNR at BER( ) when y =1 and DCT and is equal to (-0.0147 dB). The largest amount of degradation is when y=0.2 and DHT is equal to (-18.418 dB).

30 original WHT DCT DST DHT cos WHT + cos DCT+ cos DST +cos DHT +cos

PAPR

0 10

SNR at (BER =10- 4)

Figure 6.28.a 12 original WHT DCT DST DHT cos WHT + cos DCT+ cos DST +cos DHT +cos

CCDF of PAPR

0 10

SNR at (BER =10- 4)

Figure 6.28.b Figure 6.28 (a) Shows the values of the PAPR and SNR at BER = for each of the precodings, cos companding , and Hybird (precodings +cos ). b) Shows the values of the CCDF of PAPR and SNR at BER = for each of the precodings,cos companding , and Hybird (precodings +cos ). 160

Simulation Results and Analysis of Hybrid PAPR techniques 6.4.6 Pre-coding + tanhR : The following conclusion from table A.36 when comparing the proposed method with an OFDM system without PAPR reduction method:  For SNR at BER( ) The best improvement in PAPR is at k=15, y=.8 and DHT. The improvement in PAPR by = (22.7711 dB), and CCDF of PAPR = (8.9691 dB), while the SNR at BER( ) deteriorated by = (-1.1828 dB). The best improvement in CCDF of PAPR is at k=20, y=1 and DHT. The improvement in PAPR by = (22.7411 dB), and CCDF of PAPR = (9.0618 dB), while the SNR at BER( ) deteriorated by = (-1.5372 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at k= 5, y =0.2 and DCT. The improvement in PAPR by = (22.9472 dB), and CCDF of PAPR = (9.6400 dB), while the SNR at BER( ) deteriorated by = (-12.6359 dB).  For SNR at BER( ) The best improvement in PAPR and CCDF of PAPR is at k = 30, y=.2 and DCT. The improvement in PAPR by = (23.7088 dB), and CCDF of PAPR = (10.0093 dB), while the SNR at BER( ) deteriorated by = (-17.7780 dB). The following conclusion from table A.36 when comparing the proposed method with an OFDM system with a tanhR companding method:  The PAPR, there is an improvement in some of the points and the degradation the other .The largest amount of improvement is when k =5, y = 1 and DST is equal to (15.7833 dB).The largest amount of degradation is when k =20, y =1 and WHT is equal to (-19.3807dB).  The CCDF of PAPR, there is an improvement in some of the points and the degradation the other .The largest amount of improvement is when k =5, y = 1 and DST is equal to (6.7541dB).The largest amount of degradation is when k =20, y =1 and WHT is equal to (-7.703 dB).  The SNR at BER( ) at DHT, WHT, and DCT, there is an improvement in some of the points and the degradation the other .The largest amount of improvement is when k=15, y = 1 and WHT is equal to (17.8776 dB).The largest amount of degradation is when k=5, y =0.3 and DHT is equal to (-0.856 dB).  The SNR at BER( ) at DST was degraded more than 30 dB in all cases The following conclusion from table A.36 when comparing the proposed method with an OFDM system with pre-coding method:  The PAPR was improved, except when WHT and y =1, k =5 the PAPR was degraded by (-0.1807 dB).the least amount of improvement was when k=10, y = 1 and WHT and is equal to (0.0656 dB), while the vast amount of improvement is where k =40, y = 0.1 and WHT and is equal to (19.3685 dB).  The CCDF of PAPR was improved. The least amount of improvement in CCDF of PAPR when k=5, y =1 and WHT is equal to (0.2938 dB), while the vast amount of improvement is where k =40, y =0.2 and WHT and is equal to (8.3837 dB).  The SNR at BER( ) was degraded. The least amount of degradation in SNR at BER( ) when k=30, y =1 and WHT and is equal to (-0.1416 dB). The largest amount of degradation is when k=30, y =0.2 and DCT is equal to (-17.5814 dB).

161

Simulation Results and Analysis of Hybrid PAPR techniques 6.4.7 Pre-coding + logR : The following conclusion from table A.37 when comparing the proposed method with an OFDM system without PAPR reduction method:  For SNR at BER( ) The best improvement in PAPR is at k=90, y =0.5 and DHT. The improvement in PAPR by = (21.9089 dB) , and CCDF of PAPR = (8.6957 dB), while the SNR at BER( ) deteriorated by = (-1.4569 dB). The best improvement in CCDF of PAPR is at k=70, y =0.5 and DHT. The improvement in PAPR by = (21.8896 dB) , and CCDF of PAPR = (8.7080 dB), while the SNR at BER( ) deteriorated by = (-1.3793 dB).  For SNR at BER( ) The best improvement in PAPR is at k=90, y=0.2 and DHT. The improvement in PAPR by = (22.2456 dB), and CCDF of PAPR = (8.8018 dB), while the SNR at BER( ) deteriorated by = (-6.0722 dB). The best improvement in CCDF of PAPR is at k=20, y=0.2 and DHT. The improvement in PAPR by = (22.1747 dB), and CCDF of PAPR = (8.8562 dB), while the SNR at BER( ) deteriorated by = (-5.8261 dB).  For SNR at BER( ) The best improvement in PAPR is at k=90 ,y =0.2and DCT. The improvement in PAPR by = (22.8075 dB), and CCDF of PAPR = (9.5916 dB), while the SNR at BER( ) deteriorated by = (-12.8773 dB). The best improvement in CCDF of PAPR is at k=10 ,y =0.2 and DST. The improvement in PAPR by = (22.7610 dB), and CCDF of PAPR = (9.6186 dB), while the SNR at BER( ) deteriorated by = (-13.2351 dB).  For SNR at BER( ) The best improvement in PAPR is at k=90, y =0.2 and DST. The improvement in PAPR by = (23.1261 dB), and CCDF of PAPR = (9.7547 dB), while the SNR at BER( ) deteriorated by = (-14.8433 dB). The best improvement in CCDF of PAPR is at k=70 , y =0.2 and DST. The improvement in PAPR by = (22.9321 dB), and CCDF of PAPR = (9.7575 dB), while the SNR at BER( ) deteriorated by = (-14.4128 dB). The following conclusion from table A.37 when comparing the proposed method with an OFDM system with a logR companding method:  The PAPR, there is an improvement in some of the points and the degradation the other .The largest amount of improvement is when k =10, y = 1 and DHT is equal to (12.2879 dB).The largest amount of degradation is when k =70, y =1 and WHT is equal to (-7.3198 dB).  The CCDF of PAPR, there is an improvement in some of the points and the degradation the other .The largest amount of improvement is when k =10, y = 1 and DHT is equal to (4.9342 dB).The largest amount of degradation is when k =90, y =1 and WHT is equal to (-2.7575 dB).  The SNR at BER( ) at DHT, WHT, and DCT, there is an improvement in some of the points and the degradation the other .The largest amount of improvement is when k=90 , y = 1 and WHT is equal to (17.7596 dB).The largest amount of degradation is when k=70, y =0.2 and DHT is equal to (-0.7874 dB).  The SNR at BER( ) at DST was degraded, except when k =90 ,y =1 the SNR at BER( ) maintains its value .The least amount of degradation in SNR at

162

Simulation Results and Analysis of Hybrid PAPR techniques BER( ) when k =50, y =1 and is equal to (-1.012 dB). The largest amount of degradation is when k=40, y=0.2 is equal to (-14.3135dB). The following conclusion from table A.37 when comparing the proposed method with an OFDM system with pre-coding method:  The PAPR was improved, the least amount of improvement was when k=5, y = 1 and WHT and is equal to (0.4263 dB), while the vast amount of improvement is where k =40, y = 0.2 and WHT and is equal to (18.9242 dB).  The CCDF of PAPR was improved. The least amount of improvement in CCDF of PAPR when k=5, y =1 and WHT is equal to (0.5366 dB), while the vast amount of improvement is where k =90, y =0.2 and WHT and is equal to (8.2281 dB).  The SNR at BER( ) was degraded, except at WHT, y =1 and k=5, 10, 40 the SNR at BER( ) was improved and the maximum improvement is (0.0852 dB). The least amount of degradation in SNR at BER( ) when k=50, y =1 and WHT and is equal to (-0.1544 dB). The largest amount of degradation is when k=50, y =0.8 and DST is equal to (-17.2375 dB).

6.4.8 Pre-coding + NERF: This method did not work with DST and DHT, whereas the BER gives us an error. The following conclusion from table A.35 and figure 6.29 when comparing the proposed method with an OFDM system without PAPR reduction method: When using DCT and WHT with ERF, there are improved in PAPR, CCDF of PAPR. The best improvement in PAPR and CCDF of PAPR is at DCT. The improvement in PAPR by = (16.5312 dB),CCDF of PAPR = (6.8931 dB), while the SNR at BER( ) degraded by = (-1.9789 dB).

26 original WHT DCT DST DHT NERF WHT + NERF DCT+NERF

24 22 20

PAPR

18 16 14 12 10 8 6 11.5

12.5

SNR at (BER =10- 4)

Figure 6.29.a

163

13.5

Simulation Results and Analysis of Hybrid PAPR techniques 11 original WHT DCT DST DHT NERF WHT + NERF DCT+ NERF

CCDF of PAPR

9 8 7 6 5 4 3 11.5

12.5

13.5

SNR at (BER =10- 4)

Figure 6.29.b Figure 6.29 (a) Shows the values of the PAPR and SNR at BER = for each of the precodings, cos companding , and Hybird (precodings +cos ). b) Shows the values of the CCDF of PAPR and SNR at BER = for each of the precodings,cos companding , and Hybird (precodings +cos ).

164

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Conclusions and future work Chapter seven Conclusions and future work

7.1Conclusions: 1. The RFC and RCF can improve the PAPR and BER at the same time. 2. RFC is better than RCF in performance especially when I ≥ 2 while maintaining the complexity and price of RCF. 3. The performance of all kinds of proposed companding is better than the performance μ-law and A-law compandings 4. AEXP can be considered the best types of companding that we used in terms of BER and performance, followed by tanhR. 5. TanhR has better results when used in the hybrid technique and also the y, k parameters give it a kind of flexibility 6. TanhR and logR and NERF is better than tanh, log, and erf that Mohit was used in his paper, because the performance of the proposed techniques better than μ-law and A-law compandings and also the μ-law and A-law compandings better than Mohit methods. 7. The performance of logR companding asymptotic to tanhR but the tanhR have better results in most cases. 8. The performance of cos companding asymptotic to AEXP but the AEXP have better results in most cases. 9. The best type precoding in term of reduced PAPR and BER is the DFT 10. DST and DCT precodings give almost the same performance, the DST improves the PAPR more than DCT even a few percent. 11. The worst type of precoding in term of reducing the PAPR and BER is the WHT. 12. As it is clear from the results that the hybrid methods have better results but at the expense of complexity. 13. The results of hybrid pre-coding with RCF is better than the results of the RCF and pre-coding each alone, except in the case of DHT with RCF (I = 2, pilot) where the results of the DHT is better 14. For the hybrid pre-coding with RCF, the PAPR value is better when RCF (I = 1) because in this case the effect of the filter on the PAPR cancels. 15. The hybrid RCF with companding shows good results better than the results of the RCF and pre-coding each alone, because of RCF reduces the PAPR and improves the BER constant and then companding more reduces the amount of the PAPR. 16. The hybrid RCF with companding can improve the PAPR and BER at the same time with amount greater than the RCF and the best one improvement in PAPR is at (RCF + AEXP). 17. The hybrid RFC with companding shows good results better than the results of the hybrid RCF with companding, because as we demonstrated earlier RFC batter than RCF. 18. The hybrid RFC with companding can also improve the PAPR and BER at the same time, and the best one improvement in PAPR is at (RFC + AEXP). 19. The results of the hybrid precoding with companding are provides good results and the best result for the PAPR is when (DHT with tanhR) except at (DST with tanhR, DST with NERF, DHT with NERF). At DST with tanhR the BER performance significantly degraded 20. The best results are obtained at these techniques: RFC: A. 165

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Conclusions and future work

The best one improvement in PAPR and CCDF of PAPR is at I =4 and CR =1.75. The improvement in PAPR by = (18.2789 dB), CCDF of PAPR = (8.0187 dB), and the SNR at BER ( ) by = (0.6101 dB). AEXP companding: For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at d= 0.9. The improvement in PAPR by = (18.8515 dB), and CCDF of PAPR = (7.6480 dB), while the SNR at BER ( ) deteriorated by = (-4.8686 dB). LogR companding: For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k =10, y =0.3. The improvement in PAPR by = (19.6992 dB), and CCDF of PAPR = (8.2150 dB), while the SNR at BER ( ) deteriorated by = (-8.5686 dB). TanhR companding: For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k = 5, y=0.2. The improvement in PAPR by = (22.0569 dB), and CCDF of PAPR = (9.3125 dB), while the SNR at BER ( ) deteriorated by = (-13.2917 dB). LogR companding: For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at . The improvement in PAPR by = (23.5788 dB), and CCDF of PAPR = (9.9600 dB), while the SNR at BER ( ) deteriorated by = (-18.1686 dB). Hybird RFC + AEXP: The best one improvement in PAPR and CCDF of PAPR is at d = 0.6 and CR =4. The improvement in PAPR by = (21.0509dB), CCDF of PAPR = (8.7178 dB), and the SNR at BER ( ) by = (0.0116 dB). Pre-coding + tanhR : For SNR at BER( ) The best one improvement in PAPR is at k=15, y=.8 and DHT. The improvement in PAPR by = (22.7711 dB), and CCDF of PAPR = (8.9691 dB), while the SNR at BER ( ) deteriorated by = (-1.1828 dB). RFC + tanhR For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at k=40, y=0.2 and CR =3. The improvement in PAPR by = (23.7408 dB), and CCDF of PAPR = (9.9982 dB), while the SNR at BER ( ) deteriorated by = (-8.0074 dB). RFC + tanhR For SNR at BER(

)

A. 166

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Conclusions and future work

The best one improvement in PAPR and CCDF of PAPR is k=40, y=0.2 and CR =2. The improvement in PAPR by = (24.1411 dB), and CCDF of PAPR = (10.2047 dB), while the SNR at BER ( ) deteriorated by = (-13.0440 dB). RFC + RCT For SNR at BER( ) The best one improvement in PAPR and CCDF of PAPR is at R = 0.1and CR =2. The improvement in PAPR by = (24.3546 dB), and CCDF of PAPR = (10.3164 dB), while the SNR at BER ( ) deteriorated by = (-14.1974 dB).

7.2 Future work: 1. Use another type of filter that does not effect on the PAPR or have little impact with clipping. 2. Find a new type of companding to recduce the PAPR with maintaining the BER performance. 3. The proposed companding PAPR reduction methods can be combined with different PAPR reduction techniques such as PTS, SLM, TR and etc. 4. The proposed RFC can be combined with different PAPR reduction techniques such as coding, interleaving, TI and DSI etc. 5. The RCF, proposed RFC can be combined with different existing companding techniques such as airy companding, linear companding, Trapezoidal power companding and etc. 6. The proposed companding PAPR reduction methods can be combined with Zadoff-Chu matrix Transform precoding. 7. Analysis of the proposed techniques and find out its impact on the PAPR mathematically. 8. proposed new hybrid techniques by using the proposed method 9. Study the impact of these proposed techniques on bandwidth, noise , distortion and the ratio of power saving. 10. Study the impact of these proposed techniques on statistical distribution. 11. The proposed PAPR reduction methods can be used with MIMO OFDM system. 12. The proposed PAPR reduction methods can be used with other multicarrier system

A. 167

References References [1] K. Fazel and S. Kaiser, Multi-Carrier and Spread Spectrum Systems: From OFDM and MC-CDMA to LTE and WiMAX, 2nd Edition, John Wiley & Sons, 2008. [2] G. Wunder, R. F. H. Fischer, H. Boche, S. Litsyn, J,-S. No, “The PAPR Problem in OFDM Transmission: New Directions for a Long-Lasting Problem,” IEEE Signal Processing Magazine, vol. 20, no. 6, pp. 130-144, November 2013. [3] T. Jiang, and Y. Wu “An Overview: Peak-to-Average Power Ratio Reduction Techniques for OFDM Signals” IEEE Transactions on Wireless Communications, vol.54, no2, pp.257 - 268. June 2008. [4] L.M. Correia, D. Zeller, O. Blume, D. Ferling, Y. Jading , I. Gódor, G. Auer and L. Van der Perre “Challenges and Enabling Technologies for Energy Aware Mobile Radio Networks,” IEEE Communications Magazine, vol. 48, pp. 66-72, November 2010. [5] S. Litsyn. Peak Power Control in Multicarrier Communications. Cambridge University Press, January 2007. [6] S. H. Han and J. H. Lee ,“An overview of peak-to-average power ratio reduction techniques for multicarrier transmission”, IEEE Wireless Communications, ,vol.12 , no: 2 ,pp 56-65. April 2005. [7] A. Gangwar, and M. Bhardwaj. “An Overview: Peak to Average Power Ratio in OFDM system & its Effect,” International Journal of Communication and Computer Technologies, vol. 01, pp.22-25, no.2, September 2012. [8] M. M. Mowla, and S.M. M. Hasan “Performance Improvement of PAPR Reduction for OFDM Signal In LTE System,” International Journal of Wireless & Mobile Networks , vol. 5, no. 4,pp. 35-47,August 2013. [9] R.W. Bauml, R.F.H. Fischer, and J. B. Huber “Reducing the peak-to-average power ratio of multicarrier modulation by selected mapping,” IEEE Electronics Letters ,vol. 33, pp. 2056 - 2057, Oct 1996. [10] H. D. Joshi. “Performance augmentation of OFDM system,” Ph.D. dissertation, Jaypee Univ. of engineering and Technology, India, May 2012. [11] M. Park, H. Jun, J. Cho, N. Cho, D. Hong and C. Kang “PAPR Reduction in OFIIM Transmission using Hadamard Transform,” IEEE International Conference on Communications, vol.1, pp. 430-433, Jun 2000. [12] J. Armstrong. “New OFDM Peak-to-Average Power Reduction Scheme,” IEEE VTS 53rd Vehicular Technology Conference, Spring, vol. 01, pp. 756 – 760, May 2001.

168

References [13] J. Armstrong. “Peak-to-average power reduction for OFDM by repeated clipping and frequency domain filtering,” IEEE Electronics Letters, vol. 38, pp. 246 - 247, Feb 2002. [14] R. C. Hincapie and J. E. Sierra. Advanced Transmission Techniques in WiMAX. InTech, January 2012. [15] T. Jiang, Y. Yang, and Y. Song. “Companding Technique for PAPR Reduction in OFDM Systems Based on An Exponential Function,” Global Telecommunications Conference, GLOBECOM '05. IEEE , vol. 05, pp. 2798-2801, Dec. 2005. [16] W. F. Al-Azzo, B. M. Ali, S. Khatun and S. Jamalullail “Time Domain aStatistical Control for PAPR Reduction in OFDM System” Proceedings of AsiaPacific Conference on Communications 2007, pp. 141-144. [17] P. BOONSRIMUANG, K. MORI, T. PAUNGMA, and H. KOBAYASHI “Proposal of Simple PAPR Reduction Method for OFDM Signal by Using Dummy Sub -Carriers” The Institute of Electronics, Information and Communication Engineers, vol. E91–B, no. 3, pp. 784-794, march 2008. [18] C. A. Devlin, A. Zhu, and T.J. Brazil “Peak to Average Power Ratio Reduction Technique for OFDM Using Pilot Tones and Unused Carriers,” Radio and Wireless Symposium, 2008 IEEE, pp. 33 – 36, Jan. 2008. [19] K. TAKEDA and F. ADACHI “A PAPR Reduction Scheme Without Side Information For OFDM Signal Transmissions,” IEEE Transactions on Signal Processing, vol. 60, pp. 3657- 3669, July 2012. [20] Z.Wang, S. zhang, and B. qiu.“PAPR Reduction of OFDM Signal by Using Hadamard Transform in Companding Techniques,” IEEE International Conference on Communication Technology (ICCT), pp. 320 – 323, Nov. 2010. [21] I. Baig and V. Jeoti “PAPR Analysis of DHT-Precoded OFDM System for MQAM,” IEEE 2010 International Conference on Intelligent and Advanced Systems (ICIAS), pp.1-4, June 2010. [22] Z. Wang. “Combined DCT and Companding for PAPR Reduction in OFDM Signals,” Journal of Signal and Information Processing, vol. 2, pp. 100-104, 2011. [23] H. D. Joshi and R. Saxena “PAPR Reduction in OFDM Systems Using Precoding With Clipping” IEEE Communications, Computing and Control Applications, 2011, pp.1-5. [24] B. P. Lathi, Modern Digital and Analog Communication Systems. New York: Oxford University Press, 1998. [25] M. Chauhan, and A. Chobe “PAPR Reduction in OFDM system Using Tone Reservation Technique,” IJCTEE, vol. 2, pp. 57- 60, August 2012.

169

References [26] E. V. Cuteanu and D. Isar “Hybrid PAPR Reduction Scheme using Walsh Hadamard Precoding and Signal Companding” International Symposium on Electronics and Telecommunications, 2012, pp. 195-198. [27] C. Hsu, and H. Liao “PAPR Reduction Using the Combination of Precoding and Mu-Law Companding Techniques for OFDM Systems” IEEE 11th International Conference on Signal Processing, 2012, vol.1, pp.1-4. [28] S. Abouty, L. Renfa, Z. Fanzi and F. Mangone “A Novel Iterative Clipping and Filtering Technique for PAPR Reduction of OFDM Signals: System Using DCT/IDCT Transform,” International Journal of Future Generation Communication and Networking, vol. 6, no. 1 , pp. 1-8, February 2013. [29] Z. You, I. Lu, R. Yang, J. Li “Flexible Companding Design for PAPR Reduction in OFDM and FBMC Systems,” IEEE International Conference on Computing, Networking and Communications, 2013, pp. 408-412. [30] M. K. Singh, P. Siingh , A. Gupta, S. S. Sharma, and S. K. Singh “Performance Evaluation of Different Companding Techniques for Peak-to-Average Power Reduction of an OFDM Signal,” International Journal of Advances in Electrical and Electronics Engineering, vol.2, no. 1, pp. 204-209, 2013. [31] T. Jiang, W. Xiang, P. C. Richardson, D. Qu, and G. Zhu “On the Nonlinear Companding Transform for Reduction in PAPR of MCM Signals,” IEEE Transactions On Wierless Communications, vol. 6, no. 6, June 2007. [32] C. Wang and S. Ku “A Low-Complexity Companding Transform For Peak To Average Power Ratio Reduction In OFDM Systems,” IEEE International Conference on Acoustics, Speech and Signal Processing, 2006, vol. 4, pp. 329-332. [33] S. A. Aburakhia, E.F. Badran and D. A. E. Mohamed “Linear Companding Transform for the Reduction of Peak-to Average Power Ratio of OFDM Signals,” IEEE Transactions on Broadcasting, vol. 55, pp. 155 -160, February 2009. [34] O. Gazi “A New Companding Technique f or PAPR Reduction in OFDM Communication Systems,” IEEE International Congress on Ultra-Modern Telecommunications and Control Systems and Workshops, 2011, pp.1-5. [35] K.H. Sankar, K.A. Babu “Airy Function Based PAPR Reduction Method for OFDM Systems” IJMER, vol.2, pp-4295-4297, Nov-Dec. 2012. [36] J. Virdi , and S. Kumar “PAPR Reduction Based on Precoding Techniques with Companding in OFDM Systems,” International Journal of Scientific & Engineering Research, vol. 4, pp.1064 -1070, May-2013. [37] N. Kaur and L. Kansal “Peak To Average Power Ratio Reduction Of OFDM Signal By Companding Clipping With Walsh Hadamard Transform,” International Journal of Wireless & Mobile Networks , vol. 5, no. 1,pp.33-45 , February 2013.

170

References [38] M. K .Singh , A. Gupta, P. Siingh, S. S.Sharma, and S.t K. Singh “Hybrid Clipping-Companding Schemes for Peak to Average Power Reduction of OFDM Signal,” International Journal of Advances in Engineering Science and Technology, vol. 2, no. 1, pp. 114-119, 2013. [39] K. MURALIBABU, and M.SUNDHARARAJAN “DCT Based PAPR and ICI Reduction Using Companding Technique in MIMO-OFDM System ,” International Journal of Advanced Research in Computer Science and Software Engineering. vol. 3, pp. 49-53, August 2013 [40] Jijina N., and S. S. Pillai. “Linear Precoding Schemes for PAPR Reduction in Mobile WiMAX OFDMA System,” International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering, vol. 3,pp. 8873-8884, April 2014. [41] H. Zarrinkoub, Understanding LTE With MATLAB®, Massachusetts, USA: John Wiley & Sons, Ltd, 2014. [42] G. M. Kebede “Performance Evaluation of LTE Downlink with MIMO Techniques.” M.A. thesis, School of Engineering, Blekinge Institute of Technology, Sweden, November 2010. [43] X. Zhang and X. Zhou, LTE-Advanced Air Interface Technology. CRC Press, 2013. [44] S. Sesia, I. Toufik, and M. Baker, LTE – The UMTS Long Term Evolution. 2nd Edition., John Wiley & Sons, 2011. [45] M. Šoša “The challenges of LTE technologies” M.A. thesis, polytechnic university of Zagreb, June 2013. [46] 3GPP TS 36.300, “LTE; Evolved Universal Terrestrial Radio Access (E-UTRA) and Evolved Universal Terrestrial Radio Access Network (E-UTRAN); Overall Description; Stage 2 (Release 10),” version 10.7.0, April 2010. [47] H. Holma and A.Toskala, LTE for UMTS- OFDMA and SC-FDMA Based Radio Access. John Wiley & Sons, 2009. [48] E. Dahlman, S. Parkvall, J. Sköld, and P. Beming. 3G Evolution: HSPA and LTE for Mobile Broadband. Academic Press, July 2007. [49] F. KHAN. LTE for 4G Mobile Broadband: Air Interface Technologies and Performance. Cambridge University Press, April 2009. [50] Fact sheet 3GPP LTE. The "Long Term Evolution" air interface. Federal Office of Communications OFCOM, January 2011. [51] T. Bhandare, “LTE and WiMAX Comparison”, Santa Clara University, 2008.

171

References [52] A. ARUN DASH “OFDM Systems And PAPR Reduction Techniques In OFDM Systems” National Institute Of Technology, Rourkela, 2010. [53] J. Zyren “Overview of the 3GPP Long Term Evolution Physical Layer,” White Paper, freescale semiconductor, 2007. [54] S. M. Abbas, M. N. Abbas, and S. A. Mohammed “Slantlet Transform-Based OFDM Scheme,” Journal of Engineering, vol. 13,no. 3,pp. 1638-1647,September 2006. [55] V.R.Prakash, and P.Kumaraguru “Performance analysis of OFDM with QPSK using AWGN and Rayleigh Fading Channel,” Conference: ICIESP, 2012, pp.10 -15. [56] R. Prasad l and F. J. Velez. WiMAX Networks: Techno-Economic Vision and Challenges. Springer, 2010. [58] M. Sajedin, A. Ghorbani, and H. R. A. Davar “Nonlinearity Compensation for High Power Amplifiers Based on Look-Up Table Method for OFDM Transmitters”International Journal of Advanced Computer Science and Information Technology, vol. 3, no. 4, pp. 354-367, 2014. [59] F.A. Ibikunle, MNSE, and MIEEE “Broadband Wireless Access Solution based on OFDMA Technique in WiMAX IEEE 802.16,” The Pacific Journal of Science and Technology, vol. 10, no 1, pp. 286-294, May 2009. [60] Chang R. W., “Synthesis of band-limited orthogonal signals for multichannel data transmission”, Bell Systems Technical Journal, vol. 46, pp. 1775-1796, Dec. 1966. [61] Part 16: Air Interface for Fixed Broadband Wireless Access Systems Amendment 2: “Medium Access Control Modifications and Additional Physical Layer Specifications for 2-11 GHz”, IEEE, Standard IEEE Std. 802.16a-2003, 2003. [62] A. Goel “Improved PAPR Reduction in OFDM Systems,” Ph.D. dissertation, JAYPEE Institute of Information Technology, India, April 2013. [63] B. R. Saltzberg, “Performance of an efficient parallel data transmission system,”IEEE Transactions on Communication Technology, vol. COM-15, no.6, pp.805–813, Dec.1967. [64] R. W. Chang ,“Synthesis of band-limited orthogonal signals for multichannel data transmission,” Bell System Technology Journal, vol. 45, pp. 1775–1796, Dec. 1966. [65] R. W. Chang and R. A. Gibby ,“A theoretical study of performance of an orthogonal multiplexing data transmission scheme,” IEEE Transactions on Communication Technology, vol. COM-16, no.4, pp. 529–540, August 1968.

172

References [66] H. F. Marmuth, „„On the transmission of information by orthogonal time functions,‟‟ AIEE Transactions of communication and Electronics, vol. 79, pp. 248– 255, July 1960. [67] V.Tarokh, New Directions in Wireless Communications Research. Spring, September 7, 2009. [68] Despain A. M., “Very Fast Fourier Transform Algorithms Hardware for Implementation”, IEEE Trans. Comp., Vol. C-28, no. 5, pp. 333-341, May 1979. [69] Bidet E., Castelain D., Joanblanq C. , Senn P., “A fast single-chip implementation of 8192 complex point FFT”, IEEE Journal of Solid State Circ., Vol. 30, no. 3, pp. 300-305, Mar. 1995. [70] R. Peled, A. Ruiz, “Frequency domain data transmission using reduced computational complexity algorithms”, in Proceeding of the IEEE International Conference on Acoustics, Speech, and Signal Processing, ICASSP ‟80 ,USA, pp. 964–967,1980. [71] Chow P. S., Dhahir N. AL, Cioffi J.,Bingham M., “A Multicarrier E1-HDSL transceiver system with coded modulation”, J. Eur. Transactions on Telecom. Rel. Tech. (ETT) (Special Issue: Applications of Coded Modulation Techniques), vol. 4, no. 3, pp: 257 –66, June 1993. [72] Chow P.S., Tu J.C. and Cioffi J.M., “Performance Evaluation of a Multichannel Transceiver System for ADSL and VHDSL services", IEEE Journal on Selected Areas in Communications, vol. 9, no. 6, Aug. 1991. [73] ETSI, “Radio broadcasting systems; Digital Audio Broadcasting (DAB) to mobile, portable and fixed receivers”, European Telecommunication Standard, Standard EN-300- 401, May 1997. [74] ETSI, “Digital Video Broadcast (DVB); Framing Structure, Channel Coding and Modulation for Digital Terrestrial Television”, ETSI EN 300 744 v1.1.2, 1997. [75] ETSI, “Broadband Radio Access Networks (BRAN): Hiperlan Type 2; Technical Specification Part 1-Physical Layer”, DTS/BRAN 030003-1, Oct. 1999. [76] B. Sklar, Digital Communications. Fundamentals and Applications. 2nd Edition, Prentice Hall, 2001. [77] F. Tariq “Impact of PAPR on Link Adaptation Strategies of OFDM Based Systems,” M.A. thesis, Chalmers University of Technology, May 2007. [78] G. Hill, “Peak Power Reduction in Orthogonal Frequency Division Multiplexing Transmitters.” Ph.D. Thesis, Victoria University of Technology, March 2011. [79] D. Matiæ “OFDM as a possible modulation technique for multimedia applications in the range of mm waves,” CiteSeerX, 1998.

173

References [80] A. Sahu , and S. Behera “PAPR analysis and channel estimation techniques for 3GPP LTE system.” National Institute of Technology Rourkela , 2011. [81] S. Bhavsar, H. Pandey, P. Chouhan “Design and Implementation of OFDM Trans-Receiver for IEEE 802.11(WLAN),” International Journal Of Modern Engineering Research, vol. 4, pp. 55-67, Jan. 2014. [82] J. Mohanty “Analysis And Study Of Multi-Symbol Encapsulated Orthogonal Frequency Division Multiplexing.” National Institute of Technology, Rourkela, 2010. [83] E.Dahlman. S. Parkvall, and J. Sköld ,4G LTE/LTE-Advanced for Mobile Broadband. Academic Press, 2011. [84] S. L. Morancho. “Access modes based on coordinated multipoint for relay transmissions in 4G systems.” M.A. thesis, University of Catalonia, 2011. [85] S. K. Rangineni. “Multihop Concept in Cellular Systems.” M.A. thesis, Czech Technical University, Sweden, 2008. [86] J. A. Pereira “Small Cell Deployment Evaluation on LTE,” M.A. thesis, Técnico Lisboa, October 2013. [87] T. Zemen “OFDMA/SC-FDMA Basics for 3GPP LTE (E-UTRA),” Forschungszentrum Telekommunikation Wien, April 24, 2008. [88] S. Singh, M. Kaur “Physical Layer Simulation of WIMAX 802.16,” IJECT, vol. 4, pp.84-87, July - September 2013. [89] S. Singh “IMT-Advanced Requirements for 4G Technology and its Components” IJECT, vol. 4, pp. 58-60, Jan - March 2013. [90] J. Zyren “Overview of the 3GPP Long Term Evolution Physical Layer.” White Paper, Freescale Semiconductor, July 2007. [91] C. Cox. An Introduction To LTE: LTE, LTE-Advsnced, SAE And 4G Mobile Communications. John Wiley & Sons, April 2012. [92] F. E. Abd El-Samie, F. S. Al-kamali, A. Y. Al-nahari, and M. I. Dessouky .SCFDMA for Mobile Communications. CRC Press, July 2013 [93] “LTE in a Nutshell: The Physical Layer” White paper, Telesystem Innovations Inc, 2010. [94] R. Siddavaatam “Joint Graph Coloring And Ant Colony Optimization Based Sub-Channel Allocation Algorithms For Downlink LTE-A Hetnets, ” M.A. thesis, Toronto, Ontario, Canada, August 2012. [95] C. Choudhary, and V. Gupta “A Study of Performance Enhancement Schemes for Multicarrier Transmission,” International Journal of Computer Applications, vol. 68, no. 5, pp. 50-54, April 2013. 174

References [96] K. Srinivasarao, B. Prabhakararao, and M. V. S. Sairam “Peak-to-Average Power Reduction In MIMO OFDM Systems using Sub-Optimal Algorithm,”International Journal of Distributed and Parallel Systems, vol.3, no.3, pp. 261-273, May 2012. [97] N. Kaur, and A. S. Sappal “Peak to Average Power Ratio Measurement Using Oversampling Technique” International Journal for Multi Disciplinary Engineering and Business Management, vol. 2, pp. 125-129, July-September 2014. [98] Y. S. Cho, J. Kim, W. Y. Yang and C. G. Kahn, MIMO OFDM wireless communications with Matlab. Asia: John Wiley & Sons, 2010. [99] R. V. Nee and R. Prasad, OFDM for Wireless Multimedia Communication. London: Artech house Publisher, 2000. [100] S. Khalid “Peak to Average Power Ratio Reduction in Orthogonal Frequency Division Multiplexing Systems.” Ph.D. dissertation, University of Engineering and Technology Taxila, Pakistan, Dec. 2009. [101] R. D. Kanti, and R.V. Ch. Sekhar Rao “Systematic Comparison of Different PAPR Reduction Methods in OFDM Systems” International Journal of Electronics and Communication Engineering, vol. 7, no.1, pp. 21-30, 2014. [102] R. Rajbanshi “OFDM-Based Cognitive Radio for DSA Networks.” Ph.D. dissertation, The University of Kansas, September 2007. [103] V. Vijayarangan, and R. Sukanesh “An Overview Of Techniques For Reducing Peak To Average Power Ratio And Its Selection Criteria For Orthogonal Frequency Division Multiplexing Radio Systems,” Journal of Theoretical and Applied Information Technology, vol. 5, no. 1, pp.25-36, January 2009. [104] Y. G. Li and G.L. Stuber, Orthogonal Frequency Division Multiplexing for Wireless Communications. New York, NY, USA: Springer, 2006. [105] P. H. Lehne, and F. Bøhagen .OFDM(A) for wireless communication .Telenor ASA, 2008. [106] Sanjeev Saini, and O.P. Sahu “Peak to Average Power Ratio Reduction in OFDM System by Clipping and Filtering,” International Journal of Electronics Communication and Computer Technology, vol. 2, pp.105-109, May 2012. [107] A.Tiwari, and K. Markam “An Overview: Peak-to-Average Power Ratio Reduction Techniques for OFDM Signals,” International Journal of Computer & Communication Engineering Research, vol. 2, pp.23-28, January 2014. [108] T. Jiang, and Y. Wu, “An Overview: Peak-to-Average Power Ratio Reduction Techniques for OFDM Signals,” IEEE Transactions on Broadcasting, vol. 54, no. 2, pp.257-268, June 2008.

175

References [109] M. El-Tarhuni, M. Hassan, and A. B. Sediq, “A Joint Power Allocation and Adaptive Channel Coding Scheme for Image Transmission over Wireless Channels,” International Journal of Computer Networks & Communications (IJCNC), Vol.2, No.3, May 2010. [110] R. Rajbanshi “OFDM-Based Cognitive Radio for DSA Networks.” Technical Report, The University of Kansas, September 2007. [111] X. Li and L. J. Cimini, “Effects of clipping and filtering on the performance of OFDM,” IEEE Commun. Lett., vol. 2, pp. 131–133, May 1998. [112] T. A. Wilkison and A. E. Jones, “Minimization of the peak to mean envelope power ratio of multicarrier transmission schemes by block coding,” in Proc. 45th IEEE Veh. Technol. Conf., vol. 2, Chicago, IL, USA, July 1995, pp. 825 – 829. [113] H. Chen and A. M. Haimovich, “Iterative estimation and cancellation of clipping noise for OFDM signals,” IEEE Commun. Lett., vol. 7, pp. 305–307, July 2003. [114] L. Wang and C. Tellambura, “A simplified clipping and filtering technique for PAR reduction in OFDM systems,” IEEE Signal Processing Lett., vol. 12, pp. 453– 456, June 2005. [115] S.-K. Deng and M.-C. Lin, “OFDM PAPR reduction using clipping with distortion control,” in Proc. IEEE Int. Conf. Commun., vol. 4, Seoul, Korea, May 2005, pp. 2563– 2567. [116] S. C. Thompson, J. G. Proakis, and J. R. Zeidler, “The effectiveness of signal clipping for PAPR and total degradation reduction in OFDM systems,” in Proc. IEEE Global Telecommun. Conf., St. Louis, MO, USA, Nov. 2005, pp.2807–2811. [117] R. van Nee and A. de Wild, “Reducing the peak-to-average power ratio of OFDM,” in Proc. 48th IEEE Veh. Technol. Conf., Ottawa, Ontario, Canada, May 1998, pp. 2072–2076. [118] S. H. Han and J. H. Lee, “Peak-to-average power ratio reduction of an OFDM signal by signal set expansion,” in Proc. IEEE Int. Conf. Commun.,vol. 2, Paris, France, June 2004, pp. 867–971. [119] P. Foomooljareon, W. A. C. Fernando, and K. M. Ahmed, “PAPR reduction of OFDM systems using input sequence envelope scaling,” in Proc. 57th IEEE Veh. Technol. Conf. - Spring, vol. 2, Jeju, Korea, Apr. 2003, pp. 1243 – 1247. [120] B. S. Krongold and D. L. Jones, “PAR reduction in OFDM via active constellation extension,” IEEE Trans. Broadcast., vol. 49, pp. 258 – 268, Sept. 2003. [121] Y. Kou, L. Wu-Sheng, and A. Antoniou, “New peak-to-average powerratioreduction algorithms for multicarrier communications,” IEEE Trans. Circuits Syst., vol. 51, pp. 1790 – 1800, Sept. 2004. 176

References [122] S. H. Muller and J. B. Huber, “OFDM with reduced peak to average power ratio by optimum combination of partial transmit sequences,” Electron. Lett., vol. 33, pp. 368 – 369, Feb. 1997. [123] H. Nikookar and K. S. Lidsheim, “Random phase updating algorithm for OFDM transmission with low PAPR,” IEEE Trans. Broadcast., vol. 48, no. 6, pp. 123 – 128, June 2002. [124] N. Y. Ermolova and P. Vainikainen, “On the relationship between peak factor of a multicarrier signal and aperiodic autocorrelation of the generating sequence,” IEEE Commun. Lett., vol. 7, pp. 107–108, Mar. 2003. [125] S. Y. Le Goff, S.S. Al-Samahi, B.K. Khoo, C.C. Tsimenidis, and B.S. Sharif (July 2009) “Selected mapping without side information for PAPR reduction in OFDM,” IEEE Transactions on Wireless Communication, vol. 8, no. 7, pp. 33203325. [126] P. van Eetvelt, G. wade, and M. Tomlinson, “Peak to average power reduction for OFDM schemes by selective scrambling,” Electron. Lett., vol. 32, pp. 1963–1964, Oct. 1996. [127] A. D. S. Jayalath and C. Tellambura, “The use of interleaving to reduce the peak to average power ratio of an OFDM signal,” in Proc. IEEE Global Telecommun. Conf., vol. 1, San Francisco, CA, USA, Nov. 2000, pp. 82 –86. [128] S. C. Thompson, “Constant envelope OFDM phase modulation,” Ph.D. dissertation, University of California, San Diego, CA, USA, 2005. [129] K. K. Sen, and V. sahu, “PAPR Reduction Using PTS with DCT SLM Technique,” irdindia., Vol. 1, no.3, pp. 23-25, 2014. [130] B. Torun “Peak-to-Average Power Ratio Reduction Techniques for Wavelet Packet Modulation.” M.A. thesis, Department of Telecommunications at Delft University of Technology, Delft, The Netherlands. 27 October 2010. [131] Md. A. Islam, N. Ahmed, N. U.Ahamed, M. Rahman, S. A. Aljunid, “PAPR Reduction in an OFDM system using Recursive Clipping and Filtering Technique,” WSEAS TRANSACTIONS on COMMUNICATIONS, Vol. 13, pp.291-297, 2014. [132] M. Bala, M. Kumar, K. Rohilla, “PAPR Reduction in OFDM Signal Using Signal Scrambling Techniques,” International Journal of Engineering and Innovative Technology (IJEIT), Vol. 3, No. 11, pp.140-143, May 2014. [133] S. Sujitha, and R. Ramachandran “Performance Analysis of PAPR Reduction in MIMO OFDM System Using Modified Constant Modulus Algorithm,” International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering, Vol. 3, No. 2, pp520-526, April 2014.

177

References [134] M. F. Ghanim, and M.F.L.Abdullah,“ Reduction Techniques of Peak-toAverage Power Ratio in MCCDMA Systems,” International Journal of Informatics and Communication Technology (IJ-ICT), Vol.1, No.2, pp. 82-90, December 2012. [135] M. Gouda, K. A. Shehata, M. Hussien, “PAPR Reduction Performance for LTE OFDM Systems with Different Techniques,” International Journal of Scientific & Engineering Research, Vol. 4, No. 5, pp. 2335-2344, May 2013. [136] H. Sakran, M. Shokair, and A. Abou Elazm “Combined Interleaving And Companding For PAPR Reduction in OFDM Systems, ” Progress In Electromagnetics Research C, vol. 6, pp. 67–78, 2009. [137] H. Chen and H. Liang. “Combined Selective Mapping and Binary Cyclic Codes for PAPR Reduction in OFDM Systems”. IEEE Transactions on Wireless Communications, vol.6, pp. 3524 - 3528. Oct.2007. [138] K. M. K. Pervez, and M. M. Hossain “A New Proposed Scheme for PAPR Reduction of OFDM System Combining Hadamard Transform and Hann Peak Windowing,” International Journal of Computer and Information Technology, vol. 2, pp. 62-66, 2012. [139] M. M. Mowla, L. C. Paul and M. R. Hasan “COMPARATIVE PERFORMANCE ANALYSIS OF DIFFERENT MODULATION TECHNIQUES FOR PAPR REDUCTION OF OFDM SIGNAL,”, International Journal of Computer Networks & Communications (IJCNC) Vol.6, No.3,pp.63-73 , May 2014. [140] C. R. ANTHIKKAD ,and I. A. BAIG “Performance Enhancement Of OFDM In PAPR Reduction Using New Companding Transform And Adaptive AC Extension Algorithm For Next Generation Networks.” M.A. thesis, School of Engineering Blekinge, Institute of Technology, Sweden, September 2013. [141] M. Z. Parvez ,and Md. A. Al Baki “Peak To Average Power Ratio (PAPR) Reduction In OFDM Based Radio Systems,” M.A. thesis, School of Engineering, Blekinge Institute of Technology, Sweden, May 2010. [142] B. U. Rindhe, J.Digge, and S. K. Narayankhedkar, “Performance Analysis of OFDM Based System for PAPR Reduction Techniques with Optical Fiber Link,” International Journal of Engineering Science and Innovative Technology (IJESIT) Vol. 3, No. 2, pp.20-32, March 2014. [143] Van Nee, R., and Wild, A., “Reducing the peak to average power ratio of OFDM”, IEEE Vehicular Technology Conference, Vol. 3, May 1998. [144] Wilkison, T. A. and Jones A. E., "Minimization of the Peak to mean Envelope Power Ratio of Multicarrier Transmission Schemes by Block Coding," IEEE, Vehicular Conference, Vol.2, Jul. 1995. [145] Foomooljareon, P. and Fernando, W.A.C., “Input sequence envelope scaling in PAPR reduction of OFDM”, IEEE 5th international symposium on wireless personal multimedia communications, Vol. 1, Oct 2002. 178

References [146] P. Dhok, S.V. Rathkantiwar, “Performance Improvement in BER and PAPR Reduction in OFDM System Using Companding Technique,” International Journal of Application or Innovation in Engineering & Management (IJAIEM), Volume 2, No.1,pp.232-235, January 2013. [147] S. A. Aburakhia, E. F. Badran, and D. A. E. Mohamed “Linear Companding Transform for the Reduction of Peak-toAverage Power Ratio of OFDM Signals,” IEEE Transactions on Broadcasting, Vol.55 , No.1 , pp.155-160, March 2009. [148] L. Shu-hua, L. Yong-zhao1, Z. Hai-lin1, and L. Ying-ting , “Threshold-based piecewise companding transform for PAPR reduction in OFDM systems,” The Journal of China Universities of Posts and Telecommunications, No.2, April 2013. [149] B. Ragini, M. S. Babu ,and K. K. Rao , “Companding Technique for Reducing Peak-to-Average Power Ratio in OFDM Linear Coded Systems,” International Journal of Power System Operation and Energy Management, Vol.1, No. 2,pp. 48-53, 2011. [150] T. Jiang, W. Yao, P. Guo, Y. Song, and D. Qu , “Two Novel Nonlinear Companding Schemes With Iterative Receiver to Reduce PAPR in Multi-Carrier Modulation Systems,” IEEE TRANSACTIONS ON BROADCASTING, Vol.. 52, No. 2, June 2006. [151] T. Moazzeni , H. Selvaraj ,and Y. Jiang, “A Novel Multi Exponential Functionbased Companding Technique for Uniform Signal Compression over Channels with Limited Dynamic Range ,” International Journal of Electronics and Telecommunications Vol. 56, pp.125-128, Jun 2010. [152] W. F. Al-Azzo, and B. M. Ali “Adaptive Square-Rooting Companding Technique for PAPR Reduction in OFDM Systems,” World Academy of Science, Engineering and Technology, vol.5, pp. 507-511, 2011. [153] T. Jiang, Y. Yang, and Y. Song “Exponential Companding Technique for PAPR Reduction in OFDM Systems” IEEE Transactions on Broadcasting, vol. 51,pp.244247 , no. 2, June 2005. [154] Yogita1 and R. Kumar “A Review on PAPR Reduction Techniques” International Journal of Engineering, Applied and Management Sciences Paradigms, vol. 16, pp.45-48, June 2014. [155] A. E. Jones, T. A. Wilkinson, and S. K. Barton, “Block coding scheme for reduction of peak-to-average envelope power ratio of multicarrier transmission systems,” IEE Electronics Letters, vol. 30, no. 8, pp. 2098–2099, Dec. 1994. [156] D. Wulich, “Reduction of peak to mean ratio of multicarrier modulation using cyclic coding,” IEE Electronics Letters, vol. 32, no. 29, pp. 432–433, Feb. 1996.

179

References [157] S. Fragicomo, C. Matrakidis, and J. J. OReilly, “Multicarrier transmission peakto-average power reduction using simple block code,” IEEElectronics Letters, vol. 34, no. 14, pp. 953–954, May 1998. [158] T. Jiang and G. X. Zhu, “OFDM peak-to-average power ratio reduction by complement block coding scheme and its modified version,” in The 60th IEEE Vehicular Technology Conference 2004 Fall, Los Angeles, USA, pp. 448–451, Sept. 2004. [159] T. Jiang and G. X. Zhu, “Complement block coding for reduction in peak-toaverage power ratio of OFDM signals,” IEEE Communications Magazine, vol. 43, no. 9, pp. S17–S22, Sept. 2005. [160] S. Boyd, “Multitone signals with low crest factor,” IEEE Trans. Circuits Systems, vol. CAS-33, no. 10, pp. 1018–1022, Oct. 1986. [161] B. M. Popovic, “Synthesis of power efficient multitone signals with flat amplitude spectrum,” IEEE Trans. Communications, vol. 39, no. 7, pp. 1031–1033, Jul. 1991. [162] T. Jiang, G. X. Zhu, and J. B. Zheng, “A block coding scheme for reducing PAPR in OFDM systems with large number of subcarriers,” Journal of Electronics, vol. 21, no. 6, pp. 482–489, Dec. 2004. [163] J. A. Davis and J. Jedwab, “Peak-to-mean power control in OFDM,Golay complementary sequences and Reed-Muller codes,” IEEE Trans. Information Theory, vol. 45, no. 7, pp. 2397–2417, Nov. 1997. [164] C. L. Wang, S. J. Ku, and C-Ju Yang, “A low-complexity PAPR estimation scheme for OFDM signals and its application to SLM-based PAPR reduction,” IEEE Journal of Selected Topics in Signal Processing, vol. 4, no. 3, pp. 637-645, June 2010. [165] Zhongpeng Wang,” Reduction PAPR of OFDM Signals by Combining SLM with DCT Transform”, Int. J. Communications, Network and System Sciences, 2010, 3, pp. 888-892 , November 2010 . [166] A. Ghassemi and T. A. Gulliver, “Partial selective mapping OFDM with low complexity IFFTs,” IEEE Communication Letters, vol. 12, no. 1, pp. 4-6, January 2008. [167] R. J. Baxley and G. T. Zhou ,“Comparing selected mapping and partial transmit sequence for PAR reduction,” IEEE Transactions on Broadcasting, vol. 53, no. 4, pp. 797–803, December 2007. [168] Gatherer, A. and Polley, M., “Controlling clipping probability in DMT transmission,” Asilomar Conference record, vol. 1, pp. 578 - 584, Nov. 1997. [169] Tellado, J. and Cioffi, J. M., “Efficient algorithms for reducing PAR in multicarrier systems,” Proc. IEEE International Symposium on Information Theory, p. 191, Aug. 1998. 180

References [170] J. Tellado, “ Multicarrier Modulation with Low PAR: Applications to DSL and Wireless,” Kluwer Academic Publishers, 2000. [171] L. Dewangan, M. Singh, and N. Dewangan “A Survey of PAPR Reduction Techniques in LTE-OFDM System,” International Journal of Recent Technology and Engineering (IJRTE), Vol.1, No.5,pp.10-13 November 2012. [172] Mukul , and S. singh “An Overview and Comparative Study of Different PAPR Reduction Techniques in OFDM System,” National Conference on Innovative Trends in Computer Science Engineering ,pp. 118-122, April 2015. [173] D. Narendra, and P.S. Reddy “PAPR Reduction Technique in OFDM System For 4G Wireless Applications Using Partial Transmit Sequence Method,” Quest Journals, Journal of Electronics and Communication Engineering Research, Vol.1, No.1 ,pp. 38-42,2013. [174] M. Petermann, D. Wubben and K. Kammeyer “Joint Constellation Extension And Tone Reservation For PAPR Reduction In Adaptive OFDM Systems,” IEEE 10th Workshop on Signal Processing Advances in Wireless Communications, 2009. SPAWC '09, pp. 439 – 443, 21-24 June 2009. [175] Ryu, H., Lee, J., and Park, J., “Dummy Sequence insertion (DSI) for PAPR reduction in the OFDM communication system”, IEEE Transactions on Consumer Electronics, Vol. 50, Feb. 2004. [176] D. Pramudiwati “LTE System Performance In Relation To Widband Channel Properties .” M.A. thesis, Delft University of Technology. [177] ITU-R, Report M.2134, “Requirements related to technical performance for IMT-Advance Radio Interface,”2008. [178] R. Prasad. OFDM for Wireless Communications Systems. Artech House Publishers, August 2004. [179] B. Schmidt, C. Ng, P. Yien, C. Harris, U. Saripalle, A. Price, S. Brewer, G. Scelsi, R. Slaviero and J. Armstrong' “Efficient Algorithms for PAPR Reduction in OFDM Transmitters Implemented using Fixed-Point DSPs,” IEEE 63rd Vehicular Technology Conference, 2006, vol. 4 ,pp. 2023 – 2027. [180] J. Armstrong “Peak-to-Average Power Reduction in Digital Television Transmitters,” Digital Image Computing Techniques and Applications, pp.1-6, January 2002. [181] V. N. Sonawane, and S. V. Khobragade “Reduction of PAPR in OFDM System using A-law Companding Technique,” International Journal of Scientific & Engineering Research, vol. 4, pp.1450-1453 , May-2013. [182] Y. S. Reddy1, M V. K. Reddy, K Ayyanna, and G. V. Ravikumar “The Effect of NCT Techniques on SC-FDMA System in presence of HPA” International Journal

181

References of Research in Computer and Communication Technology, vol. 3,pp.844-848, August 2014. [183] V. N. Sonawane1, and S. V. Khobragade “Comparative Analysis between Alaw & μ-law Companding Technique for PAPR Reduction in OFDM” International Journal of Advanced Research in Computer and Communication Engineering, Vol. 2, PP. 2210-2214, May 2013. [184] S. S. Bhadoriya, and R. Bhatia “ Novel Method For PAPR Reduction In OFDM Using Cascading Of Companding Technique And Selective Mapping,” International Journal of Engineering Sciences and Management, vol. 2, pp. 167-169, 2012. [185] A. Vallavaraj, B.G. Stewart, D. K. Harrison, and F. G. McIntosh. “Reduction of peak-to-average power ratio of OFDM system using a companding technique,” IEEE Transactions on Broadcasting, vol. 45, pp. 303-307, Sep 1999. [186] T. Kozhakhmetov “Reduction of Peak-to-Average Power Ratio in OFDM systems using companding schemes” Helmut Schmidt University, 2008. [187] S. L. Kotgire and S. B. Deosarkar “Low Noise Technique for Reduction of Peak Power and BER in High Rate Wireless Communication System,” International Journal of Current Engineering and Technology, vol. 3,pp.1330-1334, no.4, October 2013. [188] K. K. Sen, and V. sahu “PAPR Reduction Using PTS with DCT SLM Technique” International Journal of Computer Network And Security, vol. 1, pp. 2729, 2014. [189] M. I. Ishaq., Y. A.Khan, and M. T. Gul. “Precoding in MIMO, OFDM to reduce PAPR (Peak to Average Power Ratio),” M.A. thesis, Linnaeus University, Sweden , 2012. [190] C. Ciochina, D. Mottier, and H. Sari “An Analysis of OFDMA, Precoded Ofdma and SC-FDMA for the Uplink in Cellular Systems,” Lecture Notes Electrical Engineering ,vol.1, pp. 25-36, 2007. [191] F. S. Çakar “A Study Of Precoding Schemes For OFDM Systems.” M.A. thesis, Middle East Technical University, August 2008. [192] M. A. Aboul-Dahab, E.A. A. A. Hagras, and A. A. Elhaseeb “PAPR Reduction Based on DFT Precoding for OFDM Signals,” International Journal ofFuture Computer and Communication, vol. 2, no. 4, pp. 325-328 , August 2013. [193] A. V. Sivaram and R. S. Rao “PAPR Reduction of DHT and WHT-Precoded OFDM System for M-QAM” ITSI Transactions on Electrical and Electronics Engineering, vol. 1, pp.113-117 , 2013.

182

References [194] G. D. Prakash and M.Sharmila “Analysis of PAPR in Precoded OFDM Systems for M-QAM” International Journal of Engineering Trends and Technology, vol. 4, pp.2318 - 2323, June 2013. [195] B. Pavan and Kumar.M “Comparitive Study Of PAPR Of DHT-Precoded OFDM System With DFT & WHT,” International Journal of Engineering Research and Applications, vol. 3, pp.527-532, January-February 2013. [196] D. A. Nugroho, and D. Kim “Precoded DCT and Low Complexity SLM for PAPR Reduction in OFDM Systems” International Journal of Engineering and Industries, vol. 4, no. 4, pp. 43-49, December 2013. [197] B. S. Kumari.“PAPR and Fiber Nonlinearity Mitigation of DFT-Precoded OFDM System,” International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering, vol. 3, pp. 9801- 9808, June 2014. [198] S. Chaudhari “Peak to Average Power Ratio Reduction Schemes and Synchronization Algorithms for OFDM Based Systems.” M.A. thesis, Indian Institute of Science, India, July 2004.

183

Appendix A:

Tables of Results Appendix A Tables of Results

A.1 RCF Results Table A.1 RCF Results Oversampling(I) 1

pilot

1.125

1.25

1.5

CR 4 3 2 1.75 1.5 4 3 2 1.75 1.5 4 3 2 1.75 1.5 4 3 2 1.75 1.5 4 3 2 1.75 1.5 4 3 2 1.75 1.5 4 3 2 1.75 1.5 4 3 2 1.75 1.5

PAPR 14.0866 11.5702 8.2486 7.3802 6.3838 14.1973 11.9169 9.4432 8.5889 8.0108 14.1010 11.6977 8.4887 7.4828 6.6241 14.1392 11.7044 8.5374 7.7741 6.8767 14.1728 11.8777 8.9992 8.0760 7.8858 14.4932 12.2742 9.6073 9.0189 8.3829 15.3850 13.2843 10.6525 9.8847 9.4795 15.5738 13.4298 10.7538 10.0109 9.2193 A. 1

CCDF of PAPR 6.0244 4.785 3.1186 2.594 2.9 6.248 5.225 3.8796 3.56 3.3568 6.0337 4.8436 3.2413 2.7737 2.3785 6.06 4.8772 3.43 3.023 2.618 6.145 5 3.6 3.315 2.976 6.4177 5.355 4.1216 3.785 3.6257 6.7374 5.79 4.555 4.2435 3.9 6.8674 5.8 4.4432 4.0712 3.725

SNR (BER= 11.82 12.3711 14.3756 16.72 29.5 9.36 9.8415 13 16.063 29.6( 11.3068 11.65 14.4 16.875 29.6( 10.572 11.4 13.72 16 29.6( 10 10.4765 13.7445 16.4373 29.6( 8.7679 9.414 12.128 15 29.6( 7.1838 7.665 10.418 13.1725 29.6( 5.8315 6.0725 9.6955 13 29.6(

)

Appendix A:

Tables of Results

A.2 RFC Results If (A, B, and C) positive values that's mean there is an improvement, while if the negative values this mean there is a deterioration in values The (PAPR, CCDF OF PAPR and BER) were calculated with different value of (CR (4, 3, 2, 1.75, 1.5) and I (1, pilot, 1.125, 1.25, 1.5, 2, 3, 4) These values have been placed on the table. A, B, and C also added to the table for comparison with the RCF Where A = PAPR (RCF) – PAPR (RFC) B =CCDF of PAPR (RCF) – CCDF of PAPR (RFC) C = SNR at BER ( ) (RCF) – SNR at BER ( ) (RFC) Table A.2 RFC Results I Pilot

1.125

1.25

1.5

CR 4 3 2 1.75 1.5 4 3 2 1.75 1.5 4 3 2 1.75 1.5 4 3 2 1.75 1.5 4 3 2 1.75 1.5 4 3

A 0.0232 .0906 .6703 .334 1.1464 0.0042 0.0159 -0.0866 0.1119 0.0771 0.0215 0.0583 0.2114 0.2031 0.2754 -0.0015 0.0533 0.4553 0.6558 1.3184 0.3083 0.5039 1.0271 1.4327 1.7478 1.1714 1.5283

PAPR 14.1741 11.8263 8.7729 8.2549 6.8644 14.0968 11.6818 8.5753 7.3709 6.5470 14.1177 11.6461 8.3260 7.5710 6.6013 14.1743 11.8244 8.5439 7.4202 6.5674 14.1849 11.7703 8.5802 7.5862 6.6351 14.2136 11.7560

B -0.3570 0.3490 0.5706 0.7329 0.9448 0.0069 0.0436 0.0513 0.0837 0.2372 0.0279 0.0732 0.1750 0.3118 0.3556 0.1018 0.1655 0.3289 0.2343 0.5919 0.3577 0.5010 0.7472 0.9370 1.1757 0.6830 0.9373 A. 2

CCDF of PAPR

6.605 4.876 3.309 2.8271 2.412 6.0268 4.8 3.19 2.69 2.1413 6.0321 4.804 3.255 2.7112 2.2624 6.0432 4.8345 3.2711 3.0807 2.3841 6.06 4.854 3.3744 2.848 2.45 6.0544 4.8527

-0.2400 -0.1585 0.5000 1.0630 0 0.0708 -0.1700 0.7455 -0.2390 0.9000 -0.5700 0.1900 -0.0375 0.3286 0 -0.1236 0.0510 0.5733 0.1286 0 0.1089 0.0440 -0.1548 0.3875 0 0.4163 0.2842

SNR (BER=

)

9.6 10 12.5 15 29.6( ) 11.236 11.82 13.6545 17.114 28.7 11.142 11.21 13.7575 15.6714 29.6( ) 10.1236 10.4255 13.1712 16.3087 29.6( ) 8.659 9.37 12.2828 14.6125 29.6( 6.7675 7.3808

Appendix A:

2 1.75 1.5 4 3 2 1.75 1.5

Tables of Results

2.1453 2.2903 2.9062 1.3900 1.6108 2.2738 2.6883 2.8284

8.5072 7.5944 6.5733 14.1838 11.8190 8.4800 7.3226 6.3909

1.2550 1.4123 1.5600 0.8024 0.9360 1.1382 1.2499 1.3092

3.3 2.8312 2.34 6.065 4.864 3.305 2.8213 2.4158

0.4380 1.4725 0 0.1602 0.0340 1.2685 2.1787 1.3800

9.98 11.7 29.6( ) 5.6713 6.0385 8.427 10.8213 28.22

Table A.3 Precoding Results Precoding WHT DCT DST DHT DFT

Table A.4 5 10 15 20 30 35 40 50 70 80 87.6 90 100 120

Table A.5

PAPR 25.6318 22.8377 18.1110 17.4649 6.9587 0.0200

CCDF of PAPR 10.773 9.9046 7.664 7.523 3.35 0

)

SNR (BER= 13.6 16.12 17.576 18 19.2 19.2 19.757 20.116 21.13 21.2592 21.372 21.1547 21.62 21.863

)

Companding Results PAPR 18.9061 14.6917 14.3265 14.4401 11.8545 12.9100 12.6028 12.5906 12.1503 11.8738 11.7819 11.7517 11.3543 11.4101

CCDF of PAPR 6.64 4.73 4.126 3.685 3.32 3.413 3.26 3.1 2.837 2.7542 2.723 2.739 2.58 2.535

Companding Results PAPR

5 10 20 30 40 50 60 70

SNR (BER= 11.64 11.63 11.628 11.628 11.582 11.469

CCDF of PAPR 6.416 5.77 5.078 4.4 4 3.8 3.66 3.646

17.4332 16.5470 17.2573 14.6356 13.8453 13.8622 13.7327 14.2237 A. 3

SNR (BER= 13.3363 14.64 16.25 17.165 17.75 18.27 18.777 19

)

Appendix A: 80 90 100 120 140 160 180 200 220 240 250 255 260 280 300 320 500 700 1000

Tables of Results 16.5501 12.8584 12.4142 12.1142 12.7379 12.2083 12.5858 12.2423 11.2722 15.9144 12.4358 11.7434 11.9863 11.5891 11.9073 12.3703 11.1951 10.8218 12.8953

3.33 3.3 3.12 3 3.06 2.866 2.88 2.76 2.645 2.85 2.723 2.68 2.666 2.6 2.61 2.7 2.3 2.17 2.28

19.474 19.6 20 20.28 20.5 20.475 21.05 21 21.2385 21.6 21.6 21.468 21.7 21.9 22 22.125 22.92 23.5 23.764

Table A.6 RCT Results R .9 .8 .7 .6 .5 .4 .3 .2 .1

PAPR 21.8631 21.1311 18.1291 15.3547 13.9264 11.5292 8.5529 5.9888 2.8726

CCDF of PAPR 9.55 8.6815 8.058 6.6825 5.835 4.8215 3.71 2.5745 1.268

SNR (BER= 11.6765 11.987 12.4137 13.4 14.45 16.145 18.525 22.25 28.3

CCDF of PAPR 5.1533 5.14 4.9185 4.77 4.585 4.358 4.1465 3.98 3.806 3.5995 3.374 3.192 2.9264 2.637

SNR (BER= ) 14.73 14.7 14.858 14.45 14.2 14.3138 14.5685 14.3 15.33 14.85 15.3 16.3 24.833 30 ( )

)

Table A.7 AEXP Companding Results AEXP d 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 .9 .8 .7

PAPR 13.0811 13.0240 12.1983 11.2173 10.6962 10.0664 9.6145 8.9815 8.4500 7.9523 7.3774 6.7500 6.0806 5.5253 A. 4

Appendix A: .6 .5 .4 .3 .2 .1

Tables of Results 4.7892 4.1344 3.4039 2.6518 1.8358 0.9690

2.34 2.1075 1.74 1.3272 .9425 .5116

30 ( 30 ( 30 ( 30 ( 30 ( 30 (

) ) ) ) ) )

Table A.8 Cos Companding Results y 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 .9 .8 .7 .6 .5 .4 .3 .2 .1

PAPR 26.3137 25.5112 24.6392 23.3428 22.8762 21.4160 20.1972 19.4063 18.1770 16.8064 15.6468 14.3909 13.1204 11.7106 10.2401 8.7575 7.2067 5.5700 3.8470 1.9930

CCDF of PAPR 11.355 10.858 10.47 10.129 9.682 9.256 8.875 8.5 7.919 7.48 6.9508 6.3817 5.796 5.2196 4.6249 3.9743 3.3453 2.59 1.8074 .9208

SNR (BER= 15.42 15 14.255 13.75 13.675 12.765 12.25 11.95 12 11.832 11.7031 12.1428 12.6966 13.6465 14.2953 15.7648 17.7538 29.2836 >30 >>30

)

Table A.9 tanhR Companding Results k

PAPR

5 5 5 5 10 10 10 10 10

1 .8 .5 .2 1 .9 .8 .7 .6

16.7129 13.8472 8.9043 3.5446 8.9016 8.5570 8.2624 7.7099 7.0350 A. 5

CCDF of PAPR 7.4076 6.0581 3.8861 1.5275 4.1605 3.9969 3.915 3.5129 3.2427

SNR (BER= 12.1294 12.6712 15.5877 24.7231 14.504 14.8376 15.1789 15.8577 16.8

)

Appendix A:

Tables of Results

10 10 10 10 10 15 15 15 15 20 20 20 20

.5 .4 .3 .2 .1 1 .8 .5 .2 1 .8 .5 .2

6.5160 5.4230 4.4318 3.0296 1.6412 5.2314 5.2429 4.9419 2.7437 3.3781 3.5950 3.7761 2.5710

2.9176 2.4367 1.9475 1.3133 .6988 2.5987 2.5265 2.2321 1.2027 1.8228 1.8412 1.7895 1.1315

17.8871 20.2622 22.3763 25.7854 >30 29.6 22.4656 21.4747 26.5576 30( ) 30 30 28.9392

Table A.10 tanhR Companding Results at y =1 k

PAPR

CCDF of PAPR

5 10 15 20

1 1 1 1

16.4627 8.9312 5.2314 3.3781

7.2165 4.209 2.5987 1.8228

SNR at BER 11.9245 14.6486 30 30( )

Table A.11 tanhR Companding Results at y =0.8 k

PAPR

CCDF of PAPR

5 10 15 20

.8 .8 .8 .8

13.4361 8.1480 5.2429 3.5950

5.8816 3.7579 2.5265 1.8412

SNR (BER=

)

12.817 14.9321 22.4656

Table A.12 logR Companding Results y

1 1 1 .9 .8 .7 .6 .5 .4 .3 .2

1 5 10 10 10 10 10 10 10 10 10

PAPR

23.9381 19.4187 16.7420 15.7785 14.7339 12.3933 11.1271 9.7424 8.1924 5.9023 4.0933

CCDF OF PAPR 10.14 8.54 7.3145 6.783 6.3775 5.526 4.87 4.24 3.532 2.625 1.8

SNR (BER= 11.65 12.07 12.5 12.712 13.2 13.8265 14.7521 15.75 17.27 20 23.65 A. 6

PAPR

) 1 5 20 20 20 20 20 20 20 20 20

23.9381 19.4187 14.4171 13.5847 12.8391 11.3170 10.1279 8.9142 7.4148 5.8346 4.1963

CCDF OF PAPR 10.14 8.54 6.291 6.078 5.53 5 4.4434 3.928 3.317 2.526 1.892

SNR (BER= 11.65 12.07 13.616 13.913 13.838 14.54 15.325 16.4332 18.082 20.237 23.9

)

Appendix A: .1 1 .9 .8 .7 .6 .5 .4 .3 .2 .1 1 .9 .8 .7 .6 .5 .4 .3 .2 .1 1 .9 .8 .7 .6 .5 .4 .3 .2 .1 1 .9 .8 .7 .6 .5 .4 .3 .2 .1

10 30 30 30 30 30 30 30 30 30 30 50 50 50 50 50 50 50 50 50 50 70 70 70 70 70 70 70 70 70 70 90 90 90 90 90 90 90 90 90 90

2.1028 13.7103 12.5344 11.5140 10.4388 9.6178 8.4340 7.2406 5.7490 3.7918 2.2351 11.6535 11.1455 10.2693 9.9933 9.0562 7.7808 6.5612 5.3972 3.7525 2.0379 10.3556 10.3901 9.7716 8.9933 8.8132 7.6750 6.3892 5.5562 3.9753 2.2114 9.6991 9.6399 9.0726 8.7749 8.2079 7.0926 6.6478 6.0873 3.6976 2.1337

Tables of Results .932 5.76 5.465 5.02 4.538 4.18 3.6 3.15 2.5 1.686 .882 5.08 4.803 4.492 4.2685 3.94 3.412 2.8765 2.306 1.668 .908 4.592 4.5 4.3 3.935 3.64 3.366 2.83 2.446 1.705 .9335 4.42 4.282 3.95 3.707 3.488 3.124 2.889 2.24 1.6 .89

29.6 14.7383 14.71 14.7 15.37 16 17.0828 18 20.627 23.78 30 16.58 16.82 16.26 16.478 16.93 17.424 18.893 21 24.4 29.6 20 18.3715 17.725 17.65 17.43 18.128 19.15 21.25 23.88 29.6 30 21.9 20 18.85 19.337 18.9814 19.16 21.5175 24.36 29.6

A. 7

20 40 40 40 40 40 40 40 40 40 40 60 60 60 60 60 60 60 60 60 60 80 80 80 80 80 80 80 80 80 80 100 100 100 100 100 100 100 100 100 100

2.0352 12.4590 11.7370 10.8267 10.1572 9.0205 8.3535 7.3063 5.4107 3.8688 2.1230 11.7483 10.3340 10.2586 9.5086 8.4069 7.8869 6.5492 5.2164 4.0939 2.1340 10.4009 10.2191 9.6145 8.7912 8.0294 7.5870 6.6794 5.1697 3.6822 2.0227 10.0104 9.7016 9.2385 8.2944 7.9701 7.1942 6.2888 5.0748 3.6944 2.0774

.892 5.417 5.2 4.7 4.5 4 3.62 3.05 2.4 1.6785 .885 4.6 4.515 4.483 4.0443 3.685 3.5 2.6864 2.2685 1.6474 .8863 4.6 4.2185 4.288 3.84 3.56 3.3025 3 2.28 1.626 .88 4.225 4.238 4.0666 3.65 3.504 3.2 2.755 2.275 1.63 .9

29.4665 15.6865 15.9 16 15.68 16.4 17.612 18.95 20.778 23.8868 30 18.2856 17.52 17.08 17.36 17.085 18 19.03 21.056 23.888 29.6 24.085 20 19.1 17.638 18.18 17.3734 19.4576 21.188 24.158 29.6 30 24.5 20.46 18.745 19.13 19.45 19.816 21.6 24.4 29.6

Appendix A:

Tables of Results

A.5 Hybrid RCF with companding Results:

X = PAPR (Companding ) – PAPR (Companding +RCF) Y =CCDF of PAPR (Companding) - CCDF of PAPR (AV+RCF) Z= SNR (BER= ) (Companding) – SNR (BER= ) (Companding +RCF) X1 == PAPR (RCF) – PAPR (Companding +RCF) Y1 =CCDF of PAPR (RCF) - CCDF of PAPR (Companding +RCF) Z1= SNR (BER= ) (RCF) – SNR (BER= ) (Companding +RCF) Table A.13 (RCF+A) Results and compared with the results of each of (RCF) and (A companding) A

PAPR

5 10 20 30 40 50 60 70 80 87.6 90 100 120 140 160

2 2 2 2 2

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

10.4351 7.9653 8.7683 9.3538 7.0551 8.0339 8.1579 7.8844 7.7334 7.6864 6.9071 7.3797 7.4983

6.0222 7.7668 8.9350 9.4069 9.6938 9.9243 10.0605 10.2273 10.3528 10.3977 9.6486 10.5186 10.5814 10.6902 10.7995

8.4710 6.7264 5.5582 5.0863 4.7994 4.5689 4.4327 4.2659 4.1404 4.0955 4.8446 3.9746 3.9118 3.8030 3.6937

2.7730 1.6650 1.6433 1.4150 1.1950 1.2070 1.1400 0.9340 0.9127 0.8872 0.9250 0.8150 0.8535

2.5507 3.3527 3.9350 4.1477 4.2927 4.3647 4.4577 4.5147 4.5762 4.5819 4.6037 4.6527 4.7362 4.7527 4.7847

A. 8

CCDF of PAPR 3.867 3.065 2.4827 2.27 2.125 2.053 1.96 1.903 1.8415 1.8358 1.814 1.765 1.6815 1.665 1.633

2.5095 3.1882 2.5760 1.8420 2.2000 2.3762 2.4280 2.6545 2.8042 2.7380 2.6547 2.9200 3.0930

-2.3226 -4.1639 -6.2321 -7.3901 -8.2321 -8.6129 -8.9201 -9.7076 -9.6871 -9.8661 -9.7321 -9.9321 -10.0021 -10.3171 -10.8121

SNR (BER= ) 11.0905 12.9318 15 16.158 17 17.3808 17.688 18.4755 18.455 18.634 18.5 18.7 18.77 19.085 19.58

Appendix A: 180 200 5 10 20 30 40 50 60 70 80 87.6 90 5 10 20 30 40 50 60 70 80 87.6 90 5

2 2 2

4 4 3 3 3

2 2 2

1.5

11.6867 8.8266 9.3274 9.8178 7.5657 8.5241 8.6321 8.2973 8.0553 8.0755 8.0445 13.0333 9.7802 10.0069 10.5073 8.1122 8.9745 9.0741 8.6284 8.5552 8.3071 8.4543 13.4875

10.8244 10.8726 5.0548 6.4091 7.2751 7.6519 7.9854 8.1955 8.3157 8.4212 8.4557 8.5678 8.5670 3.7345 4.6958 5.2877 5.6745 5.8650 5.9790 6.0908 6.0854 6.2887 6.1325 6.3099 2.9643

Tables of Results 3.6688 3.6206 7.2194 5.8651 4.9991 4.6223 4.2888 4.0787 3.9585 3.8530 3.8185 3.7064 3.7072 5.8728 4.9115 4.3196 3.9328 3.7423 3.6283 3.5165 3.5219 3.3186 3.4748 3.2974 5.4186

3.4560 2.1970 2.0140 1.7750 1.5435 1.5333 1.4440 1.2262 1.1762 1.1790 1.1990 4.1750 2.7300 2.5260 2.2375 1.9420 1.9300 1.8065 1.6420 1.5542 1.5230 1.5570 4.64

4.8297 4.8657 2.1710 2.8220 3.2430 3.4450 3.5785 3.6283 3.6990 3.7442 3.7770 3.8110 3.8150 1.6566 2.1216 2.5216 2.6741 2.7436 2.7916 2.8281 2.9266 2.9216 2.9216 2.9396 1.6257

1.588 1.552 3.184 2.533 2.112 1.91 1.7765 1.7267 1.656 1.6108 1.578 1.544 1.54 2.465 2 1.6 1.4475 1.378 1.33 1.2935 1.195 1.2 1.2 1.182 2

1.6000 2.1200 1.4040 0.8800 1.4000 1.2927 1.3397 2.0412 2.0229 1.8220 1.9285 -3.0400 -2.6729 -2.8240 -2.7420 -1.8000 -2.7930 -2.1605 -1.7200 -1.2408 -1.4280 -3.0453 -16.4

Table A.14 (RCF+ ) Results and compared with the results of each of (RCF) and ( companding) A. 9

-10.9921 -11.1128 -2.5860 -4.5860 -6.7580 -7.7060 -8.3860 -9.0503 -9.3623 -9.6748 -9.8223 -10.1360 -9.8122 -4.5120 -6.6649 -8.2720 -8.6140 -8.8720 -10.4220 -10.1485 -10.7220 -10.3720 -10.6720 -12.0720 0

19.76 19.8807 12 14 16.172 17.12 17.8 18.4643 18.7763 19.0888 19.2363 19.55 19.2262 16.64 18.7929 20.4 20.742 21 22.55 22.2765 22.85 22.5 22.8 24.2 30

Appendix A:

5 10 20 30 40 50 60 70 80 90 100 120 140 160 180 200 220 240 255 5 10 20 30 40 50 60 70

Tables of Results

PAPR

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

4 4 4 4

8.3398 8.7462 10.6280 8.5826 8.1186 8.3905 8.5050 9.1783 11.6283 8.0138 7.5968 7.5493 8.2890 7.8826 8.3477 8.0239 7.1092 11.8374 7.7076 9.6333 9.7698 11.4793 9.3130 8.8229 9.0029 9.0355 9.7156

5.3998 6.6924 7.8639 8.4402 8.7665 9.0215 9.2655 9.4478 9.5714 9.6486 9.6758 9.9283 10.0443 10.1675 10.2551 10.2748 10.3302 10.4162 10.4574 4.4743 5.4970 6.4962 6.9516 7.2518 7.4149 7.5770 7.7661

9.0934 7.8008 6.6293 6.0530 5.7267 5.4717 5.2277 5.0454 4.9218 4.8446 4.8174 4.5649 4.4489 4.3257 4.2381 4.2184 4.1630 4.0770 4.0358 7.7999 6.7772 5.7780 5.3226 5.0224 4.8593 4.6972 4.5081

2.3230 2.2700 2.1110 1.6640 1.4470 1.3690 1.3100 1.3860 1.1230 1.1540 1.0110 0.9738 1.0600 0.9080 1.0160 0.9200 0.8325 1.0750 0.9300 2.9530 2.8265 2.5780 2.1220 1.8440 1.7330 1.6600 1.7260

2.3247 2.9177 3.4507 3.6817 3.8647 3.9867 4.0677 4.1577 4.2107 4.2717 4.3087 4.3915 4.4177 4.4597 4.5537 4.5777 4.6052 4.6427 4.6677 1.8920 2.4115 2.8550 3.0770 3.1990 3.2880 3.3550 3.4350

3 3 3

A. 10

CCDF OF PAPR 4.093 3.5 2.967 2.736 2.553 2.431 2.35 2.26 2.207 2.146 2.109 2.0262 2 1.958 1.864 1.84 1.8125 1.775 1.75 3.463 2.9435 2.5 2.278 2.156 2.067 2 1.92

2.7063 2.8310 2.7100 2.9360 2.8084 3.0554 2.8770 2.7650 2.9880 2.7524 2.9370 2.7300 2.8215 2.7580 2.8232 2.7820 2.8935 2.9780 2.7810 1.8508 1.7762 2.1180 2.1650 1.7760 1.8700 1.8585 2.1287

-1.8621 -3.0411 -4.7721 -5.4611 -6.1737 -6.4467 -7.1321 -7.4671 -7.7181 -8.0797 -8.2951 -8.7821 -8.9106 -8.9491 -9.4589 -9.4501 -9.5771 -9.8541 -9.9191 -2.0715 -3.4498 -4.7180 -5.5860 -6.5600 -6.9860 -7.5045 -7.4573

SNR (BER= ) 10.63 11.809 13.54 14.229 14.9416 15.2146 15.9 16.235 16.486 16.8476 17.063 17.55 17.6785 17.717 18.2268 18.218 18.345 18.622 18.687 11.4855 12.8638 14.132 15 15.974 16.4 16.9185 16.8713

Appendix A: 80 90 100 120 140 160 180 200 220 240 255 5 10 20 30 40 50 60 70 80 90 100 120 140 160 180 200 220 240

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2 2 2

2 2

12.1143 8.4708 8.0750 7.9673 8.6065 8.2211 8.7098 8.4603 7.9536 12.2074 8.0323 10.9878 10.9800 12.1035 10.1735 9.4721 9.5534 9.7161 10.2669 12.6705 9.0121 8.6436 8.5096 9.1446 8.7169 8.9937 8.7313 7.9605 12.5271

7.8384 7.8866 7.9350 8.1273 8.1428 8.2870 8.3982 8.4922 8.9556 8.5672 8.5631 3.1619 4.0403 4.4535 5.1452 5.2341 5.2985 5.5907 5.6505 5.7277 5.7610 5.8367 6.0027 6.0140 6.1159 6.0152 6.0963 6.2956 6.2200

Tables of Results 4.4358 4.3876 4.3392 4.1469 4.1314 3.9872 3.8760 3.7820 3.3186 3.7070 3.7111 6.4454 5.5670 5.1538 4.4621 4.3732 4.3088 4.0166 3.9568 3.8796 3.8463 3.7706 3.6046 3.5933 3.4914 3.5921 3.5110 3.3117 3.3873

1.4350 1.4885 1.3600 1.3040 1.3750 1.2240 1.2734 1.2350 1.0450 1.3300 1.1800 3.7600 3.5200 3.1310 2.6080 2.3440 2.1870 2.1380 2.1940 1.8640 1.8850 1.7370 1.6740 1.7565 1.5840 1.6390 1.5600 1.4520 1.6650

A. 11

3.4600 3.5435 3.5950 3.6590 3.6700 3.7130 3.7484 3.8300 3.7550 3.8350 3.8550 1.4656 1.8716 2.1746 2.3296 2.4656 2.5086 2.5996 2.6696 2.6556 2.7066 2.7386 2.7956 2.8181 2.8396 2.8806 2.9216 2.9286 2.9366

1.895 1.8115 1.76 1.696 1.685 1.642 1.6066 1.525 1.6 1.52 1.5 2.656 2.25 1.947 1.792 1.656 1.613 1.522 1.452 1.466 1.415 1.383 1.326 1.3035 1.282 1.241 1.2 1.193 1.185

2.2240 2.1840 2.2374 2.2800 2.0000 1.8250 2.1320 1.6372 2.0058 2.2420 1.4680 -1.6637 -2.6100 -2.6250 -1.6850 -1.8810 -1.6875 -1.2230 -2.1380 -2.2047 -1.7000 -1.7400 -1.7200 -1.9200 -2.1100 -1.1377 -1.2575 -1.7615 -0.9850

-7.8360 -8.0020 -8.3486 - 8.5860 -9.0860 -9.2360 -9.5040 -9.9488 -9.8187 -9.9440 -10.5860 -2.8720 -5.1220 -6.7470 -6.7220 -7.5030 -7.8295 -7.8720 -9.0100 -9.5507 -9.1720 -9.6120 -9.8720 -10.2920 -10.4570 -10.0597 -10.1295 -10.8720 -10.4570

17.25 17.416 17.7626 18 18.5 18.65 18.918 19.3628 19.2327 19.358 20 15 17.25 18.875 18.85 19.631 19.9575 20 21.138 21.6787 21.3 21.74 22 22.42 22.585 22.1877 22.2575 23 22.585

Appendix A: 255 5

2 2

8.3049 11.8784

1.5

6.1688 2.8281

Tables of Results 3.4385 5.5548

1.5205 4.1715

2.9621 1.4805

1.1595 2.2445

-1.4387 -16.6637

-10.7787 0

22.9067 >30

Table A.15 (RCF+ RCT) Results and compared with the results of each of (RCF) and (RCT) R .9 .8 .7 .6 .5 .4 .3 .2 .1 .9 .8 .7 .6 .5 .4 .3 .2 .1 .9 .8 .7

I 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

PAPR

8.4451 8.8719 7.0561 5.5844 5.5235 4.5640 3.1480 2.2574 0.9227 10.5065 10.7468 8.7402 7.0046 6.7408 5.5512 3.9120 2.7595 1.1778 12.9726 12.9771 10.6546

1.0752 2.2340 3.4202 4.7229 6.0903 7.5280 9.0883 10.7618 12.5433 0.9176 1.8899 2.8853 3.9241 5.0886 6.2962 7.6333 9.0449 10.5794 0.7168 1.4533 2.1328

13.4180 12.2592 11.0730 9.7703 8.4029 6.9652 5.4049 3.7314 1.9499 11.3566 10.3843 9.3889 8.3501 7.1856 5.9780 4.6409 3.2293 1.6948 8.8905 8.1540 7.4745

3.4485 3.0435 3.0450 2.2125 2.0020 1.6643 1.2494 0.8589 0.3630 4.4625 3.8165 3.8460 2.9393 2.5280 2.0849 1.5900 1.0745 0.4965 5.5790 5.0235 4.8455

0.3162 0.7797 1.4047 1.9477 2.5847 3.2605 3.9571 4.7021 5.5127 0.2675 0.4900 1.1430 1.6118 2.0480 2.6184 3.2350 3.8550 4.5835 0.1506 0.4636 0.9091

3 3

2 2 2

A. 12

CCDF OF PAPR 6.1015 5.638 5.013 4.47 3.833 3.1572 2.4606 1.7156 0.905 5.0875 4.865 4.212 3.7432 3.307 2.7366 2.12 1.5 .7715 3.971 3.658 3.2125

2.7795 2.9000 3.0902 2.8434 2.7734 2.8350 2.7515 3.0366 2.3826 2.0289 2.2570 2.0987 2.4765 2.2000 2.3021 2.0918 2.2500 1.8866 -0.7535 -1.1263 -0.9963

-0.1291 -0.3191 -0.5556 -1.7887 -2.9087 -4.5421 -7.0056 -10.4455 -17.1495 -0.2336 -0.3160 -0.9010 -1.5095 -2.8360 -4.4289 -7.0192 -10.5860 -16.9994 -0.3020 -0.9853 -1.2820

SNR (BER= ) 8.897 9.087 9.3235 10.5566 11.6766 13.31 15.7735 19.2134 25.9174 9.6476 9.73 10.315 10.9235 12.25 13.8429 16.4332 20 26.4134 12.43 13.1133 13.41

Appendix A: .6 .5 .4 .3 .2 .1 .9

2 2 2 2 2 2 2

2 1.5

8.7222 8.2882 6.8280 4.8319 3.4152 1.5161 13.9836

2.9748 3.9691 4.9061 5.8863 7.0337 8.2508 .5034

Tables of Results 6.6325 5.6382 4.7012 3.7210 2.5736 1.3565 7.8795

3.7330 3.3380 2.7185 2.0413 1.4145 0.6680 6.201

1.1721 1.6246 2.0186 2.4529 2.9616 3.5216 .2767

2.9495 2.497 2.103 1.6687 1.16 0.6 3.349

-1.3655 -0.7285 -1.2550 -1.4750 -0.9680 -1.0090 -183235

-2.6375 -3.0505 -5.2720 -7.8720 -11.0900 -17.1810 0

14.7655 15.1785 17.4 20 23.218 29.309 > 30

Table A.16 (RCF+AEXP) Results and compared with the results of each of (RCF) and (AEXP companding) d

PAPR

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 .9 4.8 .7 .6 .5 .4

4 4 4 4 4

1.6576 2.0456 1.6168 0.9849 0.8895 0.8895 0.7767 0.6955 0.5710 0.5453 0.5075 0.4226 0.3368 0.3553 0.2150 0.2011 0.1356

3.0697 3.5148 3.9117 4.2608 4.6865 5.2963 5.6554 6.2072 6.6142 7.0862 7.6233 8.1658 8.7494 9.3232 9.9190 10.5599 11.2249

11.4235 10.9784 10.5815 10.2324 9.8067 9.1969 8.8378 8.2860 7.8790 7.4070 6.8699 6.3274 5.7438 5.1700 4.5742 3.9333 3.2683

0.3643 0.4800 0.4185 0.4080 0.3930 0.3410 0.2839 0.2534 0.2895 0.2820 0.2475 0.2760 0.2575 0.2310 0.1875 0.2485 0.1400

1.6287 1.7577 1.9177 2.0557 2.2257 2.4007 2.5551 2.6911 2.9012 3.1002 3.2912 3.5017 3.7552 4.0117 4.2652 4.5587 4.8177

A. 13

CCDF OF PAPR 4.789 4.66 4.5 4.362 4.192 4..017 3.8626 3.7266 3.5165 3.3175 3.1265 2.916 2.6625 2.406 2.1525 1.859 1.6

4.2300 3.9830 4.0281 3.7540 3.5558 3.4196 3.9685 3.9000 4.6170 4.0570 4.1153 4.0250 6.5750 15.5893 13.8000 0.7900 0

-1.7321 -1.9491 -2.0620 -1.9281 -1.8763 -2.1263 -1.8321 -1.6321 -1.9451 -2.0251 -2.4168 -3.5071 -3.9021 -5.6428 -7.4321 -20.4421 -21.2321

SNR (BER= ) 10.5 10.717 10.8299 10.696 10.6442 10.8942 10.6 10.4 10.713 10.793 11.1847 12.275 12.67 14.4107 16.2 29.21 > 30

Appendix A: 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 .9 .8 .7 .6 .5 .4 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 .9

3 3 3

2 2 2

2.6002 2.9078 2.3311 1.9316 -1.6855 1.5155 1.3912 1.2246 1.1408 1.0906 0.9998 0.8470 0.6953 0.6599 0.4746 0.4315 0.3165 3.7697 4.0878 3.2999 3.1314 2.9557 2.7710 2.6346 2.3627 2.2259 2.0747 1.8764 1.7267

1.7933 2.1580 2.4070 2.9885 3.2635 3.7233 4.0509 4.5173 4.9650 5.4125 5.8966 6.3712 6.8889 7.4088 7.9596 8.5713 9.1868 0.2959 0.6711 0.7089 1.5214 1.8668 2.3119 2.6274 2.9885 3.3832 3.7297 4.1063 4.5840

10.4809 10.1162 9.8672 9.2857 9.0107 8.5509 8.2233 7.7569 7.3092 6.8617 6.3776 5.9030 5.3853 4.8654 4.3146 3.7029 3.0874 9.3114 8.9362 8.8984 8.0859 7.7405 7.2954 6.9799 6.6188 6.2241 5.8776 5.5010 5.0233

Tables of Results 0.6733 0.8370 0.6955 0.7300 0.6150 0.5820 0.5390 0.5370 0.5380 0.4970 0.4495 0.4920 0.4200 0.3505 0.3400 0.3125 0.2740 1.3368 1.4775 1.3285 1.3250 1.2940 1.1373 1.0605 1.0460 1.0160 0.9575 0.8755 0.8580

0.8750 1.0520 1.1320 1.3150 1.3850 1.5790 1.7475 1.9120 2.0870 2.2525 2.4305 2.6550 2.8550 3.0685 3.3550 3.5600 3.8890 0.3051 0.4591 0.5316 0.6766 0.8306 0.9009 1.0356 1.1876 1.3316 1.4796 1.6231 1.7876

A. 14

4.48 4.303 4.223 4.04 3.97 3.776 3.6075 3.443 3.268 3.1025 2.9245 2.7 2.5 2.2865 2 1.795 1.466 3.8165 3.6625 3.59 3.445 3.291 3.2207 3.086 2.934 2.79 2.642 2.4985 2.334

3.2800 3.2458 3.6960 2.9420 2.4616 2.9138 3.0907 2.7642 3.5735 2.6963 3.1313 3.3500 5.4050 15.2617 9.1820 0 0 -1.0610 -3.5845 -4.1178 -4.8455 -4.3000 -5.0002 -3.6315 -4.1380 -5.6200 -7.6500 -6.3838 -4.0670

-2.0360 -2.0402 -1.7480 -2.0940 -2.3244 -1.9860 -2.0638 -2.1218 -2.3425 -2.7397 -2.7547 -3.5360 -4.4260 -5.3243 -11.4040 -20.5860 -20.5860 -3.6630 -6.1565 -6.8478 -7.1675 -6.3720 -7.1860 -6.0720 -6.3100 -8.8220 -9.5558 -10.3720 -8.2390

11.45 11.4542 11.162 11.508 11.7384 11.4 11.4778 11.5358 11.7565 12.1537 12.1687 12.95 13.84 14.7383 20.818 > 30 > 30 15.791 18.2845 18.9758 19.2955 18.5 19.314 18.2 18.438 20.95 22.5 21.6838 20.367

Appendix A: .8 .7 .6 .5 .4 2

1.5

1.4732 1.3760 1.0899 0.9386 0.7417 4.7537

4.9999 5.4580 5.9080 6.4115 6.9451 .0555

Tables of Results 4.6074 4.1493 3.6993 3.1958 2.6622 8.3274

0.7860 0.6705 0.5820 0.6000 0.4680 1.7799

1.9876 2.1551 2.3636 2.6141 2.8496 .4116

2.134 1.9665 1.758 1.5075 1.272 3.3734

-2.5675 -0.9000 0 0 0 -15.27

-9.6845 -16.9720 -17.8720 -17.8720 -17.8720 0

21.8125 29.1 >30 >30 >30 >30

Table A.17 (RCF+ cos) Results and compared with the results of each of (RCF) and (cos companding) y

PAPR

1 .9 .8 .7 .6 .5 .4 .3 .2 .1 1 .9 .8 .7 .6 .5 .4 .3

3.0167 2.7783 2.4748 2.0799 1.7684 1.4604 1.1521 0.8658

4.7448 4.3655 3.9220 3.2140 2.7977 2.2782 1.8853 1.4064

12.5439 11.6223 10.6347 9.5893 8.5374 7.3086 6.0541 4.7101 3.2819 1.7016 10.8158 10.0351 9.1875 8.4552 7.5081 6.4908 5.3209 4.1695

1.3219 1.2531 1.1279 0.9978 0.7710 0.6732 0.5450 0.4597

1.9493 2.8709 3.8585 4.9039 5.9558 7.1846 8.4391 9.7831 11.2113 12.7916 1.4584 2.2391 3.0867 3.8190 4.7661 5.7834 6.9533 8.1047

0.8896 1.2548 1.6806 2.1340 2.5857 3.1359 3.6837 4.2774 4.9241 5.6361 0.5467 0.9077 1.2406 1.5801 2.0047 2.4661 2.9620 3.4718

2.0417 1.9687 1.7506 1.5066 1.2527 1.0661 0.8860 0.7168

A. 15

CCDF OF PAPR 5.5281 5.1629 4.7371 4.2837 3.832 3.2818 2.734 2.1403 1.4936 .7816 4.8083 4.4473 4.1144 3.7749 3.3503 2.8889 2.393 1.8832

2.7245 2.9981 2.8724 3.2051 3.3736 3.2650 3.3207 4.4287

-0.3076 -0.4590 -0.9397 -1.4110 -2.3970 -3.8271 -5.4714 -8.1034 > -21.2321 > -21.2321 -0.3250 -0.4264 -1.0708 -1.4924 -2.7060 -3.8565 -5.9436 -7.7338

2.0610 2.3846 2.0952 2.4776 2.4185 2.5895 2.2024 4.1522

SNR (BER= ) 9.0755 9.2269 9.7076 10.1789 11.1649 12.595 14.2393 16.8713 29.6 30 9.739 9.8404 10.4848 10.9064 12.12 13.2705 15.3576 17.1478

Appendix A: .2 .1 1 .9 .8 .7 .6 .5 .4 .3 .2

7.1336 6.4694 5.7883 5.0089 4.4710 3.6352 2.9108 2.2328

.1 1

1.5

8.2456

Tables of Results

9.3722 10.7498 1.1803 1.6761 2.2861 2.9470 3.7725 4.4735 5.3119 6.2642 7.3113

2.9020 1.5244 8.4270 7.9312 7.3212 6.6603 5.8348 5.1338 4.2954 3.3431 2.2960

8.3921

1.2152

1.0679

7.3150

3.0466 2.8082 2.5530 2.2845 1.9571 1.6068 1.3159 1.0410

3.4937

4.0344 4.6579 0.3182 0.5138 0.8096 1.1246 1.4757 1.7734 2.1585 2.5626 3.0777

1.3206 .6971 3.8034 3.6078 3.312 2.997 2.6459 2.3482 1.9631 1.559 1.0439

3.5707

.5509

0.2694

3.3563

-17.6437 > -20.5860 -0.4563 -0.7894 -1.3324 -2.2901 -2.7810 -6.9048 -4.6413 -17.4720 > -17.8720

-0.7843 -0.6924 -0.8804 -1.0341 -0.3705 -3.1728 -0.7907 -8.3000

> -17.8720 >-18.2000

27.0577 29.6 12.5843 12.9174 13.4604 14.4181 14.909 19.0328 16.7693 29.6 29.6 29.6 29.6

Table A.18 (RCF+NERF) Results and compared with the results of each of (RCF) and (NERF companding) NERF

PAPR

4 3 2 1.5

0.9191 1.6395 2.7940 3.0791

5.2328 3.7342 2.2218 1.2825

9.2604 8.5400 7.3855 7.1004

0.4355 0.8685 1.5235 1.8955

2.4177 1.7880 1.2096 1.0857

CCDF OF PAPR 4 3.567 2.912 2.54

3.2135 2.7780 -2.3220 -16.3220

-1.6966 -1.4860 -3.8720 -0.4000

SNR (BER= ) 10.4645 10.9 16 > 30

Table A.19 (RCF+tanhR) Results and compared with the results of each of (RCF) and (tanhR companding) k

PAPR

A. 16

CCDF

SNR

Appendix A:

5 5 5 5 10 10 10 10 15 15 15 15 20 20 20 20 30 30 30 30 40 40 40 40 5 5 5

1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5

4 4 4 4

2.8874 2.4646 1.6787 0.8605 -3.0836 -1.7229 0.1334 0.6159 -4.7170 -3.0465 -0.5541 0.4601 -4.5883 -3.3107 -1.1082 0.4112

3 3 3

4.9487 4.0556 2.6254

0.6677 3.1106 7.2676 11.8091 2.4784 4.6223 8.2269 12.0589 4.5448 6.2038 8.9972 12.2096 6.5268 7.5875 9.6089 12.3334 9.4116 9.6678 10.5681 12.5057 11.1190 11.0445 11.2677 12.6433 0.5100 2.4826 5.9953

Tables of Results

13.8255 11.3826 7.2256 2.6841 12.0148 9.8709 6.2663 2.4343 9.9484 8.2894 5.4960 2.2836 7.9664 6.9057 4.8843 2.1598 5.0816 4.8254 3.9251 1.9875 3.3742 3.4487 3.2255 1.8499 11.7642 9.7916 6.2789

1.1576 0.8777 0.5861 0.2635 -1.3540 -0.7381 -0.1059 0.1921 -1.9818 -1.3095 -0.3264 0.1342 -2.0457 -1.4458 -0.5220 0.1085

2.0796 1.6441 0.9621

A. 17

0.1677 1.2373 3.1177 5.1537 0.8547 1.9217 3.4837 5.2647 1.8372 2.5817 3.8592 5.3492 2.5492 3.1307 4.1062 5.3947 3.8792 4.0477 4.5127 5.4807 4.5622 4.6592 4.8527 5.5377 0.0270 0.9410 2.4310

OF PAPR 6.25 5.1804 3.3 1.264 5.563 4.496 2.934 1.153 4.5805 3.836 2.5585 1.0685 3.8685 3.287 2.3115 1.023 2.5385 2.37 1.905 .937 1.8555 1.7585 1.565 .88 5.328 4.414 2.924

3.1158 3.3330 3.5127 4.1231 5.6486 5.2944 4.9151 5.0159 19.8030 12.2656 8.1255 4.8634 19.0682 18.3213 15.0955 7.1422

2.5768 2.8712 2.6994

- 0.2457 -0.5703 -3.3071 -11.8321 -0.2321 -0.8698 -3.9459 -12.2321 -1.0291 -1.4321 -4.5813 -12.9263 -1.7639 -2.5108 -5.7366 -13.0291 -4.7878 -4.8484 -6.9271 -14.4491 -19.5481 -10.3821 -9.1149 -14.5471 -0.1386 -0.3860 -3.4743

(BER= ) 9.0136 9.3382 12.075 20.6 9 9.6377 12.7138 21 9.797 10.2 13.3492 21.6942 10.5318 11.2787 14.5045 21.797 13.5557 13.6163 15.695 23.217 28.316 19.15 17.8828 23.315 9.5526 9.8 12.8883

Appendix A: 5 10 10 10 10 15 15 15 15 20 20 20 20 30 30 30 30 40 40 40 40 50 50 50 50 5 5 5 5

.2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2

2 2

3 3

2 2 2

1.1979 -1.6137 -0.5458 0.8648 0.9020 -3.7961 -2.2420 0.0189 0.7330 -4.1116 -2.8070 -0.6404 0.6658

7.3271 5.9194 3.7924 1.6223

9.9275 1.7293 3.5804 6.7393 10.1260 3.2467 4.7893 7.3512 10.2635 4.7845 5.8722 7.8577 10.3690 7.2887 7.6359 8.6651 10.5174 8.8946 8.8910 9.2710 10.6357 9.8817 9.7344 9.7324 9.8883 0.2215 1.6795 4.4954 7.6850

Tables of Results 2.3467 10.5449 8.6938 5.5349 2.1482 9.0275 7.4849 4.9230 2.0107 7.4897 6.4020 4.4165 1.9052 4.9855 4.6383 3.6091 1.7568 3.3796 3.3832 3.0032 1.6385 2.3925 2.5398 2.5418 2.3859 9.3858 7.9278 5.1119 1.9223

0.4465 -0.6095 -0.1761 0.2921 0.3341 -1.5233 -0.9965 -0.1039 0.2677 -1.6947 -1.1964 -0.2755 0.2270

3.2596 3.0001 1.6301 0.6385

A. 18

4.2740 0.5365 1.4210 2.8190 4.3440 1.2330 1.8320 3.0190 4.4200 1.8375 2.3174 3.2900 4.4505 2.8900 3.0680 3.6140 4.5395 3.5970 3.6190 3.9270 4.5790 4.0150 4.0445 4.0990 4.0850 -0.0264 1.0636 1.8656 3.2326

1.081 4.8185 3.934 2.536 1.011 4.122 3.523 2.336 .935 3.5175 3.0376 2.065 .9045 2.465 2.287 1.741 .8155 1.758 1.736 1.428 .776 1.34 1.3105 1.256 1.27 4.148 3.058 2.256 .889

3.0276 4.6886 3.8892 3.9912 3.4909 19.2188 11.2571 7.4021 4.5003 18.1168 17.4851 14.6743 6.0819

-0.4980 -0.1100 -0.8703 -0.9024

-12.2815 -0.5460 -1.6289 -4.2237 -13.1110 -0.9672 -1.7945 -4.6586 -12.6433 -2.0692 -2.7009 -5.5117 -13.4433 -5.1985 -5.6554 -7.1240 -14.2860 -20.5860 -13.3568 -10.2676 -15.1671 >-20.5860 >-20.5860 -15.2120 >-20.5860 -0.4994 -0.6532 -4.3300 -13.4975

21.6955 9.96 11.0429 13.6377 22.525 10.3812 11.2085 14.0726 22.0573 11.4832 12.1149 14.9257 22.8573 14.6125 15.0694 16.538 23.7 >30 22.7708 19.6816 24.5811 >30 >30 24.626 >30 12.6274 12.7812 16.458 25.6255

Appendix A: 10 10 10 10 15 15 15 15 20 20 20 20 30 30 30 30 40 40 40 40 5

1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1

0.1714 0.8666 1.7957 1.2846 -2.6181 -1.1756 0.7929 1.0662 -3.4074 -2.0768 0.0043 0.9859

1.5

8.0365

0.8475 2.3259 5.0033 7.8417 1.7578 3.1888 5.4583 7.9298 2.8218 3.9355 5.8355 8.0222 4.6925 5.2686 6.4575 8.1430 6.1219 6.2434 6.8934 8.2333 0.0685

Tables of Results 8.7598 7.2814 4.6040 1.7656 7.8495 6.4185 4.1490 1.6775 6.7855 5.6718 3.7718 1.5851 4.9148 4.3387 3.1498 1.4643 3.4854 3.3639 2.7139 1.3740 8.3144

0.3585 0.4979 0.7281 0.5421 -0.9813 -0.3955 0.3171 0.4467 -1.2977 -0.7468 -0.0085 0.3955

0.2711 0.8616 2.0216 3.3186 0.5416 1.1996 2.2066 3.3656 1.0011 1.5336 2.3236 3.3856 1.7481 1.9186 2.6191 3.4246 2.2986 2.4566 2.8286 3.4806 0.0729

3.6052

3.8505 3.26 2.1 .803 3.58 2.922 1.915 0.756 3.1205 2.588 1.798 .736 2.3735 2.203 1.5025 .697 1.823 1.665 1.293 .641 3.5528

2.0108 1.2845 -0.7796 -2.4641 15.0296 7.5001 1.1217 -2.1924 13.0242 12.6110 6.3155 -0.6608

>-18.0900

-0.5098 -1.5196 -6.2805 -16.3520 -2.4424 -2.8375 -8.2250 -16.6220 -4.4478 -4.8610 -11.1565 -17.4720 -17.8720 -14.4193 -17.3187 >-17.8720 >-17.8720 >-17.8720 >-17.8720 >-17.8720 0

12.6378 13.6476 18.4085 28.48 14.5704 14.9655 20.353 28.75 16.5758 16.989 23.2845 29.6 >30 26.5473 29.4467 >30 >>30 >>30 >30 >30 >30

Table A.20 (RCF+logR) Results and compared with the results of each of (RCF) and (logR companding) K

5 5 5

1 .8 .5

4 4 4

PAPR

1.2906 3.5653 7.4063

13.2026 10.9279 7.0869

Y1 0.5137 1.4507 3.1367

A. 19

CCDF OF PAPR 5.904 4.967 3.281

Z1 -0.2189 -0.5014 -3.8893

SNR (BER= ) 8.9868 9.2693 12.6572

Appendix A: 5 10 10 10 10 20 20 20 20 30 30 30 30 40 40 40 40 50 50 50 50 70 70 70 70 90 90 90 90

.2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

4.4581 4.5315 3.0208 1.2544 3.3770 3.5281 2.5891 1.4327 3.4707 2.7496 2.3515 1.0726 2.8036 2.4673 2.4487 1.1876 2.4698 2.2419 2.0257 1.0953 1.8443 2.2212 2.1335 1.3557 1.6931 1.8910 1.7246 1.1122

11.5744 2.2093 4.2908 7.7716 11.6543 3.4531 5.1822 8.1681 11.7296 4.2536 5.7288 8.4107 11.7740 4.8378 6.1338 8.5884 11.8120 5.3095 6.4658 8.7381 11.8360 5.9819 6.9428 8.9517 11.8736 6.4872 7.3116 9.1252 11.9078

Tables of Results 2.9188 12.2839 10.2024 6.7216 2.8389 11.0401 9.3110 6.3251 2.7636 10.2396 8.7644 6.0825 2.7192 9.6554 8.3594 5.9048 2.6812 9.1837 8.0274 5.7551 2.6572 8.5113 7.5504 5.5415 2.6196 8.0060 7.1816 5.3680 2.5854

1.7353 1.6805 1.1055 0.4720 1.1940 1.2650 1.0200 0.5980 1.0538 0.9815 0.7820 0.4210 1.0170 0.8890 0.8983 0.4235 0.8367 0.7788 0.7408 0.4230 0.7005 0.8415 0.8038 0.4858 0.7215 0.6245 0.6245 0.3890

A. 20

5.0617 0.8385 1.7207 3.2832 5.0897 1.3207 2.1527 3.5097 5.1237 1.7115 2.3792 3.5997 5.1527 2.0177 2.6067 3.6960 5.1627 2.1744 2.7045 3.7465 5.1727 2.5262 2.9592 3.8555 5.1985 2.7192 3.0922 3.9182 5.2067

1.356 5.5792 4.697 3.1345 1.328 5.097 4.265 2.908 1.294 4.7062 4.0385 2.818 1.265 4.4 3.811 2.7217 1.255 4.2433 3.7132 2.6712 1.245 3.8915 3.4585 2.5622 1.2192 3.6985 3.3255 2.4995 1.211

3.3237 3.5893 2.9930 3.2073 3.9373 3.8130 3.5467 3.3966 4.9448 4.3135 3.6750 2.9800 5.4633 5.4110 4.1120 3.2819 6.1574 5.0437 3.7357 3.5624 9.0967 6.4961 4.5018 3.0288 18.2150 7.8815 5.0814 3.7194

-11.8750 -0.4084 -0.8428 -3.9891 -11.6748 -0.9108 -1.2571 -4.1186 -11.7355 -1.0256 -1.6186 -4.6399 -12.0321 -1.4553 -1.8211 -4.7321 -11.8370 -1.6547 -2.4484 -4.9204 -12.0697 -2.1354 -2.4610 -4.8583 -12.0833 -3.0171 -3.3506 -5.1321 -11.8727

20.6429 9.1763 9.6107 12.757 20.4427 9.6787 10.025 12.8865 20.5034 9.7935 10.3865 13.4078 20.8 10.2232 10.589 13.5 20.6049 10.4226 11.2163 13.6883 20.8376 10.9033 11.2289 13.6262 20.8512 11.785 12.1185 13.9 20.6406

Appendix A: 5 5 5 5 10 10 10 10 20 20 20 20 30 30 30 30 40 40 40 40 50 50 50 50 70 70 70 70 90

1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

6.2371 5.9788 3.9290 1.6290 4.8603 4.7576 3.4143 1.7983 4.8063 3.9069 3.1544 1.4254 4.0178 3.5545 3.2209 1.5353 3.6384 3.2426 2.7517 1.4373 2.9375 3.2059 2.8683 1.6943 2.6972

1.0477 2.9539 6.1589 9.7371 1.7693 3.5191 6.4608 9.8099 2.7174 4.1927 6.7743 9.8762 3.3702 4.6671 6.9946 9.9078 3.8330 5.0020 7.1416 9.9407 4.2591 5.2475 7.2451 9.9590 4.8561 5.7085 7.4675 9.9932 5.2723

Tables of Results 11.2265 9.3203 6.1153 2.5371 10.5049 8.7551 5.8134 2.4643 9.5568 8.0815 5.4999 2.3980 8.9040 7.6071 5.2796 2.3664 8.4412 7.2722 5.1326 2.3335 8.0151 7.0267 5.0291 2.3152 7.4181 6.5657 4.8067 2.2810 7.0019

2.6175 2.4579 1.5920 0.6580 1.9060 1.8177 1.3930 0.7803 1.7293 1.4890 1.1335 0.5853 1.5754 1.4000 1.2790 0.5945 1.4450 1.2997 1.1300 -0.1060 1.1777 1.2800 1.1529 0.6430 1.1900

A. 21

0.2240 1.0718 2.5305 4.1768 0.6580 1.4354 2.7070 4.2130 0.9700 1.6427 2.8200 4.2433 1.3243 1.8240 2.8885 4.2543 1.5134 2.0550 3.0140 4.2710 1.7200 2.1627 3.0730 3.5810 1.9407 2.3350 3.1419 4.2930 2.1250

5.131 4.2832 2.8245 1.1782 4.697 3.9196 2.648 1.142 4.385 3.7123 2.535 1.1117 4.0307 3.531 2.4665 1.1007 3.8416 3.3 2.341 1.084 3.635 3.1923 2.282 1.774 3.4143 3.02 2.2131 1.062 3.23

2.7294 3.0403 2.0783 2.4248 3.1218 2.9968 2.6157 2.6000 4.1696 3.2438 2.9597 2.4858 4.3566 4.4867 3.6694 2.7652 4.5271 3.6600 3.0435 2.9000 7.5555 5.2071 3.7626 2.2988 16.9738

-0.1171 -1.1021 -3.5519 -11.7935 -0.3566 -0.7457 -4.2577 -11.8112 -1.0802 -1.4272 -4.4035 -11.8860 -1.1547 -2.0422 -4.7091 -11.8802 -1.9159 -2.0993 -4.5286 -11.7076 -2.6389 -3.1860 -4.9665 -12.0860 -3.0305 -3.1039 -4.9514 -12.1672 -3.6122

9.5311 10.5161 12.9659 21.2075 9.7706 10.1597 13.6717 21.2252 10.4942 10.8412 13.8175 21.3 10.5687 11.4562 14.1231 21.2942 11.3299 11.5133 13.9426 21.1216 12.0529 12.6 14.3805 21.5 12.4445 12.5179 14.3654 21.5812 13.0262

Appendix A: 90 90 90 5 5 5 5 10 10 10 10 20 20 20 20 30 30 30 30 40 40 40 40 50 50 50 50 70 70

.8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8

3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

2.8076 2.4246 1.4463

8.2868 7.6483 5.0356 2.1069 6.6065 6.3082 4.4669 2.2520 6.3973 5.3452 4.2164 1.9076 5.6362 4.8415 4.2284 1.9958 5.0052 4.5085 3.7058 1.8782 4.2424 4.3524

6.0092 7.6062 10.0229 0.6047 2.1294 4.6828 7.5585 1.1521 2.5217 4.9005 7.6209 1.7967 3.0764 5.1600 7.6630 2.2943 3.4385 5.3897 7.7231 2.7845 3.6221 5.4822 7.7343 2.9590 3.8465 5.5323 7.7330 3.4941 4.1881

Tables of Results 6.2650 4.6680 2.2513 9.0026 7.4779 4.9245 2.0488 8.4552 7.0856 4.7068 1.9864 7.8106 6.5309 4.4473 1.9443 7.3130 6.1688 4.2176 1.8842 6.8228 5.9852 4.1251 1.8730 6.6483 5.7608 4.0750 1.8743 6.1132 5.4192

1.0630 0.9726 0.5540

3.4773 3.2775 2.1470 0.9035 2.9073 2.6340 1.9210 1.0097 2.5700 2.2390 1.6813 0.8307 2.3297 2.0360 1.7700 0.8282 2.1132 1.9000 1.5884 0.8302 1.8590 1.8465

A. 22

2.4680 3.2036 4.3090 0.2136 0.7711 1.9060 3.2011 0.2844 1.0216 2.0286 3.2251 0.7379 1.2256 2.1146 3.2393 0.9316 1.3406 2.2029 3.2663 1.0343 1.4576 2.2716 3.2713 1.1548 1.5296 2.2980 3.2838 1.3886 1.6681

2.887 2.1514 1.046 3.908 3.3505 2.2156 0.9205 3.8372 3.1 2.093 .8965 3.3837 2.896 2.007 .8823 3.19 2.781 1.9187 .8553 3.0873 2.664 1.85 .8503 2.9668 2.592 1.8236 .8378 2.733 2.4535

6.3905 4.4034 3.1088

-0.1639 -0.9584 -2.4240 -2.1900 -0.8286 -2.3474 -1.8076 -2.2310 0.5122 -0.7515 -1.0756 -1.9290 -1.4135 -1.9750 -1.3729 -1.6906 -1.6049 -0.9329 -2.2943 -2.1000 -0.4088 -1.8350

-4.1955 -5.1640 -11.8372 -0.6420 -1.7059 -5.2076 -13.3424 -0.5359 -2.0304 -6.0460 -13.7120 -2.3166 -4.0574 -6.1128 -14.0030 -2.0981 -3.3235 -6.0304 -13.5810 -4.9720 -5.8470 -6.8569 -13.4494 -6.0569 -5.0649 -7.5903 -14.3720 -8.2808 -7.4320

13.6095 14.578 21.2512 12.77 13.8339 17.3356 25.4704 12.6639 14.1584 18.174 25.84 14.4446 16.1854 18.2408 26.131 14.2261 15.4515 18.1584 25.709 17.1 17.975 18.9849 25.5774 18.1849 17.1929 19.7183 26.5 20.4088 19.56

Appendix A: 70 70 90 90 90 90 5

.5 .2 1 .8 .5 .2 1

2 2 2 2 2 2 1.5

3.7415 2.1221 3.8737 3.9332 3.2689 1.8688 11.4446

5.6738 7.7541 3.7819 4.4679 5.7836 7.7785 0.4088

Tables of Results 3.9335 1.8532 5.8254 5.1394 3.8237 1.8288 7.9741

1.5870 0.8657 1.7740 1.6365 1.3934 0.7643 5.1875

2.3426 3.2823 1.4756 1.8081 2.3910 3.2859 0.2732

1.779 .8393 2.646 2.3135 1.7306 .8357 3.3525

-2.1860 -3.5928 0.8250 -4.0838 -2.2159 -3.1778 >-17.9300

-8.1860 -15.3448 -17.0470 -11.9558 -9.0693 -15.4098 0

20.314 27.4728 29.175 24.0838 21.1973 27.5378 >30

A.6 Hybrid RFC with companding Results:

X = PAPR (Companding ) – PAPR (Companding +RFC) Y =CCDF of PAPR (Companding) - CCDF of PAPR (Companding + RFC) Z= SNR (BER= ) (Companding) – SNR (BER= ) (Companding +RFC) X1 == PAPR (RFC) – PAPR (Companding + RFC) Y1 =CCDF of PAPR (RFC) - CCDF of PAPR (Companding + RFC) Z1= SNR (BER= ) (RFC) –SNR (BER= ) (Companding + RFC) Table A.21 (RFC+A) Results and compared with the results of each of (RFC) and (A companding) A

PAPR

5 10 20 30 40 50 60 70

4 4 4 4 4 4 4 4

10.6770 8.1290 8.8708 9.4736 7.1833 8.1376 8.2792 7.9605

5.9547 7.6211 8.7281 9.2173 9.5126 9.7186 9.8724 9.9940

8.2291 6.5627 5.4557 4.9665 4.6712 4.4652 4.3114 4.1898

2.9485 1.8080 1.7175 1.5030 1.2752 1.3045 1.2560 1.0095

2.3735 3.1430 3.6565 3.8830 4.0202 4.1095 4.2210 4.2375

A. 23

CCDF OF PAPR 3.6915 2.922 2.4085 2.182 2.0448 1.9555 1.844 1.8275

5.5216 6.2537 5.8070 5.2467 5.4416 5.6391 5.5909 6.3199

-2.4071 -4.1950 -6.0977 -7.0820 -8.0871 -8.4466 -8.8538 -9.1388

SNR (BER= ) 8.0784 9.8663 11.769 12.7533 13.7584 14.1179 14.5251 14.8101

Appendix A: 80 87.6 90 100 120 140 160 180 200 5 10 20 30 40 50 60 70 80 87.6 90 100 120 5 10 20 30 40

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2

7.7809 7.7519 7.7326 7.4070 7.5977

11.8201 8.8945 9.4051 9.9401 7.5889 8.5004 8.6499 8.2973 8.1005 8.0606 8.0332 7.5943 7.7448 13.2210 10.0137 9.3891 9.8898 8.2320

10.0909 10.1538 10.1647 10.2365 10.3714 10.4564 10.5338 9.1488 10.6436 4.7330 6.0218 6.8976 7.3190 7.5534 7.7166 7.8783 7.9660 8.0457 8.0977 8.1005 8.0590 8.1537 2.7949 3.8020 3.5426 3.9297 4.8575

Tables of Results 4.0929 4.0300 4.0191 3.9473 3.8124 3.7274 3.6500 5.0350 3.5402 7.0860 5.7972 4.9214 4.5000 4.2656 4.1024 3.9407 3.8530 3.7733 3.7213 3.7185 3.7600 3.6653 5.6851 4.6780 4.9374 4.5503 3.6225

0.9546 0.9755 1.0037 0.8784 0.8928

3.6748 2.3533 2.1496 1.9007 1.6440 1.6596 1.5366 1.3512 1.3053 1.2983 1.3020 1.1672 1.1669 4.5796 3.1030 2.1530 1.8913 2.1484

A. 24

4.2654 4.3175 4.3297 4.3634 4.4228 4.4580 4.4941 4.5362 4.5529 1.8988 2.4873 2.8876 3.0797 3.1880 3.2636 3.3006 3.3782 3.4151 3.4393 3.4270 3.4512 3.4959 1.2446 1.6780 1.3320 1.5113 2.1334

1.7996 1.7475 1.7353 1.7016 1.6422 1.607 1.5709 1.5288 1.5121 2.9652 2.3767 1.9764 1.7843 1.676 1.6004 1.5634 1.4858 1.4489 1.4247 1.437 1.4128 1.3681 2.0604 1.627 1.973 1.7937 1.1716

6.2943 6.2175 5.8577 6.1995 6.0463

4.7945 5.2186 4.5924 4.4688 5.0920 4.5339 4.8399 5.5438 5.2229 5.2748 5.1440 5.1326 5.1104 0.5206 1.2700 4.7008 3.9879 0.8444

-9.2936 -9.4832 -9.6257 -9.7492 -10.1454 -10.5325 -10.8208 -10.8888 -11.0074 -2.7670 -4.8629 -6.9451 -7.4927 -8.0695 -9.1846 -9.2376 -9.5477 -9.9978 -10.0587 -9.9722 -10.4489 -10.7141 -4.6524 -6.4230 -4.4482 -5.5851 -9.9286

14.9649 15.1545 15.297 15.4205 15.8167 16.2038 16.4921 16.5601 16.6787 8.8055 10.9014 12.9836 13.5312 14.108 15.2231 15.2761 15.5862 16. 0363 16.0972 16.0107 16.4874 16.7526 13.0794 14.85 12.8752 14.0121 18.3556

Appendix A: 50 60 70 80 87.6 90 100 120 5

4 4 4 4 4 4 4 4 4

2 2 2 2 2 2 2 2 1.5

9.0584 8.5255 8.1824 8.4834 8.1676 8.0728 7.7330 8.1811 14.3373

4.9356 4.4149 4.5121 5.0896 4.8657 4.8011 4.8587 5.2510 1.8221

Tables of Results 3.5444 4.0651 3.9679 3.3904 3.6143 3.6789 3.6213 3.2290 4.5688

2.1234 1.5636 1.3459 1.7303 1.7146 1.2991 1.1672 1.5670 5.2036

2.1684 1.7686 1.8139 2.2811 2.2966 1.8651 1.8922 2.3370 0.9794

1.1366 1.5364 1.4911 1.0239 1.0084 1.4399 1.4128 .968 1.4364

1.3836 4.9809 5.2696 1.5838 2.0854 5.3091 5.7459 2.3035 -16

-9.9464 -6.7081 -7.4334 -11.2484 -10.8596 -7.4186 -7.4471 -11.1325 -1.3800

18.3734 15.1351 15.8604 19.6754 19.2866 15.8456 15.8741 19.5595 29.6

5.5832 6.1305 6.2688 6.0706 6.3302 6.4429 6.2734 6.1287 6.2719 6.2989 6.2706 6.2921 6.1176 5.8882 6.2099

-2.0818 -2.8382 -4.3099 -5.4231 -5.7485 -6.1558 -6.8323 -7.2000 -7.5308 -7.6298 -8.0581 -8.3166 -8.7111 -8.9155 -9.1688

SNR (BER= ) 7.7531 8.5095 9.9812 11.0944 11.4198 11.8271 12.5036 12.8713 13.2021 13.3011 13.7294 13.9879 14.3824 14.5868 14.8401

Table A.22 (RFC+ ) Results and compared with the results of each of (RFC) and ( companding) MU

PAPR

5 10 20 30 40 50 60 70 80 90 100 120 140 160 180

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

8.5326 8.9351 10.1373 8.6956 8.2595 8.5378 8.5780 9.2492 11.6899 6.8796 7.7661 6.3834 8.3693 7.9319 8.3988

5.2832 6.5719 7.0638 8.2438 8.5980 8.8594 9.0291 9.2093 9.3236 8.2050 9.5357 8.4530 9.8152 9.9074 9.9968

8.9006 7.6119 7.1200 5.9400 5.5858 5.3244 5.1547 4.9745 4.8602 5.9788 4.6481 5.7308 4.3686 4.2764 4.1870

2.4646 2.3622 2.1758 1.7481 1.5164 1.4583 1.3896 1.4627 1.2103 1.2110 1.0783 1.0264 1.1766 0.9970 1.0647

2.1136 2.6572 3.1628 3.4131 3.5814 3.7233 3.7946 3.8817 3.9453 3.9760 4.0233 4.0914 4.1816 4.1960 4.2497

A. 25

CCDF OF PAPR 3.9514 3.4078 2.9022 2.6519 2.4836 2.3417 2.2704 2.1833 2.1197 2.089 2.0417 1.9736 1.8834 1.869 1.8153

Appendix A: 200 220 240 255 5 10 20 30 40 50 60 70 80 90 100 120 140 160 180 200 220 240 255 5 10 20 30 40 50

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2

8.1269 7.2315 11.9179 7.2789 9.7665 9.9560 11.5810 9.3580 8.8961 9.0117 9.0525 9.7253 12.0968 8.5998 8.2039 7.9277 8.7502 8.2815 8.7284 8.4567 7.5390 12.2278 8.0884 11.4573 11.2272 12.5123 10.1818 9.4679 9.6186

10.0684 10.1431 10.1873 9.7193 4.1523 5.2280 6.1427 6.5414 6.8698 6.9685 7.1388 7.3206 7.3657 7.5604 7.6087 7.6325 7.8313 7.8922 7.9616 8.0334 8.0858 8.1324 8.1640 2.5041 3.1602 3.7350 4.0262 4.1026 4.2364

Tables of Results 4.1154 4.0407 3.9965 4.4645 7.6667 6.5910 5.6763 5.2776 4.9492 4.8505 4.6802 4.4984 4.4533 4.2586 4.2103 4.1865 3.9877 3.9268 3.8574 3.7856 3.7332 3.6866 3.6550 5.9759 5.3198 4.7450 4.4538 4.3774 4.2436

0.9662 0.9042 1.1064 0.9756 3.2220 3.0278 2.7424 2.2537 1.9936 1.8652 1.8009 1.8636 1.5756 1.5980 1.4553 1.3889 1.4974 1.3435 1.3896 1.2902 1.1998 1.4267 1.2715 4.2422 3.8944 3.4619 2.9164 2.5916 2.4543

A. 26

4.2712 4.3242 4.3214 4.3606 1.6700 2.1218 2.5284 2.7177 2.8576 2.9292 3.0049 3.0816 3.1096 3.1620 3.1993 3.2529 3.3014 3.3415 3.3736 3.3942 3.4188 3.4407 3.4555 1.1312 1.4294 1.6889 1.8214 1.8966 1.9593

1.7938 1.7408 1.7436 1.7044 3.194 2.7422 2.3356 2.1463 2.0064 1.9348 1.8591 1.7824 1.7544 1.702 1.6647 1.6111 1.5626 1.5225 1.4904 1.4698 1.4452 1.4233 1.4085 2.1738 1.8756 1.6161 1.4836 1.4084 1.3457

5.9880 6.0007 6.2370 6.1604 4.8568 4.9688 5.0238 5.0583 5.1076 5.0674 5.4896 5.2608 5.2740 5.3629 5.8642 5.4133 5.2935 5.3149 5.2071 5.2732 5.2154 5.5058 5.3381 1.6003 1.4241 1.3077 1.2065 1.9988 2.0125

-9.3407 -9.5665 -9.6917 -9.6363 -2.4410 -3.6327 -5.1877 -6.0682 -6.6039 -7.1641 -7.2489 -7.7007 -8.1615 -8.1986 -8.0973 -8.8282 -9.1680 -9.1216 -9.8044 -9.6883 -9.9846 -10.0557 -10.0914 -3.3090 -4.7889 -6.5153 -7.5315 -7.3242 -7.8305

15.012 15.2378 15.363 15.3076 8.4795 9.6712 11.2262 12.1067 12.6424 13.2026 13.2874 13.7392 14.2 14.2371 14.1358 14.8667 15.2065 15.1601 15.8429 15.7268 16.0231 16.0942 16.1299 11.736 13.2159 14.9423 15.9585 15.7512 16.2575

Appendix A: 60 70 80 90 100 120 140 160 180 200 220 240 255 5

4 4 4 4 4 4 4 4 4 4 4 4 4 4

2 2 2 2 2 2 2 2 2 2 2 2 2 1.5

9.5917 10.1649 12.3265 8.6925 8.2980 8.0803 9.2679 8.7930 9.2168 8.9133 7.9705 12.5551 8.3361 12.4989

4.3390 4.4212 4.2564 4.3141 4.3638 4.4461 5.0100 5.0647 5.1110 5.1510 5.1783 5.1207 5.0727 1.4566

Tables of Results 4.1410 4.0588 4.2236 4.1659 4.1162 4.0339 3.4700 3.4153 3.3690 3.3290 3.3017 3.3593 3.4073 4.9343

2.3625 2.3873 2.1150 2.1121 1.9557 1.8746 1.9463 1.7787 1.8151 1.7145 1.6223 1.8317 1.6868 4.8033

2.0075 2.0463 2.0900 2.1171 2.1407 2.1796 2.1913 2.2177 2.2401 2.2595 2.2823 2.2867 2.3118 0.8031

1.2975 1.2587 1.215 1.1879 1.1643 1.1254 1.1137 1.0873 1.0649 1.0455 1.0227 1.0183 .9932 1.6127

2.1959 1.3881 1.8028 1.7990 2.1021 2.2005 2.0155 1.7904 2.1229 1.8744 2.4273 2.4166 2.1590 -16.6637

Table A.23 (RFC+ RCT) Results and compared with the results of each of (RFC) and (RCT). I CR X X1 PAPR Y Y1 CCDF OF Z R .9 .8 .7 .6 .5 .4 .3 .2 .1 .9

4 4 4 4 4 4 4 4 4

4 4 4 4 4 4 4 4 4 3

8.7058 8.2277 7.2828 5.7647 5.6633 4.7000 3.2463 2.3157 0.9577 10.8872

1.0265 1.2804 3.3375 4.5938 5.9207 7.3546 8.8772 10.5107 12.2689 0.8431

13.1573 12.9034 10.8463 9.5900 8.2631 6.8292 5.3066 3.6731 1.9149 10.9759

3.7793 3.3454 3.2633 2.4276 2.1581 1.7575 1.3323 0.9217 0.3934 4.9102

A. 27

0.2943 0.7289 1.2703 1.8101 2.3881 3.0010 3.6873 4.4122 5.1904 0.2242

PAPR 5.7707 5.3361 4.7947 4.2549 3.6769 3.064 2.3777 1.6528 .8746 4.6398

5.7109 5.8086 5.9381 6.2980 6.0880 6.0777 5.6848 5.9429 5.5206 5.2180

-8.1541 -9.1849 -9.2442 -9.3740 -9.4709 -9.6525 -10.0575 -10.2576 -10.5001 -10.6986 -10.7564 -10.3842 -10.8820 -1.7800

Z1 -0.2943 -0.5071 -0.8043 -1.4307 -2.6907 -4.3960 -7.1689 -10.6358 -17.1081 -0.4200

16.5811 17.6119 17.6712 17.801 17.8979 18.0795 18.4845 18.6846 18.9271 19.1256 18.8112 19.1834 19.309 >30

SNR (BER= ) 5.9656 6.1784 6.4756 7.102 8.362 10.0673 12.8402 16.3071 22.7794 6.4585

Appendix A: .8 .7 .6 .5 .4 .3 .2 .1 .9 .8 .7 .6 .5 .4 .3 .2 .1 .9

3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 1.5

11.0343 9.0586 7.3092 6.9813 5.7331 4.0685 2.8625 1.2197 13.9172 13.9228 11.5339 9.4644 8.8441 7.2554 5.2124 3.6394 1.6257 15.9263

1.7222 2.7485 3.7735 4.8739 6.0229 7.3346 8.6927 10.1661 0.5341 1.2717 1.8848 2.5897 3.3977 4.2062 5.1395 6.1306 7.2331 0.4541

Tables of Results 10.0968 9.0705 8.0455 6.9451 5.7961 4.4844 3.1263 1.6529 7.9459 7.2083 6.5952 5.8903 5.0823 4.2738 3.3405 2.3494 1.2469 5.9368

4.3707 4.1650 3.2110 2.8005 2.2985 1.7236 1.1770 0.5409 6.4071 5.7633 5.4034 4.2677 3.7262 3.0466 2.2784 1.5677 0.7444 7.3321

0.5532 0.9710 1.3925 1.8295 2.3410 2.8776 3.4665 4.1369 0.1621 0.3868 0.6504 0.8902 1.1962 1.5301 1.8734 2.2982 2.7814 0.1979

4.3108 3.893 3.4715 3.0345 2.523 1.9864 1.3975 .7271 3.1429 2.9182 2.6546 2.4148 2.1088 1.7749 1.4316 1.0068 .5236 2.2179

5.4803 5.2811 5.4153 5.1927 5.1535 5.5226 5.3988 4.8168 2.6339 2.8523 2.9250 3.1766 3.0342 2.3582 2.8951 3.4017 2.6712 -17.0065

-0.4682 -1.0941 -1.9462 -3.2188 -4.9530 -6.9639 -10.8127 -17.4447 -0.6156 -0.7077 -1.0617 -1.7964 -2.9888 -5.3598 -7.2029 -10.4213 -17.2018 -0.4630

6.5067 7.1326 7.9847 9.2573 10.9915 13.0024 16.8512 23.4832 9.0426 9.1347 9.4887 10.2234 11.4158 13.7868 15.6299 18.8483 25.6288 28.683

Table A.24 (RFC+AEXP) Results and compared with the results of each of (RFC) and (AEXP companding) AEXP

PAPR

2 1.9 1.8 1.7 1.6 1.5

4 4 4 4 4 4

1.9702 2.2773 1.8198 1.2434 1.1310 0.9692

3.0729 3.4371 3.8053 4.2099 4.6186 5.0866

11.1109 10.7467 10.3785 9.9739 9.5652 9.0972

0.3967 0.5200 0.4753 0.4379 0.4047 0.3714

1.3084 1.4450 1.6218 1.7329 1.8847 2.0784

A. 28

CCDF OF PAPR 4.7566 4.62 4.4432 4.3321 4.1803 3.9866

7.5446 6.7516 7.3982 6.4435 6.2825 6.8105

-1.5141 -2.2771 -1.7885 -2.3352 -2.2462 -1.8320

SNR (BER= ) 7.1854 7.9484 7.4598 8.0065 7.9175 7.5033

Appendix A: 1.4 1.3 1.2 1.1 1 .9 .8 .7 .6 .5 .4 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 .9 .8 .7 .6 .5 .4 2

4 4 4 4

4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2

0.9623 0.7656 0.7023 0.6813 0.6085 0.4981 0.3743 0.3806 0.2386 0.2162 0.1528 2.8409 3.1584 2.6683 2.0759 1.9290 1.6787 1.6218 1.4161 1.2859 1.2248 1.1195 0.9545 0.7515 0.7338 0.5422 0.4644 0.3578 4.4446

5.5316 5.9679 6.4361 6.9128 7.4149 7.9319 8.4775 9.0391 9.6332 10.2656 10.9327 1.5788 1.9534 2.2890 2.6776 3.0518 3.4313 3.8263 4.2536 4.6549 5.0915 5.5611 6.0235 6.4899 7.0275 7.5720 8.1490 8.7729 -0.1565

Tables of Results 8.6522 8.2159 7.7477 7.2710 6.7689 6.2519 5.7063 5.1447 4.5506 3.9182 3.2511 10.2402 9.8656 9.5300 9.1414 8.7672 8.3877 7.9927 7.5654 7.1641 6.7275 6.2579 5.7955 5.3291 4.7915 4.2470 3.6700 3.0461 8.6365

0.3238 0.3317 0.0659 0.3208 0.2994 0.3327 0.2977 0.2528 0.2178 0.2695 0.2031 0.8554 0.9622 0.8424 0.8486 0.8007 0.6796 0.6111 0.6129 0.5822 0.5470 0.5258 0.5417 0.5069 0.4166 0.3589 0.4074 0.3188 1.8041

2.2423 2.4167 2.3249 2.7863 2.9904 3.2057 3.4427 3.6808 3.9428 4.2270 4.5281 0.5661 0.6862 0.7879 0.9426 1.0797 1.1856 1.3286 1.4969 1.6402 1.8115 2.0158 2.2137 2.4509 2.6436 2.8829 3.1639 3.4428 -0.0442

A. 29

3.8227 3.6483 3.7401 3.2787 3.0746 2.8593 2.6223 2.3842 2.1222 1.838 1.5369 4.2979 4.1778 4.0761 3.9214 3.7843 3.6784 3.5354 3.3671 3.2238 3.0525 2.8482 2.6503 2.4131 2.2204 1.9811 1.7001 1.4212 3.3492

7.1086 7.0868 7.6358 6.8206 7.4259 7.5934 10.0902 19.5598 18.5802 15.5774 0 6.6001 6.4858 6.5050 6.2358 5.9424 5.9209 6.0119 5.7748 6.7358 6.2308 6.3021 7.0545 9.2588 19.1217 17.5183 1.3672 0 4.4356

-1.7886 -1.5419 -2.0229 -2.3581 -2.2028 -3.0353 -3.4835 -4.7689 -5.7485 -8.7513 -24.3287 -2.0914 -2.1757 -2.3145 -2.1757 -2.2191 -2.3544 -2.5181 -2.4867 -2.5557 -2.5807 -2.9594 -3.2070 -3.9477 -4.8398 -6.4432 -22.5943 -23.9615 -1.8674

7.4599 7.2132 7.6942 8.0294 7.8741 8.7066 9.1548 10.4402 11.4198 14.4226 >30 8.1299 8.2142 8.353 8.2142 8.2576 8.3929 8.5566 8.5252 8.5942 8.6192 8.9979 9.2455 9.9862 10.8783 12.4817 28.6328 >30 10.2944

Appendix A: 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 .9 .8 .7 .6 .5 .4 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1.5

4.7584 4.2805 3.5883 3.4069 3.2267 3.0173 2.7391 2.6453 2.5080 2.2867 2.0616 1.7864 1.6340 1.3458 1.1395 0.9063 5.8486

0.2144 0.5622 0.8510 1.1907 1.6403 1.8828 2.2376 2.6753 3.0357 3.3893 3.7916 4.1858 4.5887 5.0366 5.4851 5.9824 -0.8416

Tables of Results 8.2656 7.9178 7.6290 7.2893 6.8397 6.5972 6.2424 5.8047 5.4443 5.0907 4.6884 4.2942 3.8913 3.4434 2.9949 2.4976 7.2325

1.9134 1.7817 1.6806 1.5898 1.5115 1.3648 1.3171 1.3019 1.2229 1.1397 1.1004 0.9863 0.8742 0.7287 0.7272 0.5759 2.6752

0.0784 0.1682 0.2156 0.3098 0.4585 0.5233 0.6421 0.8009 0.9284 1.0707 1.2134 1.3713 1.5422 1.6937 1.9247 2.1409 -0.0623

3.2266 3.1368 3.0894 2.9952 2.8465 2.7817 2.6629 2.5041 2.3766 2.2343 2.0916 1.9337 1.7628 1.6113 1.3803 1.1641 2.4781

2.6868 2.7087 1.2358 0.5798 0.8877 0.9430 0.6155 1.5764 1.1190 0.2170 2.2184 3.6618 12.9485 1.6400 0 0 -15.2700

-3.5862 -3.7223 -4.7872 -5.1932 -4.9991 -5.1985 -5.2575 -5.3266 -5.3040 -6.6560 -5.6546 -7.1562 -8.6245 -19.9330 -21.5730 -21.5730 -1.7800

12.0132 12.1493 13.2142 13.6202 13.4261 13.6255 13.6845 13.7536 13.731 15.083 14.0816 15.5832 17.0515 28.36 >30 >30 >30

Table A.25 (RFC+ cos) Results and compared with the results of each of (RFC) and (cos companding) Cos y

PAPR

1 .9 .8 .7 .6 .5 .4

4 4 4 4 4 4 4

3.3647 2.3785 2.4468 2.3786 2.0945 1.6828 1.2488

1.9879 2.1617 3.5211 4.8932 5.9725 7.0976 8.2264

12.1959 12.0221 10.6627 9.2906 8.2113 7.0862 5.9574

1.4804 1.4216 1.2605 1.1151 0.9266 0.7674 0.6048

0.6954 1.0706 1.4605 1.8986 2.3886 2.8774 3.3908

A. 30

CCDF OF PAPR 5.3696 4.9944 4.6045 4.1664 3.6764 3.1876 2.6742

5.9268 6.1069 5.9421 6.1775 6.4601 6.2525 6.2306

-0.2019 -0.4468 -0.9666 -1.5352 -2.4071 -3.9362 -5.6581

SNR (BER= ) 5.8732 6.1181 6.6379 7.2065 8.0784 9.6075 11.3294

Appendix A:

Tables of Results

.3 .2 .1

4 4 4

1.0157

9.6236 11.0172 12.5275

4.5602 3.1666 1.6563

0.5335

3.9985 4.6068 5.3006

2.0665 1.4582 .7644

7.6706

1 .9 .8 .7 .6 .5 .4 .3 .2 .1

3 3 3 3 3 3 3 3 3 3

5.3429 4.9243 4.4042 3.7962 3.3283 2.7294 2.1840 1.6589

1.6013 2.3427 3.1137 3.9460 4.8415 5.7794 6.7968 7.9020 9.0885 10.3771

10.2177 9.4763 8.7053 7.8730 6.9775 6.0396 5.0222 3.9170 2.7305 1.4419

2.3797 2.2367 2.0116 1.8062 1.4903 1.2001 1.0164 0.8270

0.3937 0.6847 1.0106 1.3887 1.7513 2.1091 2.6014 3.0910 3.6135 4.2062

4.4703 4.1793 3.8534 3.4753 3.1127 2.7549 2.2626 1.773 1.2505 .6578

5.5047 5.4913 5.5914 5.4577 5.9005 5.6383 5.8226 7.1450

1 .9 .8 .7 .6 .5 .4 .3 .2 .1

2 2 2 2 2 2 2 2 2 2

8.4031 7.7215 6.8456 6.0573 5.2889 4.4202 3.5833 2.6995

1.3225 1.8009 2.2161 2.8681 3.4631 4.1312 4.8571 5.6036 6.4762 7.4125

7.1575 6.6791 6.2639 5.6119 5.0169 4.3488 3.6229 2.8764 2.0038 1.0675

3.7063 3.4932 3.1303 2.7832 2.3470 2.0022 1.6358 1.3013

0.1613 0.3822 0.5703 0.8067 1.0490 1.3522 1.6618 2.0063 2.3977 2.8196

3.1437 2.9228 2.7347 2.4983 2.256 1.9528 1.6432 1.2987 .9073 .4854

1.5

10.1993

1.0296

5.3613

4.5714

0.1372

2.2786

A. 31

-7.9581 >-23.9287 >>23.9287 -0.2568 -0.6952 -0.9501 -1.8878 -2.5995 -4.1832 -5.6989 -8.1165 -19.2430 -23.5615

13.6294 >30 >>30

3.3286 3.1239 2.5218 3.0995 2.6977 3.0493 2.9223 4.0627

-0.0444 -0.6741 -1.6312 -1.8575 -3.4138 -4.3837 -6.2107 -8.8103 -21.1730 -21.1730

8.4714 9.1011 10.0582 10.2845 11.8408 12.8107 14.6377 17.2373 29.6 29.6

-17.8000

-1.3800

6.2953 6.7337 6.9886 7.9263 8.638 10.2217 11.7374 14.155 25.2815 29.6

29.6

Appendix A:

Tables of Results

Table A.26 (RFC+NERF) Results and compared with the results of each of (RFC) and (NERF companding) NERF

PAPR

4 4 4 4 4

4 3 2 1.75 1.5

0.9191 1.6395 2.7940 3.7575 3.0791

5.2328 3.7342 2.2218

9.0903 8.2522 6.9153 6.4220 5.7098

0.4355 0.8685 1.5235 2.1590 1.8955

2.4177 1.7880 1.2096

1.2825

1.0857

CCDF OF PAPR 3.8875 3.4578 2.5844 2.2765 1.9541

3.2135 2.7780 -2.3220 -5.7976 -16.3220

-1.6966 -1.4860 -3.8720 -0.4000

SNR (BER= ) 7.2096 7.7814 11.883 19.4756 >30

Table A.27 (RFC+tanhR) Results and compared with the results of each of (RFC) and (tanhR companding) k

PAPR

5 5 5 5 10 10 10 10 15 15 15 15 20 20 20

1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5

4 4 4 4

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

2.6944 2.1182 1.2755 0.6910 -4.5995 -2.9960 -0.6902 0.4108 -7.5448 -5.2051 -1.6597 0.2390 -8.4594 -6.0915 -2.7713

0.1653 2.4548 6.5550 11.3302 0.6531 3.0398 7.0939 11.5444 1.4076 3.7358 7.5822 11.6791 2.3463 4.4973 7.6364

14.0185 11.7290 7.6288 2.8536 13.5307 11.1440 7.0899 2.6394 12.7762 10.4480 6.6016 2.5047 11.8375 9.6865 6.5474

1.2590 0.8791 0.4358 0.2234 -1.7440 -1.2074 -0.3799 0.1330 -3.0363 -2.1608 -0.7619 0.0530 -3.4260 -2.5352 -1.0317

-0.0836 0.8860 2.6147 4.7609 0.1120 1.0997 2.8570 4.8529 0.4300 1.3777 3.0710 4.9153 0.8162 1.6886 3.2438

4 4 4 4 4 4 4 4 4

A. 32

CCDF OF PAPR 6.1486 5.179 3.4503 1.3041 5.953 4.9653 3.208 1.2121 5.635 4.6873 2.994 1.1497 5.2488 4.3764 2.8212

6.3440 6.2949 6.8333 7.2231 8.8688 8.7788 8.5310 8.5822 23.6963 15.9471 11.7960 8.9374 23.5747 22.8019 19.6576

-0.1141 -0.7050 -3.0831 -11.8287 -0.1085 -0.4820 -3.4266 -11.7624 -0.2324 -0.8472 -4.0074 -11.9489 -0.3540 -1.1268 -4.2711

SNR (BER= ) 5.7854 6.3763 8.7544 17.5 5.7798 6.1533 9.0979 17.4337 5.9037 6.5185 9.6787 17.6202 6.0253 6.7981 9.9424

Appendix A: 20 30 30 30 30 40 40 40 40 5 5 5 5 10 10 10 10 15 15 15 15 20 20 20 20 30 30 30 30

.2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2

4 4

4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3

0.1722

5.0723 4.0340 2.4146 1.0832 -2.4226 -1.2989 0.3078 0.7564 -5.6505 -3.7325 -0.7879 0.5623 -6.7832 -4.7701 -1.5784 0.4900

11.7850 4.3979 5.9931 8.7585 11.9430 6.3037 7.3308 9.3689 12.0608 0.1784 2.0058 5.3293 9.3576 0.4652 2.3721 5.7271 9.5252 0.9371 2.8436 6.0892 9.6376 1.6577 3.4539 6.4645 9.7380 3.0272 4.4995 7.0108 9.8546

Tables of Results 2.3988 9.7859 8.1907 5.4253 2.2408 7.8801 6.8530 4.8149 2.1230 11.6406 9.8132 6.4897 2.4614 11.3538 9.4469 6.0919 2.2938 10.8819 8.9754 5.7298 2.1814 10.1613 8.3651 5.3545 2.0810 8.7918 7.3195 4.8082 1.9644

0.0269

2.4801 1.8728 1.0524 0.4255 -0.6289 -0.3228 0.1424 0.3116 -2.0568 -0.3544 -0.2991 0.2234 -2.6056 -1.8404 -0.6063 0.1826

A. 33

4.9604 1.5013 2.2362 3.5268 5.0263 2.2698 2.7876 3.7886 5.0792 -0.0635 0.6787 2.0303 3.7620 0.0261 0.7833 2.1783 3.8305 0.2085 1.9831 2.3328 3.8847 0.4356 1.1824 2.4682 3.9151 0.9312 1.5702 2.6746 3.9733

1.1046 4.5637 3.8288 2.5382 1.0387 3.7952 3.2774 2.2764 .9858 4.9275 4.1853 2.8337 1.102 4.8379 4.0807 2.6857 1.0335 4.6555 2.8809 2.5312 .9793 4.4284 3.6816 2.3958 .9489 3.9328 3.2938 2.1894 .8907

10.7485

5.9113 5.8850 6.4305 6.8634 8.6281 8.4958 7.5664 6.5865 23.2474 15.3959 11.6768 8.0920 22.9245 22.1648 19.1135 9.8503

-12.5194 -1.1109 -1.9585 -4.5051 -12.8835 -1.9663 -2.6475 -5.6472 -13.2070 -0.1796 -0.7477 -3.1187 -11.8212 0.0180 -0.3978 -4.0240 -13.3909 -0.3141 -1.0312 -3.7594 -12.4271 -0.6370 -1.3967 -4.4480 -13.0504 -0.8914 -2.1879 -5.1420 -12.8308

18.1907 6.7822 7.6298 10.1764 18.5548 7.6376 8.3188 11.3185 18.8783 6.2181 6.7862 9.1572 17.8597 6.0205 6.4363 10.0625 19.4294 6.3526 7.0697 9.7979 18.4656 6.6755 7.4352 10.4865 19.0889 6.9299 8.2264 11.1805 18.8693

Appendix A: 40 40 40 40 50 50 50 50 5 5 5 5 10 10 10 10 15 15 15 15 20 20 20 20 30 30 30

1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5

3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

8.0411 6.4911 3.9618 1.6309 0.6809 1.2029 1.8224 1.2947 -2.8631 -1.5051 0.5603 1.0594 -4.3947 -2.8497 -0.3872 0.9528

4.5177 5.5636 7.5154 9.9583 5.8206 6.4840 7.9373 10.0415 -0.1918 1.1239 3.5375 6.5663 0.2297 1.5349 3.9027 6.7245 0.3855 1.7320 4.0984 6.7957 0.7072 2.0353 4.3167 6.8618 1.4478 2.6476 4.6783

Tables of Results 7.3013 6.2554 4.3036 1.8607 5.9984 5.3350 3.8817 1.7775 8.6718 7.3561 4.9425 1.9137 8.2503 6.9451 4.5773 1.7555 8.0945 6.7480 4.3816 1.6843 7.7728 6.4447 4.1633 1.6182 7.0322 5.8324 3.8017

4.0083 3.1445 1.8462 0.7088 0.9295 0.9545 0.9068 0.5888 -0.6314 -0.2065 0.4052 0.4813 -1.3229 -0.8356 0.0055 0.4239

A. 34

1.5349 2.0098 2.9066 4.0222 2.0640 2.3953 3.0958 4.0640 -0.0943 0.3914 1.2651 2.4863 0.0255 0.5016 1.3837 2.5487 0.0749 0.5720 1.4781 2.5836 0.1593 0.6282 1.5210 2.5974 0.3891 0.8513 1.6656

3.3291 2.8542 1.9574 .8418 2.8 2.4687 1.7682 .8 3.3993 2.9136 2.0399 .8187 3.2795 2.8034 1.9213 .7563 3.2301 2.733 1.8269 .7214 3.1457 2.6768 1.784 .7076 2.9159 2.4537 1.6394

3.4383 3.3481 3.0606 3.0185 6.0106 5.1929 4.9274 4.2034 20.9592 12.6710 8.2439 3.5147 20.4398 19.6499 16.6531 5.9826

-2.1879 -2.9035 -6.1790 -13.4003 -3.7195 -4.0708 -7.0648 -14.0866 -0.2641 -0.8961 -4.1001 -13.2776 -0.2110 -1.3122 -4.2745 -13.3855 -0.2138 -1.3676 -4.8038 -14.6159 -0.7332 -1.5231 -4.5199 -14.5296 -1.0211 -2.8904 -6.2576

8.2264 8.942 12.2175 19.4388 9.758 10.1093 13.1033 20.1251 8.6911 9.3231 12.5271 21.7046 8.638 9.7392 12.7015 21.8125 8.6408 9.7946 13.2308 23.0429 9.1602 9.9501 12.9469 22.9566 9.4481 11.3174 14.6846

Appendix A: 30 40 40 40 40 5

.2 1 .8 .5 .2 1

2 2 2 2 2 1.5

10.4523

Tables of Results

6.9505 2.2898 3.2772 4.9997 7.0196 0.1303

1.5295 6.1902 5.2028 3.4803 1.4604 6.2606

4.9898

2.6311 0.6198 1.0325 1.7771 2.6697 -0.0020

.6739 2.6852 2.2725 1.5279 .6353 2.4178

-16.3460

-16.7147 -2.3539 -3.5287 -7.5339 -16.0484 -0.2554

25.1417 10.7809 11.9557 15.9609 24.4754 28.4754

Table A.28 (RFC+logR) Results and compared with the results of each of (RFC) and (logR companding) k

5 5 5 5 10 10 10 10 20 20 20 20 30 30 30 30 40

1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4

PAPR

2.7526 4.0614 2.7840 1.2286 2.3809 2.0822 1.8293 1.2155 2.3488 1.9860 2.0414 1.0410 1.6240

0.7014 3.0106 6.9250 11.2475 0.1944 3.5113 7.2254 11.3191 2.1476 3.4269 7.0989 11.2030 2.8223 4.6558 7.7912 11.4330 3.3488

13.4824 11.1732 7.2588 2.9363 13.9894 10.6725 6.9584 2.8647 12.0362 10.7569 7.0849 2.9808 11.3615 9.5280 6.3926 2.7508 10.8350

1.5031 1.6195 1.1077 0.4931 0.9625 1.0705 0.9371 0.6147 0.7103 0.7510 0.7017 0.4266 0.5382

0.1402 1.1054 2.7955 4.7255 0.2536 1.3070 2.9327 4.7581 0.7365 1.6055 3.0741 4.7877 1.0153 1.7960 3.1667 4.8056 1.1862

A. 35

CCDF OF PAPR 5.9248 4.9596 3.2695 1.3395 5.8114 4.758 3.1323 1.3069 5.3285 4.4595 2.9909 1.2773 5.0497 4.269 2.8983 1.2594 4.8788

6.6311 6.5858 6.4264 6.4737 7.5177 7.1717 6.6767 6.5743 8.5090 7.5945 7.3627 6.3249 8.9930

-0.0429 -0.7108 -3.5314 -11.5564 -0.1976 -0.9429 -3.6523 -11.5050 -0.4270 -0.9950 -4.0852 -11.6544 -0.5580 -1.4342 -4.0488 -11.7838 -1.0222

SNR (BER= ) 5.7142 6.3821 9.2027 17.2277 5.8689 6.6142 9.3236 17.1763 6.0983 6.6663 9.7565 17.3257 6.2293 7.1055 9.7201 17.4551 6.6935

Appendix A: 40 40 40 50 50 50 50 70 70 70 70 90 90 90 90 5 5 5 5 10 10 10 10 20 20 20 20 30 30

.8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 3 3 3 3 3 3 3 3 3 3 3 3 3 3

1.6777 2.1372 1.1544 1.2444 1.4152 1.6948 1.0611 0.6110 1.3851 1.7977 1.3230 0.4691 1.0355 1.3705 1.0737

5.9143 5.6943 3.8312 1.6255 4.2590 4.4077 3.2711 1.8024 4.0813 3.4342

5.0348 7.9675 11.4694 3.7747 5.3297 8.0978 11.4924 4.4392 5.7973 8.3065 11.5315 4.9538 6.1467 8.4617 11.5599 0.5230 2.4889 5.6872 9.3116 0.9913 2.7794 5.9078 9.3512 1.6609 3.3876 6.1759 9.4251 2.1900 3.7392

Tables of Results 9.1490 6.2163 2.7144 10.4091 8.8541 6.0860 2.6914 9.7446 8.3865 5.8773 2.6523 9.2300 8.0371 5.7221 2.6239 11.2960 9.3301 6.1318 2.5074 10.8277 9.0396 5.9112 2.4678 10.1581 8.4314 5.6431 2.3939 9.6290 8.0798

0.5574 0.7850 0.4373 0.4157 0.5253 0.6536 0.4371 0.2321 0.5107 0.6907 0.4884 0.2755 0.3334 0.5281 0.4024

2.6999 2.4796 1.6309 0.7015 1.9290 1.8493 1.4476 0.8319 1.5873 1.4834

A. 36

1.9224 3.2300 4.8238 1.4007 2.0983 3.3066 4.8341 1.7051 2.2757 3.3897 4.8484 1.9205 2.4484 3.4691 4.8674 0.0727 0.8322 2.1320 3.7351 0.2494 0.9661 2.2549 3.7655 0.5020 1.1833 2.3836 3.8039 0.6913 1.3274

4.1426 2.835 1.2412 4.6643 3.9667 2.7584 1.2309 4.3599 3.7893 2.6753 1.2166 4.1445 3.6166 2.5959 1.1976 4.7913 4.0318 2.732 1.1289 4.6146 3.8979 2.6091 1.0985 4.362 3.6807 2.4804 1.0601 4.1727 3.5366

8.7135 7.8613 6.3458 10.3272 9.0464 6.9655 6.8945 12.8556 10.3224 7.7414 6.1660 22.5168 12.1155 8.8508 6.5459

6.2141 6.3045 5.9720 5.5242 7.1443 6.7307 6.3066 6.2466 7.9106 6.9501

-1.6152 -4.0794 -11.8697 -0.5815 -1.5423 -4.7872 -11.8342 -1.4731 -1.7313 -4.7153 -12.0427 -1.8119 -2.2132 -4.4593 -12.1428 -0.2954 -1.0780 -3.8176 -11.9709 -0.2474 -0.8570 -3.7395 -12.0873 -0.4332 -1.0688 -4.0881 -11.6149 -0.7892 -1.7114

7.2865 9.7507 17.541 6.2528 7.2136 10.4585 17.5055 7.1444 7.4026 10.3866 17.7149 7.4832 7.8845 10.1306 17.8141 6.3339 7.1165 9.8561 18.0094 6.2859 6.8955 9.778 18.1258 6.4717 7.1073 10.1266 17.6534 6.8277 7.7499

Appendix A: 30 30 40 40 40 40 50 50 50 50 70 70 70 70 90 90 90 90 5 5 5 5 10 10 10 10 20 20

.5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8

3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 2 2

2.9839 1.4195 3.1671 3.0396 3.0029 1.5260 2.7298 2.6866 2.5838 1.4481 1.9549 2.6206 2.6296 1.7008 1.7931 2.1595 2.1755 1.4527

8.8239 8.1036 5.3311 2.2409 6.7883 6.4508

6.3689 9.4467 2.5271 4.0319 6.4684 9.4762 2.8953 4.2363 6.6220 9.5146 3.4183 4.6680 6.7736 9.5445 3.9130 4.9059 6.9019 9.5741 0.2534 1.5881 3.8544 6.5880 0.5619 1.8497 4.0687 6.6276 0.8512 2.0917

Tables of Results 5.4501 2.3723 9.2919 7.7871 5.3506 2.3428 8.9237 7.5827 5.1970 2.3044 8.4007 7.1510 5.0454 2.2745 7.9060 6.9131 4.9171 2.2449 8.2266 6.8919 4.6256 1.8920 7.9181 6.6303 4.4113 1.8524 7.6288 6.3883

1.2057 0.6260 1.3860 1.2730 1.2607 0.6341 1.2264 1.1700 1.1024 0.6294 0.9412 1.1170 1.1271 0.6822 0.9073 0.9197 0.9443 0.5923

4.1088 3.6671 2.3911 1.0076 3.2716 2.9631

A. 37

2.4697 3.8040 0.8330 1.4370 2.5047 3.8196 1.0104 1.5420 2.5544 3.8254 1.2132 1.6810 2.6251 3.8412 1.3513 1.8337 2.6843 3.8563 -0.0018 0.5126 1.3946 2.4961 0.0993 0.5946 1.4561 2.5126 0.2856 0.7381

2.3943 1.06 4.031 3.427 2.3593 1.0444 3.8536 3.322 2.3096 1.0386 3.6508 3.183 2.2389 1.0228 3.5127 3.0303 2.1797 1.0077 3.3068 2.7924 1.9104 .8089 3.2057 2.7104 1.8489 .7924 3.0194 2.5669

6.8281 5.7628 8.5830 8.2716 6.8099 5.7427 9.2743 8.1233 6.8099 6.4055 12.6132 9.5620 6.9099 5.7203 22.0765 11.0954 7.7612 6.0773

3.8634 3.3864 2.5828 2.6103 4.5313 3.2202

-4.2162 -11.9787 -1.0650 -1.6899 -4.7636 -12.1056 -1.2672 -2.0982 -4.5756 -11.9560 -1.3483 -2.1245 -5.1796 -12.1212 -1.8850 -2.8661 -5.1817 -12.2442 0.1415 -0.8028 -4.3434 -13.2386 -0.2096 -1.3866 -4.7402 -12.6127 -0.6577 -2.1908

10.2547 18.0172 7.1035 7.7284 10.8021 18.1441 7.3057 8.1367 10.6141 17.9945 7.3868 8.163 11.2181 18.1597 7.9235 8.9046 11.2202 18.2827 8.2855 9.2298 12.7704 21.6656 8.6366 9.8136 13.1672 21.0397 9.0847 10.6178

Appendix A: 20 20 30 30 30 30 40 40 40 40 50 50 50 50 70 70 70 70 90 90 90 90 5

.5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 1.5

4.6377 2.3661 6.3349 5.3149 4.2482 1.9742 5.2026 4.7025 4.3637 2.0398 4.9773 4.5914 3.8064 1.9923 3.9401 4.3097 3.8191 2.1731 3.5714 3.7503 3.3643 1.9762

4.2035 6.6498 1.1046 2.2809 4.2942 6.6624 1.2236 2.3558 4.4902 6.6510 1.8038 2.8021 4.5056 6.7198 2.0645 3.0181 4.6241 6.6778 2.3523 3.1577 4.7517 6.7586 0.0422

Tables of Results 4.2765 1.8302 7.3754 6.1991 4.1858 1.8176 7.2564 6.1242 3.9898 1.8290 6.6762 5.6779 3.9744 1.7602 6.4155 5.4619 3.8559 1.8022 6.1277 5.3223 3.7283 1.7214 6.3487

2.1625 1.1165 2.8383 2.5214 1.8696 0.9202 2.5925 2.2848 1.9292 0.9285 2.3391 2.1381 1.7476 0.9215 1.9476 2.0283 1.7605 0.9599 1.9041 1.7519 1.5721 0.8578

A. 38

1.5395 2.5295 0.3833 0.8064 1.5746 2.5392 0.4805 0.8898 1.6142 2.5550 0.5641 0.9511 1.6406 2.5585 0.6606 1.0333 1.6995 2.5599 0.7891 1.1069 1.7531 2.5628 0.1114

1.7655 .7755 2.9217 2.4986 1.7304 .7658 2.8245 2.4152 1.6908 .75 2.7409 2.3539 1.6644 .7465 2.6444 2.2717 1.6055 .7451 2.5159 2.1981 1.5519 .7422 2.3044

2.9487 2.0485 4.4838 3.8255 3.0074 1.9224 5.9716 4.2817 3.3653 1.5957 6.0264 4.6848 3.3485 2.5252 7.8528 5.0058 3.1049 2.0422 16.2341 7.4641 3.3538 1.4370

-5.0575 -13.4245 -1.8275 -2.4475 -5.6484 -13.4306 -1.2879 -3.2913 -5.8197 -13.8641 -2.1266 -3.1482 -5.6485 -13.4478 -3.7202 -4.2922 -6.5961 -13.4108 -5.3389 -4.1089 -7.2006 -14.4960 8.7803

13.4845 21.8515 10.2545 10.8745 14.0754 21.8576 9.7149 11.7183 14.2467 22.2911 10.5536 11.5752 14.0755 21.8748 12.1472 12.7192 15.0231 21.8378 13.7659 12.5359 15.6276 22.923 19.4397

Appendix A:

Tables of Results

A.7 Hybrids RCF with companding Results: X = PAPR (Pre-coding ) – PAPR (Pre-coding +RCF) Y =CCDF of PAPR (Pre-coding) - CCDF of PAPR (Pre-coding +RCF) Z= SNR (BER= ) (Pre-coding) – SNR (BER= ) (Pre-coding +RCF) X1 == PAPR (RCF) – PAPR (Pre-coding +RCF) Y1 =CCDF of PAPR (RCF) - CCDF of PAPR (Pre-coding +RCF) Z1= SNR (BER= ) (RCF) – SNR (BER= ) (Pre-coding +RCF) Table A.29 (Pre-coding +RCF) Results and compared with the results of each of (Pre-coding) and (RCF). Precoding

PAPR

CCDF OF PAPR

WHT WHT WHT WHT WHT WHT DCT DCT DCT DCT DCT DCT DST DST DST

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

4 3 2 1.5 1.3 1.1 4 3 2 1.5 1.3 1.1 4 3 2

8.8659 11.4801 14.8677 16.6813 17.6205 18.3732 4.2276 6.9879 10.3525 12.1038 12.7949 13.3286 3.5817 6.3420 9.7467

0.1148 0.2126 0.2786 0.2274

13.9718 11.3576 7.9700 6.1564 5.2172 4.4645 13.8834 11.1231 7.7585 6.0072 5.3161 4.7824 13.8832 11.1229 7.7182

3.8833 5.1262 6.8155 7.8561 8.2340 8.6977 1.6434 2.8918 4.6089 5.6285 6.0657 6.4410 1.5024 2.7508 4.4656

0.0031 0.0066 0.0295 0.8515

6.0213 4.7784 3.0891 2.0485 1.6706 1.2069 6.0206 4.7722 3.0551 2.0355 1.5983 1.223 6.0206 4.7722 3.0574

-0.1058 -0.6262 -2.5798 -13.3450 -18.3700 -18.3700 0.0853 0.0641 -1.8862 -5.2472 -10.2117 -18.3720 -0.1290 -0.3210 -1.4552

0.0842 0.1149 0.1658 4.5250

0.2032 0.4471 0.4901 0.3766

0.2034 0.4473 0.5304

A. 39

0.0038 0.0128 0.0635 0.8645

0.0038 0.0128 0.0612

0.2773 0.8072 0.8614 12.6248

0.0630 0.4221 1.2924

SNR (BER= ) 11.7358 12.2562 14.2098 24.975 >30 >30 11.5427 11.5639 13.5142 16.8752 21.8397 >30 11.757 11.949 13.0832

Appendix A: DST DST DST DHT DHT DHT DHT DHT DHT DCT DCT DCT DCT DST DST DST DST DHT DHT DHT DHT WHT WHT WHT WHT DCT DCT DCT DCT

1 1 1 1 1 1 1 1 1 Pilot Pilot Pilot Pilot Pilot Pilot Pilot Pilot Pilot Pilot Pilot Pilot 2 2 2 2 2 2 2 2

1.5 1.3 1.1 4 3 2 1.5 1.3 1.1 4 3 2 1.5 4 3 2 1.5 4 3 2 1.5 4 3 2 1.5 4 3 2 1.5

11.6978 12.7190 13.0431 0.0069 0.0115 0.0008 1.7911 2.3945 0.0323 4.1673 6.7675 9.3391 10.5003 3.5404 6.1023 8.8583 9.9215 -6.9472 -4.2539 -1.3648 -0.5168 8.4655 10.7834 13.4396 14.6473 3.9078 6.3048 9.1329 10.1414

0.6167

7.1348 4.6230 1.2907 1.2162

0.2536 0.5734 0.6713 0.4001 0.2728 0.5543 0.8366 0.4674 0.2914 0.7043 1.1197 0.5353 0.1210 0.2199 0.2092 0.1925 0.2900 0.4680 0.6292 0.4133

Tables of Results 5.7671 4.7459 4.4218 6.9518 6.9472 6.9579 5.1676 4.5642 6.9264 13.9437 11.3435 8.7719 7.6107 13.9245 11.3626 8.6066 7.5434 13.9059 11.2126 8.3235 7.4755 14.3722 12.0543 9.3981 8.1904 14.2032 11.8062 8.9781 7.9696

5.4920 5.9291 6.3389 0.3397 0.3397 0.3397 1.4546 1.6229 1.4687 1.5259 2.6731 3.9837 4.5440 1.3769 2.4963 3.8617 4.4768 -2.7662 -1.6302 -0.2838 0.1727 3.5038 4.6331 5.9192 6.3965 1.3830 2.4546 3.6787 4.2608

A. 40

0.8690

3.0141 1.7747 0.1083 1.0046

0.1099 0.2341 0.1993 0.2368 0.1019 0.1983 0.2183 0.3106 0.1318 0.2448 0.2458 0.1795 0.0169 0.0835 0.1362 0.1176 0.1367 0.1456 0.1363 0.2225

2.031 1.5939 1.1841 3.0103 3.0103 3.0103 1.8954 1.7271 1.8813 6.1381 4.9909 3.6803 3.12 6.1461 5.0267 3.6613 3.0462 6.1162 4.9802 3.6338 3.1773 6.4008 5.2715 3.9854 3.5081 6.281 5.2094 3.9853 3.4032

-5.1509 -10.4894 -18.3720 0.0759 0.0272 -0.0615 -1.0130 -1.4779 -1.5015 2.4153 2.0345 -1.2236 -10.8087 2.2559 1.7912 -1.1227 -8.0414 1.9934 1.8664 0.2611 -2.5626 2.8937 1.9358 -0.8971 -18.3700 2.7806 2.3883 -0.3089 -7.3951

12.7211

0.3139 0.8163 2.7321 16.9050

0.1473 0.2480 0.1484 7.1633 -0.0121 0.0047 0.2493 9.9306 -0.2286 0.1259 1.6791 15.4554 0.0316 -0.2802 -0.3991 -0.4000 -0.0795 0.1743 0.1911 10.5769

16.7789 22.1174 >30 11.5061 11.5548 11.6435 12.595 13.0599 13.0835 9.2127 9.5935 12.8516 22.4367 9.3721 9. 8368 12.7507 19.6694 9.5886 9.7156 11.3209 14.1446 8.7363 9.6942 12.5271 >30 8.8474 9.2397 11.9369 19.0231

Appendix A: DST DST DST DST DHT DHT DHT DHT

2 2 2 2 2 2 2 2

4 3 2 1.5 4 3 2 1.5

3.2174 5.7121 8.4680 9.5290 -7.2690 -4.7567 -1.9188 -0.5578

0.2457 0.5214 0.6104 0.4470 0.2655 0.5588 0.7298 0.8664

Tables of Results 14.2475 11.7528 8.9969 7.9359 14.2277 11.7154 8.8775 7.5165

1.2382 2.3688 3.6055 4.1101 -2.9015 -1.7979 -0.6022 -0.0681

0.1329 0.2008 0.2041 0.2128 0.1662 0.2071 0.1694 0.2076

6.2848 5.1542 3.9175 3.4129 6.2515 5.1479 3.9522 3.4181

2.8864 2.6270 -0.3350 -7.2475 2.7701 2.6878 0.7115 -0.8762

0.0263 0.4130 0.1650 10.7245 -0.0440 0.5198 1.2575 17.1418

8.7416 9.001 11.963 18.8755 8.8119 8.8942 10.8705 12.4582

A.8 Hybrid Pre-coding with Companding Results: X = PAPR (Pre-coding ) – PAPR (Pre-coding + Companding) Y =CCDF of PAPR (Pre-coding) - CCDF of PAPR(Pre-coding + Companding) Z= SNR (BER= ) (Pre-coding) – SNR (BER= ) (Pre-coding + Companding) X1 == PAPR (Companding) – PAPR (Pre-coding + Companding) Y1 =CCDF of PAPR (Companding) - CCDF of PAPR (Pre-coding + Companding) Z1= SNR (BER= ) (Companding) – SNR (BER= ) (Pre-coding + Companding) Table A.30 (Pre-coding +A) Results and compared with the results of each of (Pre-coding) and (A companding). Precoding

PAPR

WHT WHT WHT WHT WHT WHT WHT

5 10 15 20 30 35 40

6.4003 8.2527 9.0269 9.4809 10.0185 10.1382 10.2848

2.4687 0.1067 0.5157 1.0833 -0.9647 0.2105 0.0499

16.4374 14.5850 13.8108 13.3568 12.8192 12.6995 12.5529

4.2596 5.5140 6.0383 6.3435 6.7024 6.9423 7.0335

0.9950 0.3394 0.2597 0.1239 0.1178 0.4507 0.3889

A. 41

CCDF OF PAPR 5.645 4.3906 3.8663 3.5611 3.2022 2.9623 2.8711

-2.2393 -4.7003 -6.2322 -6.6714 -8.0292 -8.1607 -8.5875

-0.2693 -0.2103 -0.2862 -0.3014 -0.4592 -0.5907 -0.4605

SNR (BER= ) 13.8693 16.3303 17.8622 18.3014 19.6592 19.7907 20.2175

Appendix A: WHT WHT WHT WHT WHT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DST DST DST DST DST DST DST DST DST DST DST

50 70 87.6 100 120 5 10 15 20 30 35 40 50 70 87.6 100 120 5 10 15 20 30 35 40 50 70 87.6 100

10.5114 10.8155 10.9967 11.0811 11.2105 6.2514 7.7579 8.3981 8.7766 9.5223 9.6724 9.5424 9.9838 9.9556 10.1086 10.6923 10.3032 6.1302 7.6475 8.2865 8.6628 9.1103 8.2173 8.3394 8.5285 8.7829 8.9348 9.1724

0.2643 0.1281 -0.0591 -0.4023 -0.2171 7.0465 4.3386 4.6136 5.1057 3.2658 4.4714 4.0342 4.4634 3.9949 3.7795 3.9356 3.6023 7.5714 4.8743 5.1481 5.6380 3.4999 3.6624 3.4773 3.6542 3.4683 3.2518 3.0618

Tables of Results 12.3263 12.0222 11.8410 11.7566 11.6272 11.8596 10.3531 9.7129 9.3344 8.5887 8.4386 8.5686 8.1272 8.1554 8.0024 7.4187 7.8078 11.3347 9.8174 9.1784 8.8021 8.3546 9.2476 9.1255 8.9364 8.6820 8.5301 8.2925

7.1740 7.3616 7.4729 7.5333 7.6154 3.4636 4.3642 4.7366 4.9509 5.2077 5.2937 5.4200 5.4722 5.5693 5.6578 5.7713 5.7703 3.3458 4.2613 4.6392 4.8599 5.1202 5.1115 5.1855 5.2998 5.4526 5.5434 5.6106

A. 42

0.3694 0.2940 0.2913 0.2087 0.2458 2.4396 1.4302 1.1986 0.9719 0.8637 1.0427 1.0160 0.9082 0.7423 0.7168 0.6873 0.6413 2.4628 1.4683 1.2422 1.0219 0.9172 1.0015 0.9225 0.8768 0.7666 0.7434 0.6676

2.7306 2.543 2.4317 2.3713 2.2892 4.2004 3.2998 2.9274 2.7131 2.4563 2.3703 2.244 2.1918 2.0947 2.0062 1.8927 1.8937 4.1772 3.2617 2.8838 2.6631 2.4028 2.4115 2.3375 2.2232 2.0704 1.9796 1.9124

-10.0027 -9.5360 -10.0027 -10.6669 -10.9249 -2.4773 -4.6072 -6.0249 -6.7160 -7.3912 -7.7982 -8.0486 -8.6099 -9.0955 -9.5582 -9.7262 -10.0745 -2.7785 -4.8245 -6.4536 -7.0850 -8.3413 -8.2486 -8.4777 -9.0263 -9.6570 -10.1584 -10.2688

-1.5167 -0.0360 -0.2607 -0.6769 -0.6919 -0.5053 -0.1152 -0.0769 -0.3440 0.1808 -0.2262 0.0804 -0.1219 0.4065 0.1858 0.2658 0.1605 -0.8065 -0.3325 -0.5056 -0.7130 -0.7693 -0.6766 -0.3487 -0.5383 -0.1550 -0.4144 -0.2768

21.6327 21.166 21.6327 22.2969 22.5549 14.1053 16.2352 17.6529 18.344 19.0192 19.4262 19.6766 20.2379 20.7235 21.1862 21.3542 21.7025 14.4065 16.4525 18.0816 18.713 19.9693 19.8766 20.1057 20.6543 21.285 21.7864 21.8968

Appendix A: DST DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT

120 5 10 15 20 30 35 40 50 70 87.6 100 120

9.2816 2.2752 2.6637 2.8227 2.9158 3.0324 3.0690 3.0988 3.1450 3.2239 3.2613 3.2819 3.3088

3.2268 14.2226 10.3967 10.1905 10.3972 7.9282 9.0203 8.7429 8.7769 8.4155 8.0845 7.6775 7.7602

Tables of Results 8.1833 4.6835 4.2950 4.1360 4.0429 3.9263 3.8897 3.8599 3.8137 3.7348 3.6974 3.6768 3.6499

5.6740 0.9293 1.0712 1.1343 1.1712 1.1238 1.1358 1.1456 1.1608 1.1915 1.2050 1.2125 1.2224

0.6860 4.2193 2.4512 1.9103 1.5062 1.0938 1.1988 1.0556 0.9108 0.6785 0.5780 0.4425 0.4074

1.849 2.4207 2.2788 2.2157 2.1788 2.2262 2.2142 2.2044 2.1892 2.1585 2.145 2.1375 2.1276

-10.5360 -0.8663 -1.5606 -2.2378 -2.7303 -3.4581 -3.6582 -3.7728 -4.0772 -4.4893 -4.6584 -4.7606 -4.9040

-0.3010 1.1517 2.9774 3.7562 3.6877 4.1599 3.9598 4.4022 4.4568 5.0587 5.1316 5.2774 5.3770

22.164 12.4483 13.1426 13.8198 14.3123 15.0401 15.2402 15.3548 15.6592 16.0713 16.2404 16.3426 16.486

Table A.31(Pre-coding + ) Results and compared with the results of each of (Pre-coding) and ( companding). Precoding WHT WHT WHT WHT WHT WHT WHT

5 10 20 30 40 50 60

PAPR

6.3192 8.7224 9.9641 8.8772 9.2710 10.2718 8.9088

0.9147 2.4317 4.3837 0.6751 0.2786 1.2963 -0.1962

16.5185 14.1153 12.8736 13.9605 13.5667 12.5659 13.9289

4.1987 5.1385 5.8731 6.1212 6.3771 6.6101 6.4984

0.7101 1.0039 1.0465 0.6166 0.4725 0.5055 0.2538

A. 43

CCDF OF PAPR 5.7059 4.7661 4.0315 3.7834 3.5275 3.2945 3.4062

-1.8052 -3.1978 -4.9404 -5.8312 -6.7931 -7.0780 -7.2855

-0.0989 -0.1878 -0.3204 -0.2962 -0.6731 -0.4380 -0.1385

SNR (BER= ) 13.4352 14.8278 16.5704 17.4612 18.4231 18.708 18.9155

Appendix A: WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT DCT DCT DCT DCT DCT DCT DCT DCT

70 80 90 100 120 140 160 180 200 220 240 250 255 260 280 300 320 500 700 1000 5 10 20 30 40 50 60 70

9.0814 11.7882 11.9079 10.9096 11.5372 11.6736 9.8688 9.9644 11.3965 11.5960 11.6595 12.1288 12.1429 11.3810 11.8364 11.8831 11.8574 12.1312 12.4720 13.5600 5.2168 6.4678 6.0880 9.0769 8.2610 8.7019 8.8635 8.3644

0.4674 5.5006 1.9286 0.4861 0.8137 1.5738 -0.7606 -0.2875 0.8011 0.0305 4.7362 1.7269 1.0486 0.5296 0.5878 0.9527 1.3900 0.4886 0.4561 3.6176 4.5390 4.9038 5.2343 5.6015 3.9953 4.4531 4.4852 4.4771

Tables of Results 13.7563 11.0495 10.9298 11.9281 11.3005 11.1641 12.9689 12.8733 11.4412 11.2417 11.1782 10.7089 10.6948 11.4567 11.0013 10.9546 10.9803 10.7065 10.3657 9.2777 12.8942 11.6432 12.0230 9.0341 9.8500 9.4091 9.2475 9.7466

6.6125 7.0090 7.0824 6.9954 7.1759 7.2552 7.1305 7.1930 7.3336 7.3780 7.4171 7.5244 7.5329 7.5448 7.4979 7.5278 7.5393 7.7131 7.8745 8.0035 3.2318 3.8305 4.1559 4.7366 4.8099 5.0522 5.1270 5.1100

A. 44

0.3539 0.4344 0.4778 0.2108 0.2713 0.4106 0.0919 0.1684 0.1890 0.1184 0.3625 0.3428 0.3083 0.3062 0.1933 0.2332 0.3347 0.1085 0.1399 0.3789 1.9838 1.9365 1.5699 1.4726 1.1459 1.1882 1.1230 1.0920

3.2921 2.8956 2.8222 2.9092 2.7287 2.6494 2.7741 2.7116 2.571 2.5266 2.4875 2.3802 2.3717 2.3598 2.4067 2.3768 2.3653 2.1915 2.0301 1.9011 4.4322 3.8335 3.5081 2.9274 2.8541 2.6118 2.537 2.554

-7.4804 -8.0466 -8.1737 -8.4410 -9.1505 -9.4242 -9.2081 -9.8181 -10.0901 -10.3356 -10.5607 -10.3413 -10.4272 -10.3295 -10.6842 -10.8583 -11.0382 -11.5297 -11.9464 -12.5842 -1.8527 -3.3271 -4.7831 -5.5879 -6.2800 -6.7103 -7.2375 -7.4138

-0.1104 -0.2026 -0.2037 -0.0710 -0.5005 -0.5542 -0.3631 -0.3981 -0.7201 -0.7271 -0.5907 -0.3713 -0.5892 -0.2595 -0.4142 -0.4883 -0.5432 -0.2397 -0.0764 -0.4502 -0.1444 -0.3151 -0.1611 -0.0509 -0.1580 -0.0683 -0.0885 -0.0418

19.1104 19.6766 19.8037 20.071 20.7805 21.0542 20.8381 21.4481 21.7201 21.9656 22.1907 21.9713 22.0572 21.9595 22.3142 22.4883 22.6682 23.1597 23.5764 24.2142 13.4807 14.9551 16.4111 17.2159 17.908 18.3383 18.8655 19.0418

Appendix A: DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DST DST DST DST DST DST DST DST DST

80 90 100 120 140 160 180 200 220 240 250 255 260 280 300 320 500 700 1000 5 10 20 30 40 50 60 70 80

9.4263 9.7652 9.4938 8.9284 10.3110 9.7695 10.1595 10.2138 9.3948 10.3095 9.0514 10.4596 9.5897 10.6507 9.8587 10.7427 10.1111 10.8348 11.1737 5.4486 6.8133 6.7980 8.3677 7.6474 9.2676 9.0030 9.2551 9.2642

7.8654 4.5126 3.7970 2.9316 4.9379 3.8668 4.6343 4.3451 2.5560 8.1129 3.3762 4.0920 3.4650 4.1288 3.6550 5.0020 3.1952 3.5456 5.9580 5.4169 5.8954 6.5904 5.5384 4.0278 5.6649 5.2708 6.0139 8.3494

Tables of Results 8.6847 8.3458 8.6172 9.1826 7.8000 8.3415 7.9515 7.8972 8.7162 7.8015 9.0596 7.6514 8.5213 7.4603 8.2523 7.3683 7.9999 7.2762 6.9373 12.0163 10.6516 10.6669 9.0972 9.8175 8.1973 8.4619 8.2098 8.2007

5.3400 5.3276 5.3421 5.3968 5.5709 5.5726 5.6011 5.5775 5.6864 5.6913 5.7476 5.7487 5.6988 5.8074 5.7534 5.8207 5.9902 6.0670 6.1554 3.0512 3.7000 4.1850 4.6087 4.7031 5.0013 5.0522 5.1230 5.2023

A. 45

1.0060 0.9636 0.7981 0.7328 0.9669 0.7746 0.8171 0.6735 0.6674 0.8773 0.8066 0.7647 0.7008 0.7434 0.6994 0.8567 0.6262 0.5730 0.7714 1.9442 1.9470 1.7400 1.4857 1.1801 1.2783 1.1892 1.2460 1.0093

2.324 2.3364 2.3219 2.2672 2.0931 2.0914 2.0629 2.0865 1.9776 1.9727 1.9164 1.9153 1.9652 1.8566 1.9106 1.8433 1.6738 1.597 1.5086 4.4718 3.823 3.338 2.9143 2.8199 2.5217 2.4708 2.4 2.3207

-7.4558 -8.0063 -8.2689 -8.6476 -8.9126 -8.9073 -9.0835 -9.6093 -9.1604 -9.7126 -9.8144 -9.8207 -10.0232 -10.0021 -10.2307 -10.4537 -10.8432 -11.7915 -12.0486 -2.3432 -3.5551 -5.0565 -6.3507 -6.8122 -6.9744 -7.7744 -8.1161 -8.4165

0.3902 -0.0343 0.1031 0.0044 -0.0406 -0.0603 0.3385 -0.2373 0.4501 0.2594 0.1576 0.0193 0.0488 0.2699 0.1413 0.0433 0.4488 0.0805 0.0874 -0.6349 -0.5431 -0.4345 -0.8137 -0.6902 -0.3324 -0.6254 -0.7441 -0.5705

19.0838 19.6343 19.8969 20.2756 20.5406 20.5353 20.7115 21.2373 20.7884 21.3406 21.4424 21.4487 21.6512 21.6301 21.8587 22.0817 22.4712 23.4195 23.6766 13.9712 15.1831 16.6845 17.9787 18.4402 18.6024 19.4024 19.7441 20.0445

Appendix A: DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT

90 100 120 140 160 180 200 220 240 250 255 260 280 300 320 500 700 1000 5 10 20 30 40 50 60 70 80 90

9.4738 9.3820 9.7971 9.7060 10.0042 9.8794 10.1197 9.5615 10.2651 9.6395 9.5708 8.1344 9.7061 8.2172 9.1446 10.3526 8.6398 10.6545 1.9728 2.3552 2.6599 2.8047 2.8756 2.9382 2.9855 3.0230 3.0564 3.0824

4.8673 4.3313 4.4464 4.9790 4.7476 5.0003 4.8971 3.3688 8.7146 4.6104 3.8493 2.6558 3.8303 2.6596 4.0500 4.0828 1.9967 6.0849 12.4473 11.9435 12.9585 10.4816 9.7622 9.8417 9.7595 10.2880 12.6478 8.9821

Tables of Results 7.9911 8.0829 7.6678 7.7589 7.4607 7.5855 7.3452 7.9034 7.1998 7.8254 7.8941 9.3305 7.7588 9.2477 8.3203 7.1123 8.8251 6.8104 4.9859 4.6035 4.2988 4.1540 4.0831 4.0205 3.9732 3.9357 3.9023 3.8763

5.2485 5.3318 5.3680 5.4550 5.4857 5.5538 5.5763 5.6042 5.6338 5.6467 5.6370 5.5533 5.6830 5.6017 5.6361 5.8364 5.8487 6.0095 0.7742 0.9217 1.0222 1.0703 1.0526 1.0762 1.0940 1.1081 1.0898 1.0982

A. 46

1.0255 0.9288 0.8450 0.9920 0.8287 0.9108 0.8133 0.7262 0.9608 0.8467 0.7940 0.6963 0.7600 0.6887 0.8131 0.6134 0.4957 0.7665 3.8402 3.3417 2.7502 2.1203 1.7026 1.5262 1.4040 1.4041 1.0698 1.0482

2.2745 2.1912 2.155 2.068 2.0373 1.9692 1.9467 1.9188 1.8892 1.8763 1.886 1.9697 1.84 1.9213 1.8869 1.6866 1.6743 1.5135 2.5758 2.4283 2.3278 2.2797 2.2974 2.2738 2.256 2.2419 2.2602 2.2518

-8.5426 -8.6108 -9.0312 -9.3167 -9.6425 -9.7710 -10.0101 -10.1189 -10.4585 -10.3842 -10.4538 -10.7142 -10.7494 -10.8662 -10.8145 -11.6766 -11.6894 -12.5300 -0.7449 -1.2059 -2.1132 -2.4565 -2.7943 -3.3194 -3.5662 -3.7728 -3.7046 -3.7978

-0.5706 -0.2388 -0.3792 -0.4447 -0.7955 -0.3490 -0.6381 -0.5084 -0.4865 -0.4122 -0.6138 -0.6422 -0.4774 -0.4942 -0.3175 -0.3846 0.1826 -0.3940 1.0094 1.8521 2.5548 3.1265 3.3737 3.3686 3.6288 3.6452 4.1874 4.2202

20.1706 20.2388 20.6592 20.9447 21.2705 21.399 21.6381 21.7469 22.0865 22.0122 22.0818 22.3422 22.3774 22.4942 22.4425 23.3046 23.3174 24.158 12.3269 12.7879 13.6952 14.0385 14.3763 14.9014 15.1482 15.3548 15.2866 15.3798

Appendix A: DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT

100 120 140 160 180 200 220 240 250 255 260 280 300 320 500 700 1000

3.0929 3.1412 3.1703 3.1929 3.2132 3.2306 3.2458 3.2399 3.2541 3.2490 3.2519 3.2708 3.2715 3.2898 3.3475 3.3767 3.4307

8.5484 8.2967 8.9495 8.4425 8.8403 8.5142 7.5593 12.1956 8.7312 8.0337 8.2795 7.9012 8.2201 8.7014 7.5839 7.2398 9.3673

Tables of Results 3.8658 3.8175 3.7884 3.7658 3.7455 3.7281 3.7129 3.7188 3.7046 3.7097 3.7068 3.6879 3.6872 3.6689 3.6112 3.5820 3.5280

1.1892 1.1173 1.1267 1.2065 1.2134 1.2193 1.2244 1.1561 1.2272 1.1591 1.1600 1.2337 1.2504 1.2412 1.2639 1.2848 1.3010

0.9592 0.7673 0.8367 0.7225 0.7434 0.6293 0.5194 0.6561 0.6002 0.4891 0.4760 0.4837 0.5104 0.5912 0.2139 0.1048 0.2310

2.1608 2.2327 2.2233 2.1435 2.1366 2.1307 2.1256 2.1939 2.1228 2.1909 2.19 2.1163 2.0996 2.1088 2.0861 2.0652 2.049

-3.7443 -4.0950 -4.2397 -4.2496 -4.3751 -4.4275 -4.5322 -4.8847 -4.9731 -5.2350 -5.2350 -5.0874 -5.0789 -5.3548 -5.7347 -5.7148 -6.0038

4.6737 4.6030 4.6783 4.6434 5.0929 4.9905 5.1243 5.1333 5.0449 4.6510 4.8830 5.2306 5.3391 5.1882 5.6033 6.2032 6.1782

15.3263 15.677 15.8217 15.8316 15.9571 16.0095 16.1142 16.4667 16.5551 16.817 16.817 16.6694 16.6609 16.9368 17.3167 17.2968 17.5858

Table A.32 (Pre-coding + RCT) Results and compared with the results of each of (Pre-coding) and (RCT) Pre-coding

PAPR

WHT WHT WHT WHT WHT WHT WHT

.9 .8 .7 .6 .5 .4 .3

3.3459 5.1820 6.1784 8.2743 10.1461 13.2863 15.4834

2.3713 3.4754 1.4698 0.7913 1.2348 1.9778 1.1986

19.4918 17.6557 16.6593 14.5634 12.6916 9.5514 7.3543

1.3932 2.1560 2.6885 3.5811 4.3915 5.6906 6.6295

1.0386 0.9329 0.8419 0.3590 0.3219 0.6075 0.4349

A. 47

CCDF OF PAPR 8.5114 7.7486 7.2161 6.3235 5.5131 4.214 3.2751

-0.1006 -0.2737 -1.0924 -1.9051 -3.0635 -4.9407 -7.1992

-0.0541 0.0833 -0.3087 -0.1351 -0.2435 -0.4257 -0.3042

SNR (BER= ) 11.7306 11. 9037 12.7224 13.5351 14.6935 16.5707 18.8292

Appendix A: WHT WHT DCT DCT DCT DCT DCT DCT DCT DCT DCT DST DST DST DST DST DST DST DST DST DHT DHT DHT DHT DHT DHT DHT DHT DHT

.2 .1 .9 .8 .7 .6 .5 .4 .3 .2 .1 .9 .8 .7 .6 .5 .4 .3 .2 .1 .9 .8 .7 .6 .5 .4 .3 .2 .1

17.7489 20.1831 2.6159 4.0365 5.5106 6.0660 7.8308 9.6726 11.9638 13.6369 15.8025 1.7433 3.1975 4.7059 6.2744 7.9097 9.1755 11.0822 13.0856 15.2166 0.3560 0.7383 1.1257 1.5174 1.9183 2.3380 2.7565 3.1938 3.6504

0.9000 0.2180 6.3680 7.0566 5.5287 3.3097 3.6462 3.0908 2.4057 1.5147 0.5641 6.1415 6.8637 5.3701 4.1642 4.3712 3.2398 2.1702 1.6095 0.6243 15.2604 14.9107 12.2961 9.9134 8.8860 6.9085 4.3507 2.2239 -0.4357

Tables of Results 5.0888 2.6546 15.4951 14.0745 12.6004 12.0450 10.2802 8.4384 6.1472 4.4741 2.3085 15.7216 14.2674 12.7590 11.1905 9.5552 8.2894 6.3827 4.3793 2.2483 6.6027 6.2204 5.8330 5.4413 5.0404 4.6207 4.2022 3.7649 3.3083

7.6444 8.7120 0.8902 1.5129 2.1557 2.5025 3.2498 4.0337 4.9843 5.7340 6.6729 0.7555 1.3692 2.0010 2.6500 3.3369 3.9292 4.7562 5.6273 6.5535 0.1052 0.2788 0.5084 0.6703 0.7518 0.8374 1.0510 1.0568 1.2249

A. 48

0.3143 0.0754 2.7762 2.5304 2.5497 1.5210 1.4208 1.1912 1.0303 0.6445 0.2769 2.7825 2.5277 2.5360 1.8095 1.6489 1.2277 0.9432 0.6788 0.2985 6.3052 5.6103 5.2164 4.0028 3.2368 2.3089 1.4110 0.2813 -0.8571

2.2602 1.1926 6.7738 6.1511 5.5083 5.1615 4.4142 3.6303 2.6797 1.93 .9911 6.7675 6.1538 5.522 4.873 4.1861 3.5938 2.7668 1.8957 .9695 3.2448 3.0712 2.8416 2.6797 2.5982 2.5126 2.299 2.2932 2.1251

-10.7564 -17.0412 -0.0147 -0.3815 -1.0401 -1.8911 -3.0857 -4.7014 -6.6401 -10.2162 -16.5112 -0.0393 -0.3827 -0.9126 -1.8332 -2.9499 -4.6498 -7.0794 -10.5597 -16.8057 0.0718 -0.2792 -0.6722 -1.1254 -1.5191 -2.5360 -3.4451 -5.8003 -9.5778

-0.1364 -0.3712 0.0338 -0.0225 -0.2544 -0.1191 -0.2637 -0.1844 0.2569 0.4058 0.1608 0.0092 -0.0237 -0.1269 -0.0612 -0.1279 -0.1328 -0.1824 0.0623 -0.1337 0.1663 0.1258 0.1595 0.6926 1.3489 2.0270 3.4979 4.8677 7.1402

22.3864 28.6712 11.6427 12.0095 12.6681 13.5191 14.7137 16.3294 18.2681 21.8442 28.1392 11.6673 12.0107 12.5406 13.4612 14.5779 16.2778 18.7074 22.1877 28.4337 11.5102 11.8612 12.2542 12.7074 13.1011 14.118 15.0271 17.3823 21.1598

Appendix A:

Tables of Results

Table A.33(Pre-coding +AEXP) Results and compared with the results of each of (Pre-coding) and (AEXP companding). Precoding WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT DCT DCT DCT DCT

AEXP d

PAPR

2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 .9 .8 .7 .6 .5 .4 .3 .2 .1 2 1.9 1.8 1.7

10.7804 11.3253 11.7057 12.2595 12.3909 12.9152 13.4360 14.2743 14.8045 15.2924 15.8627 16.4192 17.0353 17.4907 18.2253 18.8350 19.5373 20.2553 21.0167 21.8494 6.7007 7.1704 7.6401 8.2400

1.0238 1.5116 1.0663 0.6391 0.2494 0.1439 0.2128 0.4181 0.4168 0.4070 0.4024 0.3315 0.2782 0.1783 0.1768 0.1317 0.1035 0.0694 0.0148 -0.0193 1.6708 2.0834 1.7274 1.3463

12.0573 11.5124 11.1320 10.5782 10.4468 9.9225 9.4017 8.5634 8.0332 7.5453 6.9750 6.4185 5.8024 5.3470 4.6124 4.0027 3.3004 2.5824 1.8210 0.9883 11.4103 10.9406 10.4709 9.8710

4.9111 5.0575 5.2111 5.4254 5.5148 5.6605 5.8436 6.0031 6.2983 6.4313 6.6393 6.8279 7.1031 7.2846 7.6039 7.9506 8.2612 8.5546 8.9118 9.3313 2.9261 3.0746 3.2279 3.4745

0.1598 0.2929 0.2250 0.2908 0.1952 0.1139 0.0855 0.0785 0.1997 0.1262 0.1087 0.1153 0.1185 0.0170 0.0393 0.1535 0.0966 -0.0228 -0.0503 -0.0617 0.4154 0.5506 0.4824 0.5805

A. 49

CCDF OF PAPR 4.9935 4.8471 4.6935 4.4792 4.3898 4.2441 4.061 3.9015 3.6063 3.4733 3.2653 3.0767 2.8015 2.62 2.3007 1.954 1.6434 1.35 .9928 .5733 4.7379 4.5894 4.4361 4.1895

-4.3415 -4.5962 -5.3536 -4.7950 -4.3508 -5.2656 -4.0073 -4.0536 -4.1271 -4.4413 -4.2999 -4.2078 -5.5993 -6.5697 -9.5950 -18.3700 -18.3700 -18.3700 -18.3700 -18.3700 -3.9507 -3.9175 -3.8548 -3.1582

-1.2415 -1.5262 -2.1256 -1.9750 -1.7808 -2.5818 -1.0688 -1.3836 -0.4271 -1.2213 -0.6299 0.4622 2.0157 11.8003 8.7750 0 0 0 0 0 -0.8487 -0.8455 -0.6248 -0.3362

SNR (BER= ) 15.9715 16.2262 16.9836 16.425 15.9808 16.8956 15.6373 15.6836 15.7571 16.0713 15.9299 15.8378 17.2293 18.1997 21.225 >30 >30 >>30 >>30 >>>30 15.5787 15.5455 15.4828 14.7862

Appendix A: DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DST DST DST DST DST DST DST DST DST DST DST DST DST

1.6 1.5 1.4 1.3 1.2 1.1 1 .9 .8 .7 .6 .5 .4 .3 .2 .1 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 .9 .8

8.5743 9.0476 9.5246 10.0065 10.4319 10.9577 11.4605 11.9813 12.5214 13.1121 13.6710 14.2768 14.9229 15.6162 16.3628 17.1819 5.7603 6.2487 6.7376 7.4201 7.9659 8.2102 8.6838 9.2006 9.6836 10.2053 10.6868 11.3528 11.9005

1.1595 1.0030 1.0281 0.8770 0.7709 0.7990 0.7269 0.6203 0.4910 0.5264 0.3492 0.3002 0.2158 0.1570 0.0876 0.0399 1.3765 1.8078 1.4710 1.1725 1.1972 0.8117 0.8334 0.7172 0.6687 0.6927 0.5993 0.6379 0.5162

Tables of Results 9.5367 9.0634 8.5864 8.1045 7.6791 7.1533 6.6505 6.1297 5.5896 4.9989 4.4400 3.8342 3.1881 2.4948 1.7482 0.9291 11.7046 11.2162 10.7273 10.0448 9.4990 9.2547 8.7811 8.2643 7.7813 7.2596 6.7781 6.1121 5.5644

3.6095 3.7703 3.9270 4.0957 4.5142 4.3928 4.5868 4.8018 5.0146 5.3103 5.5512 5.8225 6.1132 6.4346 6.8036 7.1850 2.8291 2.9692 3.1148 3.3569 3.4962 3.5812 3.7896 3.9663 4.1170 4.3442 4.5007 4.7532 4.9746

A. 50

0.5305 0.4643 0.4095 0.4117 0.6562 0.3283 0.2968 0.3298 0.2706 0.2833 0.2272 0.2660 0.1892 0.0978 0.0821 0.0326 0.4594 0.5862 0.5103 0.6039 0.5582 0.4162 0.4131 0.4233 0.4000 0.4207 0.3517 0.4222 0.3716

4.0545 3.8937 3.737 3.5683 3.1498 3.2712 3.0772 2.8622 2.6494 2.3537 2.1128 1.8415 1.5508 1.2294 .8604 .479 4.6939 4.5538 4.4082 4.1661 4.0268 3.9418 3.7334 3.5567 3.406 3.1788 3.0223 2.7698 2.5484

-3.2536 -2.9112 -2.7500 -2.6924 -2.4536 -2.4922 -3.0850 -3.7744 -5.2869 -6.8777 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -3.4924 -2.9424 -1.7942 -2.1349 -1.5678 -1.8026 -2.0358 -2.4312 -3.5155 -4.6118 -17.9720 -18.3720 -18.3720

-0.6816 -0.2254 0.1905 -0.0204 1.2484 0.7298 0.5870 0.8976 2.3301 11.4943 0 0 0 0 0 0 -0.3904 0.1296 1.4358 0.6871 1.0042 0.8832 0.9047 0.2408 0.1865 -1.3898 -14.3000 -13.7000 -10.7550

14.8816 14.5392 14.378 14.3204 14.0816 14.1202 14.713 15.4024 16.9149 18.5057 >30 >30 >30 >>30 >>30 >>>30 15.1204 14.5704 13.4222 13.7629 13.1958 13.4306 13.6638 14.0592 15.1435 16.2398 29.6 >30 >30

Appendix A: DST DST DST DST DST DST DST DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT

.7 .6 .5 .4 .3 .2 .1 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1.1 1 .9 .8 .7 .6 .5 .4 .3 .2 .1

12.4544 13.0419 13.6703 14.3023 15.0261 15.7624 16.5640 -0.6937 -0.4865 -0.3933 -0.2155 0.0648 0.3050 0.2838 0.5355 0.8006 1.1528 1.5750 1.8437 2.1033 2.3210 2.5943 2.8462 3.1118 3.3695 3.6525 3.8711

0.5148 0.3662 0.3398 0.2413 0.2130 0.1333 0.0681 5.4287 5.5788 4.8463 4.0431 3.8023 3.4127 2.9396 2.5583 2.2919 2.1464 1.9937 1.6350 1.2252 0.8876 0.4248 0.0219 -0.4430 -0.9374 -1.4704 -2.1186

Tables of Results 5.0105 4.4230 3.7946 3.1626 2.4388 1.7025 0.9009 7.6524 7.4452 7.3520 7.1742 6.8939 6.6537 6.6749 6.4232 6.1581 5.8059 5.3837 5.1150 4.8554 4.6377 4.3644 4.1125 3.8469 3.5892 3.3062 3.0876

5.2316 5.4339 5.6985 5.9822 6.3203 6.6784 7.0644 0.1988 0.2950 0.3265 0.3825 0.5006 0.5780 0.6027 0.6680 0.7343 0.7608 0.9006 0.9706 1.0415 1.0391 1.1280 1.1485 1.2126 1.2777 1.3818 1.3298

A. 51

0.3456 0.2509 0.2830 0.1992 0.1245 0.0979 0.0530 2.0021 2.0850 1.8950 1.8025 1.7356 1.5860 1.3992 1.2980 1.1903 1.0103 0.9246 0.8126 0.6115 0.3261 0.1180 -0.0940 -0.3974 -0.7451 -1.0257 -1.5086

2.2914 2.0891 1.8245 1.5408 1.2027 .8446 .4586 3.1512 3.055 3.0235 2.9675 2.8494 2.772 2.7473 2.682 2.6157 2.5892 2.4494 2.3794 2.3085 2.3109 2.222 2.2015 2.1374 2.0723 1.9682 2.0202

-18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -1.9173 -1.5431 -1.5752 -1.0598 -0.8207 -0.8038 -1.0845 -1.0560 -0.8581 -0.3896 -0.5745 -0.6731 -0.8873 -1.4496 -1.7265 -2.3714 -3.0057 -4.1479 -18.4180 -18.4180

0 0 0 0 0 0 0 1.2307 1.5749 1.7008 1.8082 1.7973 1.9280 1.9020 1.6620 2.8899 2.8784 3.1435 4.0449 6.7757 6.9684 16.6915 16.0466 15.4123 14.2701 0 0

>30 >>30 >>30 >>30 >>30 >>>30 >>>30 13.4993 13.1251 13.1572 12.6418 12.4027 12.3858 12.6665 12.638 12.4401 11.9716 12.1565 12.2551 12.4693 13.0316 13.3085 13.9534 14.5877 15.7299 17.8183 -30 >>30

Appendix A:

Tables of Results

Table A.34 (Pre-coding +cos) Results and compared with the results of each of (Pre-coding) and (cos companding). Precoding WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DST DST DST DST DST

Cos

PAPR

1 .9 .8 .7 .6 .5 .4 .3 .2 .1 1 .9 .8 .7 .6 .5 .4 .3 .2 .1 1 .9 .8 .7 .6

7.3889 8.6546 9.9471 11.3571 12.8150 14.2197 15.7226 17.3259 19.0411 20.8281 4.5103 5.5939 6.7188 7.8898 9.1125 10.0773 11.4935 12.9874 14.5824 16.2663 3.4964 5.0333 6.0390 7.0375 8.3999

0.1118 0.2175 0.2189 0.1886 0.2831 0.1510 0.0911 0.0641

15.4488 14.1831 12.8906 11.4806 10.0227 8.6180 7.1151 5.5118 3.7966 2.0096 13.6007 12.5171 11.3922 10.2212 8.9985 8.0337 6.6175 5.1236 3.5286 1.8447 13.9685 12.4316 11.4259 10.4274 9.0650

3.0367 3.5637 4.2290 4.7941 5.4976 6.0483 6.6971 7.4495 8.1231 8.9201 1.7336 2.2109 2.6944 3.1956 3.7198 4.1538 4.7682 5.4192 6.0790 6.8477 1.4699 2.0815 2.5870 3.0940 3.5805

-0.0179 0.0751 0.1894 0.1710 0.1960 0.0987 0.0715 0.1449

1.9599 1.8835 1.7173 1.4480 1.3073 0.7353 0.5887 0.4523

1.5921 1.9690 1.6836 1.2418 1.2408

A. 52

0.9196 0.9629 0.8954 0.8131 0.6588 0.4448 0.3832 0.3552

0.7969 0.9745 0.9290 0.8525 0.6605

CCDF of PAPR 6.8679 6.3409 5.6756 5.1105 4.407 3.8563 3.2075 2.4551 1.7815 .9845 5.9304 5.4531 4.9696 4.4684 3.9442 3.5102 2.8958 2.2448 1.585 .8163 6.0531 5.4415 4.936 4.429 3.9425

-0.0936 -0.3785 -0.8393 -1.5116 -2.6935 -3.9494 -6.0413 -9.0243 -18.3700 -18.3700 -0.0147 -0.3795 -1.0026 -1.5364 -2.5236 -3.9940 -5.9277 -11.3055 -18.3720 -18.3720 -0.5719 -0.6383 -1.3140 -2.1014 -3.0753

0.0764 0.2165 0.1107 0.2424 0.2150 0.2806 -0.1113 0.6457

0.1573 0.2175 -0.0506 0.2196 0.3869 0.2380 0.0043 -1.6335

-0.3999 -0.0413 -0.3620 -0.3454 -0.1648

SNR (BER= ) 11.7236 12.0085 12.4693 13.1416 14.3235 15.5794 17.6713 20.6543 >30 >>30 11.6427 12.0075 12.6306 13.1644 14.1516 15.622 17.5557 22.9335 >30 >>30 12.1999 12.2663 12.942 13.7294 14.7033

Appendix A: DST DST DST DST DST DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT

.5 .4 .3 .2 .1 1 .9 .8 .7 .6 .5 .4 .3 .2 .1

9.6431 11.1108 12.4423 13.9884 15.6603 -0.8692 -0.3893 0.0275 0.5943 1.2213 1.6803 2.1705 2.6262 3.1364 3.6225

Tables of Results

0.9472 0.8521 0.5533

7.8218 6.3541 5.0226 3.4765 1.8046 7.8279 7.3480 6.9312 6.3644 5.7374 5.2784 4.7882 4.3325 3.8223 3.3362

7.7327 7.0526 6.1783 5.3048 4.5684 3.4906 2.4180 1.2434

4.1717 4.6548 5.3188 5.9826 6.7346 0.2092 0.3310 0.4508 0.5863 0.6807 0.7570 0.8625 1.0832 1.2095 1.3423

0.6037 0.4108 0.3958

3.7092 3.3970 2.9658 2.5178 1.9337 1.3620 0.7915 0.3332

3.3513 2.8682 2.2042 1.5404 .7884 3.1408 3.019 2.8992 2.7637 2.6693 2.593 2.4875 2.2668 2.1405 2.0077

-4.0155 -6.3518 -8.6413 -17.8447 -18.3720 -0.1667 -0.3262 -0.5046 -0.9311 -1.1133 -2.2444 -2.6397 -4.0128 -18.4180 -18.4180

0.2165 -0.4198 1.0307

0.0513 0.3168 0.4934 0.8709 1.8432 2.0336 3.3383 5.7052

15.6435 17.9798 20.2693 29.4727 >>>30 11.7487 11.9082 12.0866 12.5131 12.6953 13.8264 14.2217 15.5948 30 >>30

Table A.35 (Pre-coding +NERF) Results and compared with the results of each of (Pre-coding) and (NERF companding). Precoding

NERF

WHT DCT DST DHT

PAPR

12.7937 9.0407 8.4569 1.1145

0.3968 1.3705 1.4328 4.5966

10.0440 9.0703 9.0080 5.8442

5.6684 3.7171 3.6090 0.6177

0.2838 0.5731 0.6060 1.7877

CCDF of PAPR 4.2362 3.9469 3.914 2.7323

-2.1831 -1.7823

-0.4681 -0.0653

SNR (BER= ) 13.8131 13.4103 non-NaN. non-NaN

Table A.36 (Pre-coding + tanhR) Results and compared with the results of each of (Pre-coding) and (tanhR companding). Precoding

PAPR

A. 53

CCDF of

SNR

Appendix A:

WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT DCT DCT DCT DCT

5 5 5 5 10 10 10 10 15 15 15 15 20 20 20 20 30 30 30 30 40 40 40 40 5 5 5 5

1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2

-0.1807 3.7573 10.6855 18.3499 0.0656 4.2484 11.1658 18.6064 0.3235 4.9015 10.8964 18.7511 0.0789 5.7588 12.0332 19.1134 2.9821 6.1658 12.9867 18.9871 4.2670 7.5802 13.8960 19.3685 4.9948 7.4490 11.3493 15.4567

-6.3055 -5.2332 -3.2479 -0.9432 -13.8409 -10.4413 -5.2722 -1.1811 -17.2828 -12.6933 -6.9994 -1.3429 -19.3807 -13.4839 -7.0284 -1.1533

3.5967 3.1852 2.1426 0.8903

Tables of Results

23.0184 19.0804 12.1522 4.4878 22.7721 18.5893 11.6719 4.2313 22.5142 17.9362 11.9413 4.0866 22.7588 17.0789 10.8045 3.7243 19.8556 16.6719 9.8510 3.8506 18.5707 15.2575 8.9417 3.4692 13.1162 10.6620 6.7617 2.6543

0.2938 2.2354 4.7480 7.9541 0.3874 2.1231 4.9280 8.0609 0.4487 2.3646 5.1420 8.1767 0.3788 2.6213 5.1936 8.2956 1.4998 2.7222 5.6462 8.2285 1.8924 3.4669 5.9673 8.3837 1.8882 2.9477 4.6478 6.4640

A. 54

-2.2032 -1.6111 -1.2705 -0.4230 -5.3082 -4.0236 -2.1485 -0.4986 -6.8572 -5.0135 -2.5305 -0.5252 -7.7030 -5.4421 -2.9215 -0.4775

1.6318 1.3418 0.8699 0.3275

PAPR 9.6108 7.6692 5.1566 1.9505 9.5172 7.7815 4.9766 1.8437 9.4559 7.54 4.7626 1.7279 9.5258 7.2833 4.711 1.609 8.4048 7.1824 4.2584 1.6761 8.0122 6.4377 3.9373 1.5209 5.7758 4.7163 3.0162 1.2

0.1112 -0.6251 -3.0382 -10.9634 0.1112 -0.4102 -3.1630 -11.2273 -0.0924 -0.6197 -3.2753 -11.4066 -0.2736 -0.7083 -3.4580 -11.7669 0.1416 -0.6555 -3.4154 -12.3505 0.0095 -0.6978 -3.9522 -12.5536 -0.2843 -1.2305 -4.0362 -12.4393

0.6106 0.4161 0.9195 2.1297 3.1298 2.8919 2.8359 3.1586 17.8776 10.2159 6.5694 3.5210 17.6964 17.2617 14.5120 5.5423

0.2171 -0.1873 -0.0765 0.6558

(BER= ) 11.5188 12.2551 14.6682 22.5934 11.5188 12.0402 14.793 22.8573 11.7224 12.2497 14.9053 23.0366 11.9036 12.3383 15.088 23.3969 11.4884 12.2855 15.0454 23.9805 11.6205 12.3278 15.5822 24.1836 11.9123 12.8585 15.6642 24.0673

Appendix A: DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DST DST DST DST DST DST DST DST DST

10 10 10 10 15 15 15 15 20 20 20 20 30 30 30 30 40 40 40 40 .5 .5 .5 .5 1 1 1 1 5

1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1

10.0154 10.8270 12.7260 15.6243 13.2826 13.3063 13.9101 15.8469 14.9735 14.7766 14.7322 16.0073 16.4535 16.2397 15.7594 16.2183 17.0593 16.9063 16.3777 16.3828 2.7791 5.6384 10.1178 14.6923 7.7531 9.1948 11.8357 15.0636 16.5353

0.8356 0.8640 1.0147 0.5635 0.4030 0.4382 0.7410 0.4796 0.2406 0.2606 0.3973 0.4673

15.7833

Tables of Results 8.0956 7.2840 5.3850 2.4867 4.8284 4.8047 4.2009 2.2641 3.1375 3.3344 3.3788 2.1037 1.6575 1.8713 2.3516 1.8927 1.0517 1.2047 1.7333 1.7282 14.6858 11.8265 7.3471 2.7726 9.7118 8.2701 5.6292 2.4013 0.9296

4.0445 4.4516 5.3042 6.5815 5.2743 5.3592 5.7489 6.6630 5.9965 5.9706 6.1017 6.7400 6.6685 6.5711 6.5149 6.8333 6.9437 6.9363 6.7772 6.8898 1.2087 2.4243 4.3530 6.3053 3.1540 3.7915 4.9741 6.4409 6.8695

A. 55

0.5895 0.5455 0.4683 0.2626 0.2090 0.2217 0.3170 0.2017 0.1553 0.1478 0.2272 0.2075

6.7541

3.6195 3.2124 2.3598 1.0825 2.3897 2.3048 1.9151 1.001 1.6675 1.6934 1.5623 .924 .9955 1.0929 1.1491 .8307 .7203 .7277 .8868 .7742 6.3143 5.0987 3.17 1.2177 4.369 3.7315 2.5489 1.0821 .6535

-2.1407 -2.6784 -6.2448 -13.8424 -7.3080 -7.1678 -8.8231 -14.9256 -18.3720 -18.3720 -13.4514 -16.1143 -18.3720 -18.3720 -18.3720 -17.5814 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720

0.8799 0.6257 -0.2439 0.5455 10.6640 3.6698 1.0236 0.0040 -0.4000 -0.4000 4.5206 1.1969

-17.8706

13.7687 14.3064 17.8728 25.4704 18.936 18.7958 20.4511 26.5536 >30 >30 25.0794 27.7423 >>30 >>30 >30 29.2094 >>30 >>30 >>30 >30 >30 >>30 >>>30 >>>30 >30 >30 >>>30 >>>30 >>>30

Appendix A: DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DHT DHT DHT DHT DHT DHT

5 5 5 10 10 10 10 15 15 15 15 20 20 20 20 30 30 30 30 40 40 40 40 5 5 5 5 10 10

.8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8

16.3710 15.8264 15.8076 17.1973 17.1359 16.8045 16.1937 17.3352 17.3044 17.1086 16.3513 17.3829 17.3648 17.2385 16.4904 17.4153 17.4074 17.3451 16.6910 17.4283 17.4237 17.3863 16.7987 4.0819 4.0969 3.9269 3.8097 4.1283 4.1282

12.7533 7.2658 1.8873 8.6636 7.8190 5.7393 1.7790 5.1017 5.0824 4.5856 1.6301 3.2961 3.4949 3.5497 1.5965

13.8361 10.9854 5.8725 0.3956 6.1008 5.3175

Tables of Results 1.0939 1.6385 1.6573 0.2676 0.3290 0.6604 1.2712 0.1297 0.1605 0.3563 1.1136 0.0820 0.1001 0.2264 0.9745 0.0496 0.0575 0.1198 0.7739 0.0366 0.0412 0.0786 0.6662 2.8768 2.8618 3.0318 3.1490 2.8304 2.8305

6.8535 6.6730 6.7862 7.2727 7.2537 7.1194 6.9533 7.3498 7.3447 7.2769 7.0354 7.4045 7.3987 7.3456 7.0845 7.4273 7.4277 7.3967 7.1582 7.4550 7.4562 7.4420 7.2140 1.4368 1.4315 1.4552 1.3754 1.4791 1.4791

A. 56

5.3886 3.0361 0.7907 3.9587 3.4886 2.4245 0.7754 2.4255 2.3482 1.9860 0.7151 1.7043 1.7169 1.6121 0.6930

5.4944 4.1396 1.9913 -0.4471 2.3381 1.8870

.6695 .85 .7368 .2503 .2693 .4036 .5697 .1732 .1783 .2461 .4876 .1185 .1243 .1774 .4385 .0957 .0953 .1263 .3648 .068 .0668 .081 .309 1.9132 1.9185 1.8948 1.9746 1.8709 1.8709

-18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -18.3720 -1.4034 -0.9347 -1.3122 -5.7323 -1.5015 -1.0322

-17.3288 -14.4123 -5.2769 -15.3514 -15.0679 -12.3711 -3.9841 -0.4000 -7.5344 -8.5253 -3.4424 -0.4000 -0.4000 -0.4000 -1.0608

-0.8560 0.1545 2.6935 7.4088 1.5651 2.3179

>>>30 >>30 >30 >>>30 >>>30 >>>30 >>30 >>>30 >>>30 >>>30 >>30 >>>30 >>>30 >>>30 >>30 >>>30 >>>30 >>>30 >>>30 >>>30 >>>30 >>>30 >>>30 12.9854 12.5167 12.8942 17.3143 13.0835 12.6142

Appendix A: DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT

10 10 15 15 15 15 20 20 20 20 30 30 30 30 40 40 40 40

.5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2

4.0944 3.8978 4.1283 4.1283 4.1193 3.9387 4.0983 4.0983 4.0954 3.9368 4.1032 4.1264 4.1028 3.9758 4.1264 4.1024 4.1263 4.0197

Tables of Results

3.5354 -0.0107 2.4010 2.4125 2.1025 -0.2763 0.5177 0.7346 0.9128 -0.4509

2.8643 3.0609 2.8304 2.8304 2.8394 3.0200 2.8604 2.8604 2.8633 3.0219 2.8555 2.8323 2.8559 2.9829 2.8323 2.8563 2.8324 2.9390

1.4683 1.4054 1.4791 1.4791 1.4762 1.4185 1.5718 1.5619 1.5609 1.5058 1.5600 1.5207 1.4790 1.4365 1.5207 1.5619 1.5207 1.4816

0.9464 -0.5995 0.7278 0.6556 0.3583 -0.7288 0.0446 0.0531 0.0004 -0.7127

1.8817 1.9446 1.8709 1.8709 1.8738 1.9315 1.7782 1.7881 1.7891 1.8442 1.79 1.8293 1.871 1.9135 1.8293 1.7881 1.8293 1.8684

-1.1244 -5.7624 -1.5015 -1.0322 -1.0754 -5.7564 -1.3866 -0.6494 -1.2818 -5.5015 -1.3244 -0.8384 -1.0972 -5.7159 -1.4180 -0.8458 -1.2255 -5.5194

4.9225 8.6715 16.5165 9.8514 8.8173 9.2192 16.6314 17.3686 16.7362 11.8557

12.7064 17.3444 13.0835 12.6142 12.6574 17.3384 12.9686 12.2314 12.8638 17.0835 12.9064 12.4204 12.6792 17.2979 13 12.4278 12.8075 17.1014

Table A.37 (Pre-coding +logR) Results and compared with the results of each of (Pre-coding) and (logR companding). Precoding

WHT WHT WHT WHT WHT

5 5 5 5 10

1 .8 .5 .2 1

0.4263 4.6287 11.9168 18.3833 1.0356

PAPR

-5.0601

22.4114 18.2090 10.9209 4.4544 21.8021

0.5366 2.2852 5.1927 7.9769 0.7807

A. 57

-1.8094

CCDF of PAPR 9.368 7.6194 4.7119 1.9277 9.1239

0.0852 -0.4767 -3.2966 -11.4059 0.0587

0.9287

SNR (BER= ) 11.5448 12.1067 14.9266 23.0359 11.5713

Appendix A: WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT WHT DCT

10 10 10 20 20 20 20 30 30 30 30 40 40 40 40 50 50 50 50 70 70 70 70 90 90 90 90 5

.8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1

5.2030 11.7711 18.4830 1.9256 5.9356 12.1732 18.5564 2.8015 6.5904 12.4999 18.6257 4.7516 8.0905 13.3591 18.9242 4.2553 7.6258 12.9890 18.7335 5.1623 8.2282 13.2299 18.7585 5.9939 8.7922 13.4752 18.8058 4.0783

-2.9008 -1.3242 -0.2614 -6.4950 -4.0630 -1.7503 -0.0850 -6.3259 -4.7333 -1.9038 -0.4202 -5.6271 -3.9205 -1.1251 -0.0447 -6.9289 -4.9426 -2.0679 -0.3517 -7.3198 -4.8379 -1.9328 -0.1039 -7.1447 -4.9729 -2.2699 -0.3343

Tables of Results 17.6347 11.0666 4.3547 20.9121 16.9021 10.6645 4.2813 20.0362 16.2473 10.3378 4.2120 18.0861 14.7472 9.4786 3.9135 18.5824 15.2119 9.8487 4.1042 17.6754 14.6095 9.6078 4.0792 16.8438 14.0455 9.3625 4.0319 14.0327

2.5204 5.1833 8.0193 1.4846 3.0211 5.4977 8.1109 1.4262 3.0154 5.5439 8.1529 2.1352 3.5446 5.7738 8.2183 2.0956 3.4779 5.6726 8.1261 2.6547 3.8796 5.9123 8.1934 2.7271 3.9383 5.9566 8.2281 1.5751

A. 58

-1.0067 -0.4813 -0.0853 -2.1290 -1.3535 -0.4789 0.0983 -2.7184 -1.8692 -0.7607 -0.0657 -2.3524 -1.6600 -0.5108 -0.0078 -2.7290 -1.9347 -0.8200 -0.1105 -2.6579 -1.7250 -0.6263 -0.0062 -2.7575 -2.0163 -0.8240 -0.0765

7.3842 4.7213 1.8853 8.42 6.8835 4.4069 1.7937 8.4784 6.8892 4.3607 1.7517 7.7694 6.36 4.1308 1.6863 7.809 6.4267 4.232 1.7785 7.2499 6.025 3.9923 1.7112 7.1775 5.9663 3.948 1.6765 6.0889

-0.5679 -3.3075 -11.4272 -0.2059 -0.7820 -3.4292 -11.4118 -0.2365 -0.8880 -3.4371 -11.8373 0.0341 -0.8033 -3.5819 -11.7032 -0.1544 -0.9347 -3.9124 -11.5775 -0.4487 -1.1765 -4.1543 -11.6655 -0.6104 -1.3420 -4.0355 -12.0414 -0.4586

1.0021 0.8125 0.5928 1.7801 1.4260 1.3740 0.8582 2.8718 2.1820 2.0157 0.3127 4.0906 3.5667 2.4001 0.5536 4.7956 3.6953 1.8816 1.1925 7.9213 4.9185 2.3437 0.5845 17.7596 7.0280 3.3159 0.6886

12.1979 14.9375 23.0572 11.8359 12.412 15.0592 23.0418 11.8665 12.518 15.0671 23.4673 11.5959 12.4333 15.2119 23.3332 11.7844 12.5647 15.5424 23.2075 12.0787 12.8065 15.7843 23.2955 12.2404 12.972 15.6655 23.6714 12.0866

Appendix A: DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT DCT

5 5 5 10 10 10 10 20 20 20 20 30 30 30 30 40 40 40 40 50 50 50 50 70 70 70 70 90 90

.8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8

5.8776 10.1251 14.7651 5.2700 7.5462 10.9668 14.9984 6.4419 8.8030 11.5302 15.1147 8.4747 9.6024 11.9149 14.9918 8.4754 9.4797 11.6821 15.0467 9.6001 9.8944 11.8863 15.0848 9.7691 10.4531 12.1597 15.2596 10.5879 11.1396

3.9010 4.1691 2.5982 0.9807 2.7480 3.5311 2.3334 1.2000 4.0740 3.0054 2.2379 0.6726 2.8234 2.1954 1.9246 0.8045 3.1426 2.0527 1.5561 0.7263 2.0137 2.1137 1.7237 1.1239 2.1760 2.1012

Tables of Results 12.2334 7.9859 3.3459 12.8410 10.5648 7.1442 3.1126 11.6691 9.3080 6.5808 2.9963 9.6363 8.5086 6.1961 3.1192 9.6356 8.6313 6.4289 3.0643 8.5109 8.2166 6.2247 3.0262 8.3419 7.6579 5.9513 2.8514 7.5231 6.9714

2.4715 4.2709 6.2418 2.0919 3.0042 4.5075 6.2912 2.7293 3.5713 4.7449 6.3263 3.3960 3.8900 4.9083 6.3167 3.4950 3.9296 4.8823 6.3449 3.8828 4.0978 4.9704 6.3745 4.0838 4.3864 5.1309 6.4068 4.3312 4.5770

A. 59

1.7424 1.7177 1.0835 0.4272 1.3563 1.4373 1.0089 0.5543 1.4920 1.2460 0.8443 0.3387 1.2480 0.9656 0.8383 0.3594 1.2988 0.9258 0.7184 0.3785 1.0118 1.0224 0.8329 0.4478 1.0872 0.8630

5.1925 3.3931 1.4222 5.5721 4.6598 3.1565 1.3728 4.9347 4.0927 2.9191 1.3377 4.268 3.774 2.7557 1.3473 4.169 3.7344 2.7817 1.3191 3.7812 3.5662 2.6936 1.2895 3.5802 3.2776 2.5331 1.2572 3.3328 3.087

-0.8859 -4.2414 -11.6890 -0.7392 -1.4409 -4.2249 -11.9552 -1.6394 -2.3940 -5.0332 -12.1126 -2.4175 -2.9251 -5.0536 -12.0250 -3.5951 -3.7052 -5.4937 -12.5112 -3.6787 -3.9254 -6.2297 -12.2911 -5.0376 -4.8662 -6.3668 -13.0394 -6.3649 -5.9901

0.1328 0.1311 -0.1029 0.0668 0.3486 -0.1840 -0.2280 0.1594 0.6928 0.1469 0.4012 0.1270 0.4634 0.6668 0.4903 -0.2524 1.2733 0.7066 -0.4337 0.4809 3.3344 1.2308 0.1332 -0.7874 12.0071 2.3819

12.5139 15.8694 23.317 12.3672 13.0689 15.8529 23.5832 13.2674 14.022 16.6612 23.7406 14.0455 14.5531 16.6816 23.653 15.2231 15.3332 17.1217 24.1392 15.3067 15.5534 17.8577 23.9191 16.6656 16.4942 17.9948 24.6674 17.9929 17.6181

Appendix A: DCT DCT DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST DST

90 90 5 5 5 5 10 10 10 10 15 15 15 15 20 20 20 20 30 30 30 30 40 40 40 40 50 50 50

.5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5

12.5958 15.3170 8.5202 9.3524 11.4249 14.5077 9.8718 10.0125 11.7969 14.6244 10.4876 10.8296 12.3484 14.6892 10.9548 11.2190 12.3938 14.6429 11.6232 11.7005 12.5336 14.7947 11.9575 12.1610 12.9059 14.8824 11.9685 12.1433 12.8274

1.5774 0.9036

9.1489 7.2815 4.0744 1.2528

7.9070 6.5932 3.8431 1.3743 7.8686 5.7496 3.5027 1.1216 6.9516 5.5228 3.7945 1.2863 6.1571 4.9477 3.1433

Tables of Results 5.5152 2.7940 8.9447 8.1125 6.0400 2.9572 7.5931 7.4524 5.6680 2.8405 6.9773 6.6353 5.1165 2.7757 6.5101 6.2459 5.0711 2.8220 5.8417 5.7644 4.9313 2.6702 5.5074 5.3039 4.5590 2.5825 5.4964 5.3216 4.6375

5.2242 6.4156 3.6111 4.0014 4.8296 6.2632 4.1997 4.3860 5.0592 6.3016 4.5158 4.6424 5.2379 6.2822 4.6516 4.8452 5.2302 6.3268 4.9818 5.0121 5.3564 6.3350 5.0939 5.1844 5.5148 6.3855 5.1368 5.2135 5.5130

A. 60

0.6842 0.3516

3.9912 3.2405 1.7762 0.5786

3.4196 2.8522 1.6352 0.6958 3.2188 2.5091 1.4334 0.4980 2.9879 2.3614 1.6118 0.5410 2.6938 2.1825 1.4020

2.4398 1.2484 3.9119 3.5216 2.6934 1.2598 3.3233 3.137 2.4638 1.2214 3.0072 2.8806 2.2851 1.2408 2.8714 2.6778 2.2928 1.1962 2.5412 2.5109 2.1666 1.188 2.4291 2.3386 2.0082 1.1375 2.3862 2.3095 2.01

-6.7500 -12.6807 -2.6156 -3.4388 -5.9240 -13.2083 -5.1839 -5.3699 -7.6083 -13.0385 -7.2293 -6.8672 -8.0515 -13.6972 -8.9870 -8.1981 -8.7103 -13.8951 -15.0476 -11.0759 -9.2805 -13.9518 -18.3720 -13.6810 -9.9412 -13.8786 -18.3720 -17.2375 -10.5717

0.6034 0.0513

-4.3119 -3.7979 -3.4863 -1.0165

-6.9990 -5.9881 -3.9051 -1.6231 -11.9373 -8.0039 -3.8257 -1.7998 -14.3135 -9.3090 -3.9572 -1.6198 -13.4200 -12.6055 -4.7757

18.378 24.3087 14.2436 15.0668 17.552 24.8363 16.8119 16.9979 19.2363 24.6665 18.8573 18.4952 19.6795 25.3252 20.615 19.8261 20.3383 25.5231 26.6756 22.7039 20.9085 25.5798 >30 25.309 21.5692 25.5066 >30 28.8655 22.1997

Appendix A: DST DST DST DST DST DST DST DST DST DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT

50 70 70 70 70 90 90 90 90 5 5 5 5 10 10 10 10 20 20 20 20 30 30 30 30 40 40 40 40

.2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2

14.8421 12.3682 12.5000 13.0636 14.7955 12.8291 12.9326 13.4032 14.9895 2.1510 2.3800 2.8097 3.4943 2.5046 2.6241 2.9457 3.5365 2.7535 2.8192 3.0364 3.5319 2.8996 2.9317 3.1032 3.5497 2.9635 3.0145 3.1604 3.5777

1.1297 5.2589 4.8067 3.2737 1.3059 5.0633 4.5403 3.0309 1.2222

12.2879 10.3993 5.7294 0.6711 10.2119 8.6996 4.9919 0.7695 9.6512 7.4870 4.5785 0.3828 8.4638 6.8825 4.5552 0.4878

Tables of Results 2.6228 5.0967 4.9649 4.4013 2.6694 4.6358 4.5323 4.0617 2.4754 4.8077 4.5787 4.1490 3.4644 4.4541 4.3346 4.0130 3.4222 4.2052 4.1395 3.9223 3.4268 4.0591 4.0270 3.8555 3.4090 3.9952 3.9442 3.7983 3.3810

6.3588 5.2927 5.3530 5.6055 6.4405 5.4648 5.5119 5.7247 6.4377 0.8595 0.9299 1.0939 1.3200 0.9697 1.0064 1.1058 1.2908 1.1068 1.1329 1.2124 1.3662 1.0915 1.0909 1.1471 1.2944 1.1013 1.1159 1.1686 1.3201

A. 61

0.5038 2.3617 2.1300 1.4485 0.6225 2.3618 1.9389 1.3257 0.5147

4.9342 4.0339 1.9958 -0.2592 4.0478 3.3129 1.7904 -0.0918 3.5015 2.7609 1.3971 -0.3696 3.1683 2.4659 1.4386 -0.3514

1.1642 2.2303 2.17 1.9175 1.0825 2.0582 2.0111 1.7983 1.0853 2.4905 2.4201 2.2561 2.03 2.3803 2.3436 2.2442 2.0592 2.2432 2.2171 2.1376 1.9838 2.2585 2.2591 2.2029 2.0556 2.2487 2.2341 2.1814 2.0299

-13.7840 -18.3720 -18.3720 -11.9018 -14.2162 -18.3720 -18.3720 -11.9831 -14.6467 -0.3995 -0.4631 -1.8985 -6.1537 -0.8641 -0.5591 -1.5443 -6.0508 -0.5381 -0.4235 -1.4934 -5.6755 -1.0229 -0.5152 -1.3132 -5.8967 -1.0149 -0.4055 -1.4070 -5.5958

-1.0120 -10.0000 -12.2750 -5.4018 -1.9642 0 -10.0000 -4.6297 -1.9147

0.0539 1.0589 2.6237 6.0172 1.4959 1.8325 3.3578 6.6425 2.1334 2.6028 4.1876 6.3013 3.0896 4.0125 4.6230 6.7090

25.412 >30 >30 23.5298 25.8442 >30 >30 23.6111 26.2747 11.9815 12.0451 13.4805 17.7357 12.4461 12.1411 13.1263 17.6328 12.1201 12.0055 13.0754 17.2575 12.6049 12.0972 12.8952 17.4787 12.5969 11.9875 12.989 17.1778

Appendix A: DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT DHT

50 50 50 50 70 70 70 70 90 90 90 90

1 .8 .5 .2 1 .8 .5 .2 1 .8 .5 .2

3.0338 3.0455 3.1724 3.5636 3.0908 3.1424 3.2468 3.6069 3.1589 3.1763 3.2661 3.6028

7.7286 6.3561 3.9945 0.3574 6.4877 5.9553 3.9631 0.6235 5.8993 5.2902 3.4000 0.3417

Tables of Results 3.9249 3.9132 3.7863 3.3951 3.8679 3.8163 3.7119 3.3518 3.7998 3.7824 3.6926 3.3559

1.1228 1.1738 1.2150 1.3427 1.1885 1.1830 1.2180 1.3394 1.1722 1.1776 1.2057 1.3118

A. 62

2.8528 2.3158 1.2770 -0.3393 2.4305 2.1330 1.2340 -0.3056 2.2422 1.7776 0.9797 -0.4382

2.2272 2.1762 2.135 2.0073 2.1615 2.167 2.132 2.0106 2.1778 2.1724 2.1443 2.0382

-0.9711 -0.5152 -1.3591 -6.0558 -0.8581 -0.6713 -1.2287 -5.5747 -1.1746 -0.6321 -1.3063 -5.9216

4.0269 4.1628 4.4829 6.7622 7.5599 5.4717 5.3173 6.7233 17.2434 7.7859 6.0931 6.8564

12.5531 12.0972 12.9411 17.6378 12.4401 12.2533 12.8107 17.1567 12.7566 12.2141 12.8883 17.5036

Appendix B

MATLAB Code Appendix B MATLAB Code

&& OFDM CODE: clear all clc close % --------------% Parameters % --------------M = 4; fft_size = 128; fspacing=15000; fs=15000*128; Ts = 1/fs; Fd = 0; nsym = 1000; snr = 0:0.8:30;

% QPSK signal constellatio % have 128 data point

% Sampling period of channel % Max Doppler frequency shift

Appendix B

MATLAB Code

cp(j,1) = ifft_data_matrix(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data = vertcat(cp,ifft_data_matrix); % s/p for transmission [rows_ifft_data, cols_ifft_data]=size(ifft_data); length_ofdm_data = rows_ifft_data*cols_ifft_data; pilot = zeros(1,length_ofdm_data); ofdm_signal = reshape(ifft_data, 1, length(ifft_data)); %p/s Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); tx = [tx ofdm_signal]; end [cdf0, PAPR0] = ecdf(PAPR_Orignal); %-------------------------------------------------------------------------semilogy(PAPR0,1-cdf0,'-b') legend('Orignal') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)) % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 4 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1; % --------------% RECEIVER % -----------no_of_error=[]; ratio=[]; B. 2

% Path delays % Avg path

Appendix B

MATLAB Code

for ii=1:length(snr) rx= []; rx_signal_matrix = reshape(tt,length(ifft_data),nsym); [~, c] = size(rx_signal_matrix); for j = 2: nsym-1 hx = filter(h,rx_signal_matrix(:,j).'); % Pass through Rayleigh channel a = h.PathGains; AM = h.channelFilter.alphaMatrix; g = a*AM; % Channel coefficients G(j,:) = fft(g,fft_size); % DFT of channel coefficients % Add AWGN no rx_signal = awgn(hx,snr(ii),'measured'); d=size(rx_signal); % Convert Data back to "parallel" form to perform FFT con=reshape( rx_signal , length(ifft_data),1); % Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); % p/s rx_serial_data = reshape(fft_data_matrix, 1,fft_size); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

B. 3

Appendix B

MATLAB Code

&& RCF CODE: clear all clc close % --------------% Parameters % --------------M = 4; % QPSK signal constellatio fft_size = 128; % have 128 data point fspacing=15000; fs=15000*128; Ts = 1/fs; % Sampling period of channel Fd = 0; % Max Doppler frequency shift nsym = 1000; snr = 0:0.8:30; CR = 3; ITERATE_NUM = 4; K = 128; % SIZE OF OFDM Symbol IF = 2; % Interpolation factor (Oversamplingfactor) fft_size = K*IF; % SIZE OF FFT mm = 193 %when IF =1.125 =81 ;when IF =1.25 =97 ;when IF =1.5 =129; when IF = 2 =193; when IF =3 =321; when IF = 4=449 % -----------------% TRANSMITTER % -----------------% Generate 1 x 128 vector of random data points source = randsrc(1, K*(nsym-2), 0:M-1); %the data source cp_length = .25*K; % length of cyclic prefix sp = reshape(source , K , nsym-2);% s/p s = size (sp); tx = []; PAPR_Orignal = zeros(1,nsym); PAPR_RCF = zeros(ITERATE_NUM,nsym); for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); % making s/p serial_to_paralle = reshape(qpsk_mod, K , 1);% s/p xy = [serial_to_paralle(1:K/2) ; zeros(fft_size-K,1); serial_to_paralle(K/2+1:K)]; ifft_data_matrix = ifft(xy,fft_size); % s/p for transmission pilot = zeros(1,length(ifft_data_matrix));

B. 4

Appendix B

MATLAB Code

ofdm_signal = reshape(ifft_data_matrix, 1, length(ifft_data_matrix));%p/s Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); for nIter=1:ITERATE_NUM % Clipping x_tmp = ofdm_signal(Signal_Power>CR*Mean_Power); x_tmp = sqrt(CR*Mean_Power)*x_tmp./abs(x_tmp); ofdm_signal(Signal_Power>CR*Mean_Power) = x_tmp; % Filtering XX = fft(ofdm_signal,[],2); XX(K/2+(1:fft_size-K)) = zeros(1,fft_size-K); ofdm_signal = ifft(XX,[],2); % PAPR Compute Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_RCF(nIter,i) = 10*log10(Peak_Power./Mean_Power); end % make ifft to each block and add CP serial_to_paralle2 = reshape(ofdm_signal,fft_size , 1);% s/p % to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) =serial_to_paralle2(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data_cp = vertcat(cp,serial_to_paralle2); ofdm = reshape(ifft_data_cp, 1, length(ifft_data_cp));%p/s pilot = zeros(1,length(ifft_data_cp)); tx = [tx ofdm]; end [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_RCF(1,:)); [cdf2, PAPR2] = ecdf(PAPR_RCF(2,:)); [cdf3, PAPR3] = ecdf(PAPR_RCF(3,:)); [cdf4, PAPR4] = ecdf(PAPR_RCF(4,:)); %-------------------------------------------------------------------------B. 5

Appendix B

MATLAB Code

semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r',PAPR2,1-cdf2,'-g',PAPR3,1-cdf3,'c',PAPR4,1-cdf4,'-m') legend('Orignal','One clip and filter','Two clip and filter','Three clip and filter','Four clip and filter') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)) % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 6 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Path delays % Avg path power

Remove CP B. 6

Appendix B

MATLAB Code

con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); du = [fft_data_matrix(1:K/2) ; [];fft_data_matrix(mm:fft_size)]; % p/s rx_serial_data = reshape(du, 1,K); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end ofdm_signal = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('SNR'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

&& RCF I=1 CODE: clear all clc close % --------------% Parameters % --------------M = 4; % QPSK signal constellatio fft_size = 128; % have 128 data point fspacing=15000; fs=15000*128; Ts = 1/fs; % Sampling period of channel Fd = 0; % Max Doppler frequency shift nsym = 1000; snr = 0:0.8:30; CR = 2; ITERATE_NUM = 4; % -----------------B. 7

Appendix B

MATLAB Code

% TRANSMITTER % -----------------% Generate 1 x 128 vector of random data points source = randsrc(1, fft_size*(nsym-2), 0:M-1); %the data source cp_length = .25*fft_size; % length of cyclic prefix sp = reshape(source , fft_size , nsym-2);% s/p s = size (sp); tx = []; PAPR_Orignal = zeros(1,nsym); PAPR_RCF = zeros(ITERATE_NUM,nsym); for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); % making s/p serial_to_paralle = reshape(qpsk_mod, fft_size , 1);% s/p

% to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % make ifft to each block and add CP ifft_data_matrix = ifft(serial_to_paralle ,fft_size); % s/p for transmission pilot = zeros(1,length(ifft_data_matrix)); ofdm_signal = reshape(ifft_data_matrix, 1, length(ifft_data_matrix));%p/s Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); for nIter=1:ITERATE_NUM % Clipping x_tmp = ofdm_signal(Signal_Power>CR*Mean_Power); x_tmp = sqrt(CR*Mean_Power)*x_tmp./abs(x_tmp); ofdm_signal(Signal_Power>CR*Mean_Power) = x_tmp;

% PAPR Compute Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_RCF(nIter,i) = 10*log10(Peak_Power./Mean_Power); end % make ifft to each block and add CP serial_to_paralle2 = reshape(ofdm_signal,fft_size , 1);% s/p B. 8

Appendix B

MATLAB Code

% to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) =serial_to_paralle2(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data_cp = vertcat(cp,serial_to_paralle2); ofdm = reshape(ifft_data_cp, 1, length(ifft_data_cp));%p/s pilot = zeros(1,length(ifft_data_cp)); tx = [tx ofdm]; end figure(1) [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_RCF(1,:)); [cdf2, PAPR2] = ecdf(PAPR_RCF(2,:)); [cdf3, PAPR3] = ecdf(PAPR_RCF(3,:)); [cdf4, PAPR4] = ecdf(PAPR_RCF(4,:)); %-------------------------------------------------------------------------semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r',PAPR2,1-cdf2,'-g',PAPR3,1-cdf3,'c',PAPR4,1-cdf4,'-m') legend('Orignal','One clip and filter','Two clip and filter','Three clip and filter','Four clip and filter') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; t = size (tx); Q = size(tt); x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)) % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 6 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; B. 9

% Path delays % Avg path

Appendix B

MATLAB Code

h.StorePathGains = 1; h.ResetBeforeFiltering = 1; % --------------% RECEIVER % -----------no_of_error=[]; ratio=[]; for ii=1:length(snr) rx= []; rx_signal_matrix =reshape(tt,length( ifft_data_cp),nsym); [~, c] = size(rx_signal_matrix); for j = 2: nsym-1 hx = filter(h,rx_signal_matrix(:,j).'); % Pass through Rayleigh channel a = h.PathGains; AM = h.channelFilter.alphaMatrix; g = a*AM; % Channel coefficients G(j,:) = fft(g,fft_size); % DFT of channel coefficients % Add AWGN no rx_signal = awgn(hx ,snr(ii),'measured'); % Convert Data back to "parallel" form to perform FFT con=reshape( rx_signal, length( ifft_data_cp),1); % Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); % p/s rx_serial_data = reshape(fft_data_matrix, 1,fft_size); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end ofdm_signal = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') B. 10

Appendix B

MATLAB Code

grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

&& RCF( I =pilot =76 in this case )CODE: clear all clc close % --------------% Parameters % --------------M = 4; % QPSK signal constellatio fft_size = 128; % have 128 data point fspacing=15000; fs=15000*128; Ts = 1/fs; % Sampling period of channel Fd = 0; % Max Doppler frequency shift nsym = 1000; snr = 0:0.8:30; CR = 4; ITERATE_NUM = 4; K = 76; % SIZE OF OFDM Symbol % -----------------% TRANSMITTER % -----------------% Generate 1 x 128 vector of random data points source = randsrc(1, K*(nsym-2), 0:M-1); %the data source cp_length = .25*K; % length of cyclic prefix sp = reshape(source , K , nsym-2);% s/p s = size (sp); tx = []; PAPR_Orignal = zeros(1,nsym); PAPR_RCF = zeros(ITERATE_NUM,nsym); for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); % making s/p serial_to_paralle = reshape(qpsk_mod, K , 1);% s/p xy = [serial_to_paralle(1:K/2) ; zeros(fft_size-K,1); serial_to_paralle(K/2+1:K)]; ifft_data_matrix = ifft(xy,fft_size); % s/p for transmission pilot = zeros(1,length(ifft_data_matrix)); ofdm_signal = reshape(ifft_data_matrix, 1, length(ifft_data_matrix));%p/s B. 11

Appendix B

MATLAB Code

Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); for nIter=1:ITERATE_NUM % Clipping x_tmp = ofdm_signal(Signal_Power>CR*Mean_Power); x_tmp = sqrt(CR*Mean_Power)*x_tmp./abs(x_tmp); ofdm_signal(Signal_Power>CR*Mean_Power) = x_tmp; % Filtering XX = fft(ofdm_signal,[],2); XX(K/2+(1:fft_size-K)) = zeros(1,fft_size-K); ofdm_signal = ifft(XX,[],2); % PAPR Compute Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_RCF(nIter,i) = 10*log10(Peak_Power./Mean_Power); end % make ifft to each block and add CP serial_to_paralle2 = reshape(ofdm_signal,fft_size , 1);% s/p % to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) =serial_to_paralle2(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data_cp = vertcat(cp,serial_to_paralle2); ofdm = reshape(ifft_data_cp, 1, length(ifft_data_cp));%p/s pilot = zeros(1,length(ifft_data_cp)); tx = [tx ofdm]; end figure(1) [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_RCF(1,:)); [cdf2, PAPR2] = ecdf(PAPR_RCF(2,:)); [cdf3, PAPR3] = ecdf(PAPR_RCF(3,:)); [cdf4, PAPR4] = ecdf(PAPR_RCF(4,:)); %-------------------------------------------------------------------------B. 12

Appendix B

MATLAB Code

semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r',PAPR2,1-cdf2,'-g',PAPR3,1-cdf3,'c',PAPR4,1-cdf4,'-m') legend('Orignal','One clip and filter','Two clip and filter','Three clip and filter','Four clip and filter') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; t = size (tx); Q = size(tt); x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)) % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 6 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Path delays % Avg path

Appendix B

MATLAB Code

% Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); du = [fft_data_matrix(1:K/2) ; [];fft_data_matrix(91:fft_size)]; % p/s rx_serial_data = reshape(du, 1,K); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end % r = rx(1,(K+1:length(rx)-K)); figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end ofdm_signal = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

&& RFC CODE : clear all clc close % --------------% Parameters % --------------M = 4; fft_size = 128; fspacing=15000; fs=15000*128; Ts = 1/fs; Fd = 0; nsym = 1000; snr = 0:0.8:30; CR = 3;

% QPSK signal constellatio % have 128 data point

% Sampling period of channel % Max Doppler frequency shift

B. 14

Appendix B

MATLAB Code

ITERATE_NUM = 4; K = 76; % SIZE OF OFDM Symbol % -----------------% TRANSMITTER % -----------------% Generate 1 x 128 vector of random data points source = randsrc(1, K*(nsym-2), 0:M-1); %the data source cp_length = .25*K; % length of cyclic prefix sp = reshape(source , K , nsym-2);% s/p s = size (sp); tx = []; PAPR_Orignal = zeros(1,nsym); PAPR_RCF = zeros(ITERATE_NUM,nsym); for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); % making s/p serial_to_paralle = reshape(qpsk_mod, K , 1);% s/p xy = [serial_to_paralle(1:K/2) ; zeros(fft_size-K,1); serial_to_paralle(K/2+1:K)]; ifft_data_matrix = ifft(xy,fft_size); % s/p for transmission pilot = zeros(1,length(ifft_data_matrix)); ofdm_signal = reshape(ifft_data_matrix, 1, length(ifft_data_matrix));%p/s Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); for nIter=1:ITERATE_NUM % Filtering XX = fft(ofdm_signal,[],2); XX(K/2+(1:fft_size-K)) = zeros(1,fft_size-K); ofdm_signal = ifft(XX,[],2); % Clipping x_tmp = ofdm_signal(Signal_Power>CR*Mean_Power); x_tmp = sqrt(CR*Mean_Power)*x_tmp./abs(x_tmp); ofdm_signal(Signal_Power>CR*Mean_Power) = x_tmp; % PAPR Compute Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); B. 15

Appendix B

MATLAB Code

PAPR_RCF(nIter,i) = 10*log10(Peak_Power./Mean_Power); end % make ifft to each block and add CP serial_to_paralle2 = reshape(ofdm_signal,fft_size , 1);% s/p % to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) =serial_to_paralle2(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data_cp = vertcat(cp,serial_to_paralle2); ofdm = reshape(ifft_data_cp, 1, length(ifft_data_cp));%p/s pilot = zeros(1,length(ifft_data_cp)); tx = [tx ofdm]; end [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_RCF(1,:)); [cdf2, PAPR2] = ecdf(PAPR_RCF(2,:)); [cdf3, PAPR3] = ecdf(PAPR_RCF(3,:)); [cdf4, PAPR4] = ecdf(PAPR_RCF(4,:)); figure(1) %-------------------------------------------------------------------------semilogy(PAPR4,1-cdf4,'-b') legend('I =pilot ','1.125','I =1.25 ','I =1.5 ','I= 2','I =3','I =4') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; t = size (tx); Q = size(tt); figure(2) plot(real(tt)); xlabel('Time'); ylabel('Amplitude'); title('OFDM Signal');grid on; x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)) % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 6 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9];

% Path delays B. 16

Appendix B

MATLAB Code

pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Avg path

% --------------% RECEIVER % -----------no_of_error=[]; ratio=[]; for ii=1:length(snr) rx= []; rx_signal_matrix =reshape(tt,length( ifft_data_cp),nsym); [~, c] = size(rx_signal_matrix); for j = 2: nsym-1 hx = filter(h,rx_signal_matrix(:,j).'); % Pass through Rayleigh channel a = h.PathGains; AM = h.channelFilter.alphaMatrix; g = a*AM; % Channel coefficients G(j,:) = fft(g,K); % DFT of channel coefficients % Add AWGN no rx_signal = awgn(hx ,snr(ii),'measured'); % Convert Data back to "parallel" form to perform FFT con=reshape( rx_signal, length( ifft_data_cp),1); % Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); du = [fft_data_matrix(1:K/2) ; [];fft_data_matrix(91:fft_size)]; % p/s rx_serial_data = reshape(du, 1,K); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end ofdm_signal = size (source); B. 17

Appendix B

MATLAB Code

v = size(rx); semilogy(snr,ratio,'--*b','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('I =pilot ','1.125','I =1.25 ','I =1.5 ','I= 2','I =3','I =4') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

&& A_ law CODE: clear all clc close % --------------% Parameters % --------------M = 4; fft_size = 128; fspacing=15000; fs=15000*128; Ts = 1/fs; Fd = 0; nsym = 1000; snr = 0:0.8:30;

% QPSK signal constellatio % have 128 data point

% Sampling period of channel % Max Doppler frequency shift

% -----------------% TRANSMITTER % -----------------% Generate 1 x 128 vector of random data points source = randsrc(1, 128*(nsym-2), 0:M-1); %the data source %figure(1) cp_length = .25*128 ; % length of cyclic prefix sp = reshape(source , 128 , nsym-2);% s/p s = size (sp); tx = []; for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); % making s/p serial_to_paralle = reshape(qpsk_mod, fft_size, 1); % s/p B. 18

Appendix B

MATLAB Code

% to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % make ifft to each block and add CP ifft_data_matrix = ifft(serial_to_paralle,fft_size); % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) = ifft_data_matrix(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data = vertcat(cp,ifft_data_matrix); % s/p for transmission [rows_ifft_data, cols_ifft_data]=size(ifft_data); length_ofdm_data = rows_ifft_data*cols_ifft_data; pilot = zeros(1,length_ofdm_data); ofdm_signal = reshape(ifft_data, 1, length(ifft_data));%p/s Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); A = 90; % Parameter for A-law compander V = max(abs(ofdm_signal)); compsig = compand(ofdm_signal,A,V,'A/compressor'); Signal_Power = abs(compsig.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal1(i) = 10*log10(Peak_Power./Mean_Power); tx = [tx compsig]; end figure(1) [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_Orignal1); %-------------------------------------------------------------------------semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; t = size (tx); Q = size(tt); x_abs=abs(tt); B. 19

Appendix B

MATLAB Code

papr=10*log(max(x_abs.^2)/mean(x_abs.^2)); disp('PAPR of original signal in dB'); disp(papr); % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 4 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Path delays % Avg path

% --------------% RECEIVER % -----------no_of_error=[]; ratio=[]; for ii=1:length(snr) rx= []; rx_signal_matrix = reshape(tt,length(ifft_data),nsym); [~, c] = size(rx_signal_matrix); for j = 2: nsym-1 hx = filter(h,rx_signal_matrix(:,j).'); % Pass through Rayleigh channel a = h.PathGains; AM = h.channelFilter.alphaMatrix; g = a*AM; % Channel coefficients G(j,:) = fft(g,fft_size); % DFT of channel coefficients % Add AWGN no rx_signal = awgn(hx ,snr(ii),'measured'); V = max(abs(rx_signal)); compsig = compand(rx_signal,A, V,'A/expander'); % Convert Data back to "parallel" form to perform FFT con=reshape( compsig, length(ifft_data),1); % Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); % p/s B. 20

Appendix B

MATLAB Code

semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

&& CODE clear all clc close % --------------% Parameters % --------------M = 4; fft_size = 128; fspacing=15000; fs=15000*128; Ts = 1/fs; Fd = 0; nsym = 1000; snr = 0:0.8:30;

% QPSK signal constellatio % have 128 data point

% Sampling period of channel % Max Doppler frequency shift

Appendix B

MATLAB Code

for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); %scatterplot(qpsk_mod); %title('MODULATED TRANSMITTED DATA'); % making s/p serial_to_paralle = reshape(qpsk_mod, fft_size, 1); % s/p % to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % make ifft to each block and add CP ifft_data_matrix = ifft(serial_to_paralle,fft_size); % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) = ifft_data_matrix(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data = vertcat(cp,ifft_data_matrix); % s/p for transmission [rows_ifft_data, cols_ifft_data]=size(ifft_data); length_ofdm_data = rows_ifft_data*cols_ifft_data; pilot = zeros(1,length_ofdm_data); ofdm_signal = reshape(ifft_data, 1, length(ifft_data)); %p/s ofdm_signal = reshape(ifft_data, 1, length(ifft_data));%p/ Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); Mu = 700; % Parameter for mu-law compander V = max(abs(ofdm_signal)); x = compand(ofdm_signal,Mu,V,'mu/compressor'); Signal_Power = abs(x.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal1(i) = 10*log10(Peak_Power./Mean_Power); tx = [tx x]; end figure(1) [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_Orignal1); %-------------------------------------------------------------------------B. 22

Appendix B

MATLAB Code

semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)); disp('PAPR of original signal in dB'); disp(papr); % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 6 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Path delays % Avg path

Remove CP B. 23

Appendix B

MATLAB Code

con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); % p/s rx_serial_data = reshape(fft_data_matrix, 1,fft_size); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end x = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

&& Rooting CODE: clear all clc close % --------------% Parameters % --------------M = 4; % QPSK signal constellatio fft_size = 128; % have 128 data point fspacing=15000; fs=15000*128; Ts = 1/fs; % Sampling period of channel Fd = 0; % Max Doppler frequency shift nsym = 1000; snr = 0:0.8:30; y = .5; % -----------------% TRANSMITTER % -----------------% Generate 1 x 128 vector of random data points B. 24

Appendix B

MATLAB Code

source = randsrc(1, 128*(nsym-2), 0:M-1); %the data source cp_length = .25*fft_size; % length of cyclic prefix sp = reshape(source , 128 , nsym-2);% s/p s = size (sp); tx = []; for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); % making s/p serial_to_paralle = reshape(qpsk_mod, fft_size, 1); % s/p % to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % make ifft to each block and add CP ifft_data_matrix = ifft(serial_to_paralle,fft_size); % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) = ifft_data_matrix(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data = vertcat(cp,ifft_data_matrix); % s/p for transmission [rows_ifft_data, cols_ifft_data]=size(ifft_data); length_ofdm_data = rows_ifft_data*cols_ifft_data; pilot = zeros(1,length_ofdm_data); ofdm_signal = reshape(ifft_data, 1, length(ifft_data));%p/ Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); a =abs (ofdm_signal ); b =a.^y; x= complex(b.*cos(angle(ofdm_signal )),b.*sin(angle(ofdm_signal ))); Signal_Power = abs(x.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal1(i) = 10*log10(Peak_Power./Mean_Power); tx = [tx x]; end [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_Orignal1); %-------------------------------------------------------------------------B. 25

Appendix B

MATLAB Code

semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; t = size (tx); Q = size(tt); x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)); disp('PAPR of original signal in dB'); disp(papr); % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 4 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Path delays % Avg path

Appendix B

MATLAB Code

con=reshape(xx , length(ifft_data),1); % Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); % p/s rx_serial_data = reshape(fft_data_matrix, 1,fft_size); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end x = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

&& NERF CODE : clear all clc close % --------------% Parameters % --------------M = 4; fft_size = 128; fspacing=15000; fs=15000*128; Ts = 1/fs; Fd = 0; nsym = 1000; snr = 0:0.8:30;

% QPSK signal constellatio % have 128 data point

% Sampling period of channel % Max Doppler frequency shift

% -----------------B. 27

Appendix B

MATLAB Code

% TRANSMITTER % -----------------% Generate 1 x 128 vector of random data points source = randsrc(1, 128*(nsym-2), 0:M-1); %the data source cp_length = .25*fft_size; % length of cyclic prefix sp = reshape(source , 128 , nsym-2);% s/p s = size (sp); tx = []; for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); % making s/p serial_to_paralle = reshape(qpsk_mod, fft_size, 1); % s/p % to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % make ifft to each block and add CP ifft_data_matrix = ifft(serial_to_paralle,fft_size); % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) = ifft_data_matrix(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data = vertcat(cp,ifft_data_matrix); % s/p for transmission [rows_ifft_data, cols_ifft_data]=size(ifft_data); length_ofdm_data = rows_ifft_data*cols_ifft_data; pilot = zeros(1,length_ofdm_data); ofdm_signal = reshape(ifft_data, 1, length(ifft_data)); %p/s Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); b =erf(((abs(ofdm_signal)))./(sqrt(2).*std(ofdm_signal))); f= (((2).*std(ofdm_signal).*b)); h= sign(ofdm_signal).*f; Signal_Power = abs(h.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal1(i) = 10*log10(Peak_Power./Mean_Power); B. 28

Appendix B

MATLAB Code

tx = [tx h]; end tt =[ pilot tx pilot]; t = size (tx); [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_Orignal1); %-------------------------------------------------------------------------semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)); disp('PAPR of original signal in dB'); disp(papr); % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 6 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Path delays % Avg path

Appendix B

MATLAB Code

rx_signal = awgn(hx,snr(ii),'measured'); a =abs(rx_signal); dd=erfinv((a)./(2).*std(rx_signal)); v =sqrt(2).*std(rx_signal); s =(v.*dd); ff=abs(s); rr =sign(rx_signal).*ff; % Convert Data back to "parallel" form to perform FFT con=reshape( rr , length(ifft_data),1); % Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); % p/s rx_serial_data = reshape(fft_data_matrix, 1,fft_size); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end x = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

&& AEXP CODE: clear all clc close % --------------% Parameters % --------------M = 4;

% QPSK signal constellatio B. 30

Appendix B fft_size = 128; fspacing=15000; fs=15000*128; Ts = 1/fs; Fd = 0; nsym = 1000; snr = 0:0.8:30;

MATLAB Code % have 128 data point

% Sampling period of channel % Max Doppler frequency shift

% -----------------% TRANSMITTER % -----------------% Generate 1 x 128 vector of random data points source = randsrc(1, 128*(nsym-2), 0:M-1); %the data source cp_length = .25*fft_size; % length of cyclic prefix sp = reshape(source , 128 , nsym-2);% s/p s = size (sp); tx = []; for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); % making s/p serial_to_paralle = reshape(qpsk_mod, fft_size, 1); % s/p % to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % make ifft to each block and add CP ifft_data_matrix = ifft(serial_to_paralle,fft_size); % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) = ifft_data_matrix(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data = vertcat(cp,ifft_data_matrix); % s/p for transmission [rows_ifft_data, cols_ifft_data]=size(ifft_data); length_ofdm_data = rows_ifft_data*cols_ifft_data; pilot = zeros(1,length_ofdm_data); ofdm_signal = reshape(ifft_data, 1, length(ifft_data)); %p/s Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); B. 31

Appendix B

MATLAB Code

d =1.4; a =var(abs(ofdm_signal))+ mean(abs(ofdm_signal)); b =exp(-((abs(ofdm_signal)).^2)./var(ofdm_signal)); c =(1-b).^2; e =(c).^(d/2); E =( a./mean(e)).^(d/2); f= (E.*(1-b)).^(d/2); h= sign(ofdm_signal).*f; Signal_Power = abs(h.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal1(i) = 10*log10(Peak_Power./Mean_Power); tx = [tx h]; end tt =[ pilot tx pilot]; t = size (tx); [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_Orignal1); %-------------------------------------------------------------------------semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)); disp('PAPR of original signal in dB'); disp(papr); % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 6 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1; % --------------% RECEIVER % -----------no_of_error=[]; B. 32

% Path delays % Avg path

Appendix B

MATLAB Code

ratio=[]; for ii=1:length(snr) rx= []; rx_signal_matrix = reshape(tt,length(ifft_data),nsym); [~, c] = size(rx_signal_matrix); for j = 2: nsym-1 hx = filter(h,rx_signal_matrix(:,j).'); % Pass through Rayleigh channel a = h.PathGains; AM = h.channelFilter.alphaMatrix; g = a*AM; % Channel coefficients G(j,:) = fft(g,fft_size); % DFT of channel coefficients % Add AWGN no rx_signal = awgn(hx,snr(ii),'measured'); a =abs(rx_signal); c =a.^(2/d); aa =var(abs(rx_signal))+ mean(abs(rx_signal)); b =exp(-((abs(rx_signal)).^2)./var(rx_signal)); cc =(1-b).^2; e =(cc).^(d/2); E =( aa./mean(e)).^(d/2); dd=log(1-(c./E)); v =var(rx_signal); s =sqrt(-v.*dd); ff=abs(s); rr =sign(rx_signal).*ff;

% Convert Data back to "parallel" form to perform FFT con=reshape( rr , length(ifft_data),1); % Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); % p/s rx_serial_data = reshape(fft_data_matrix, 1,fft_size); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end x = size (source); B. 33

Appendix B

MATLAB Code

v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

&& COS CODE : clear all clc close % --------------% Parameters % --------------M = 4; fft_size = 128; fspacing=15000; fs=15000*128; Ts = 1/fs; Fd = 0; nsym = 1000; snr = 0:0.8:30;

% QPSK signal constellatio % have 128 data point

% Sampling period of channel % Max Doppler frequency shift

Appendix B

MATLAB Code

ifft_data_matrix = ifft(serial_to_paralle,fft_size); % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) = ifft_data_matrix(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data = vertcat(cp,ifft_data_matrix); % s/p for transmission [rows_ifft_data, cols_ifft_data]=size(ifft_data); length_ofdm_data = rows_ifft_data*cols_ifft_data; pilot = zeros(1,length_ofdm_data); ofdm_signal = reshape(ifft_data, 1, length(ifft_data));%p/ Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); a =abs (ofdm_signal ); b =sqrt(a); x= complex(b.*cos(angle(ofdm_signal )),b.*sin(angle(ofdm_signal ))); Signal_Power = abs(x.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal1(i) = 10*log10(Peak_Power./Mean_Power); tx = [tx x]; end [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_Orignal1); %-------------------------------------------------------------------------semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; t = size (tx); Q = size(tt); x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)); disp('PAPR of original signal in dB'); disp(papr); % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 6 taps B. 35

Appendix B

MATLAB Code

tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Path delays % Avg path

% --------------% RECEIVER % -----------no_of_error=[]; ratio=[]; for ii=1:length(snr) rx= []; rx_signal_matrix = reshape(tt,length(ifft_data),nsym); [~, c] = size(rx_signal_matrix); for j = 2: nsym-1 hx = filter(h,rx_signal_matrix(:,j).'); % Pass through Rayleigh channel a = h.PathGains; AM = h.channelFilter.alphaMatrix; g = a*AM; % Channel coefficients G(j,:) = fft(g,fft_size); % DFT of channel coefficients % Add AWGN no rx_signal = awgn(hx ,snr(ii),'measured'); aa =abs (rx_signal); bb =aa.^2; xx= complex(bb.*cos(angle(rx_signal)),bb.*sin(angle(rx_signal))); % Convert Data back to "parallel" form to perform FFT con=reshape(xx , length(ifft_data),1); % Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); % p/s rx_serial_data = reshape(fft_data_matrix, 1,fft_size); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end figure(2) B. 36

Appendix B

MATLAB Code

[no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end x = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

$$ tanhR CODE: clear all clc close % --------------% Parameters % --------------M = 4; fft_size = 128; fspacing=15000; fs=15000*128; Ts = 1/fs; Fd = 0; nsym = 10; snr = 0:0.8:30; k=5; k1 =1; y = 1;

% QPSK signal constellatio % have 128 data point

% Sampling period of channel % Max Doppler frequency shift

Appendix B

MATLAB Code

% to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % make ifft to each block and add CP ifft_data_matrix = ifft(serial_to_paralle,fft_size); % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) = ifft_data_matrix(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data = vertcat(cp,ifft_data_matrix); % s/p for transmission [rows_ifft_data, cols_ifft_data]=size(ifft_data); length_ofdm_data = rows_ifft_data*cols_ifft_data; pilot = zeros(1,length_ofdm_data); ofdm_signal = reshape(ifft_data, 1, length(ifft_data)); %p/s Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); x = k1.*tanh((((abs(ofdm_signal).*k).^(y)))).* sign(ofdm_signal); Signal_Power = abs(x.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal1(i) = 10*log10(Peak_Power./Mean_Power); tx = [tx x]; end figure(1) [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_Orignal1); %-------------------------------------------------------------------------semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)); disp('PAPR of original signal in dB'); disp(papr); % -----------B. 38

Appendix B

MATLAB Code

% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 4 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Path delays % Avg path

% --------------% RECEIVER % -----------no_of_error=[]; ratio=[]; for ii=1:length(snr) rx= []; rx_signal_matrix = reshape(tt,length(ifft_data),nsym); [~, c] = size(rx_signal_matrix); for j = 2: nsym-1 hx = filter(h,rx_signal_matrix(:,j).'); % Pass through Rayleigh channel a = h.PathGains; AM = h.channelFilter.alphaMatrix; g = a*AM; % Channel coefficients G(j,:) = fft(g,fft_size); % DFT of channel coefficients % Add AWGN no rx_signal = awgn(hx,snr(ii),'measured'); xx = abs((atanh(abs(rx_signal)./(k))).^(1/y))./((k1).^(1/y)) .* sign(rx_signal); % Convert Data back to "parallel" form to perform FFT con=reshape(xx , length(ifft_data),1); % Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); % p/s rx_serial_data = reshape( fft_data_matrix, 1,fft_size); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; B. 39

Appendix B

MATLAB Code

end figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end x = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

&& logR CODE: clear all clc close % --------------% Parameters % --------------M = 4; fft_size = 128; fspacing=15000; fs=15000*128; Ts = 1/fs; Fd = 0; nsym = 1000; snr = 0:0.8:30;

% QPSK signal constellatio % have 128 data point

% Sampling period of channel % Max Doppler frequency shift

Appendix B

MATLAB Code

% to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % make ifft to each block and add CP ifft_data_matrix = ifft(serial_to_paralle,fft_size); % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) = ifft_data_matrix(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data = vertcat(cp,ifft_data_matrix); % s/p for transmission [rows_ifft_data, cols_ifft_data]=size(ifft_data); length_ofdm_data = rows_ifft_data*cols_ifft_data; pilot = zeros(1,length_ofdm_data); ofdm_signal = reshape(ifft_data, 1, length(ifft_data)); %p/s ofdm_signal = reshape(ifft_data, 1, length(ifft_data));%p/ Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); k=10; k1 =1; y = .6; x = k1.*log((((abs(ofdm_signal).*k).^(y))+1)).* sign(ofdm_signal); Signal_Power = abs(x.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal1(i) = 10*log10(Peak_Power./Mean_Power); tx = [tx x]; end figure(1) [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_Orignal1); %-------------------------------------------------------------------------semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; t = size (tx); Q = size(tt); B. 41

Appendix B

MATLAB Code

x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)); disp('PAPR of original signal in dB'); disp(papr); % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 6 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Path delays % Avg path

% --------------% RECEIVER % -----------no_of_error=[]; ratio=[]; for ii=1:length(snr) rx= []; rx_signal_matrix = reshape(tt,length(ifft_data),nsym); [~, c] = size(rx_signal_matrix); for j = 2: nsym-1 hx = filter(h,rx_signal_matrix(:,j).'); % Pass through Rayleigh channel a = h.PathGains; AM = h.channelFilter.alphaMatrix; g = a*AM; % Channel coefficients G(j,:) = fft(g,fft_size); % DFT of channel coefficients % Add AWGN no rx_signal = awgn(hx ,snr(ii),'measured');h; xx = abs((exp(abs(rx_signal)./k)-1).^(1/y))./((k1).^(1/y)) .* sign(rx_signal); % Convert Data back to "parallel" form to perform FFT con=reshape(xx , length(ifft_data),1); % Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); % p/s B. 42

Appendix B

MATLAB Code

rx_serial_data = reshape(fft_data_matrix, 1,fft_size); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end x = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

&& dht function CODE: function X=dht(x) N=length(x); X=zeros(size(x)); i=sqrt(-1); for k=0:N-1 for n=0:N-1 X(k+1)=X(k+1)+x(n+1)*(cos(2*pi*k*n/N)+sin(2*pi*k*n/N)); end end &&idht function CODE: function x=idht(X) N=length(X); x=zeros(size(X)); i=sqrt(-1); for k=0:N-1 for n=0:N-1, x(k+1)=x(k+1)+X(n+1)*(cos(2*pi*k*n/N)+sin(2*pi*k*n/N)); end end x=x/N;

B. 43

Appendix B

MATLAB Code

&& precoding CODE: clear all clc close % --------------% Parameters % --------------M = 4; fft_size = 128; fspacing=15000; fs=15000*128; Ts = 1/fs; Fd = 0; nsym = 1000; snr = 0:0.8:30;

% QPSK signal constellatio % have 128 data point

% Sampling period of channel % Max Doppler frequency shift

% -----------------% TRANSMITTER % -----------------% Generate 1 x 128 vector of random data points source = randsrc(1, 128*(nsym-2), 0:M-1); %the data source cp_length = .25*fft_size; % length of cyclic prefix sp = reshape(source , 128 , nsym-2);% s/p s = size (sp); tx = []; for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); % making s/p serial_to_paralle = reshape(qpsk_mod, fft_size, 1);% s/p ddg = dht(serial_to_paralle); % ddg = fft(serial_to_paralle); % for DFT precoding % ddg = dct(serial_to_paralle); % for DCT precoding % ddg = dst(serial_to_paralle); % for DST precoding % ddg = fwht(serial_to_paralle); % for WHT precoding % to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % make ifft to each block and add CP ifft_data_matrix = ifft(ddg,fft_size); % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) = ifft_data_matrix(j+cp_start,1); end B. 44

Appendix B

MATLAB Code

% Append the CP to the existing block to create the actual OFDM block ifft_data = vertcat(cp,ifft_data_matrix); % s/p for transmission [rows_ifft_data, cols_ifft_data]=size(ifft_data); length_ofdm_data = rows_ifft_data*cols_ifft_data; pilot = zeros(1,length_ofdm_data); ofdm_signal = reshape(ifft_data, 1, length(ifft_data)); %p/s Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); tx = [tx ofdm_signal]; end [cdf0, PAPR0] = ecdf(PAPR_Orignal); %-------------------------------------------------------------------------semilogy(PAPR0,1-cdf0,'-b') legend('Orignal') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)) % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 6 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1; % --------------% RECEIVER % -----------no_of_error=[]; ratio=[]; for ii=1:length(snr) rx= []; rx_signal_matrix = reshape(tt,length(ifft_data),nsym); B. 45

% Path delays % Avg path

Appendix B

MATLAB Code

[~, c] = size(rx_signal_matrix); for j = 2: nsym-1 hx = filter(h,rx_signal_matrix(:,j).'); % Pass through Rayleigh channel a = h.PathGains; AM = h.channelFilter.alphaMatrix; g = a*AM; % Channel coefficients G(j,:) = fft(g,fft_size); % DFT of channel coefficients % Add AWGN no rx_signal = awgn(hx,snr(ii),'measured'); d=size(rx_signal); % Convert Data back to "parallel" form to perform FFT con=reshape( rx_signal , length(ifft_data),1); % Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); dg = idht(fft_data_matrix); % dg = ifft(fft_data_matrix); % for DFT precoding % ddg = idct(serial_to_paralle); % for DCT precoding % ddg = idst(serial_to_paralle); % for DST precoding % ddg = ifwht(serial_to_paralle); % for WHT precoding % p/s rx_serial_data = reshape(dg, 1,fft_size); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end x = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM'); B. 46

Appendix B

MATLAB Code

hybrid && precoding + RCF CODE: clear all clc close % --------------% Parameters % --------------M = 4; % QPSK signal constellatio fft_size = 128; % have 128 data point fspacing=15000; fs=15000*128; Ts = 1/fs; % Sampling period of channel Fd = 0; % Max Doppler frequency shift nsym = 1000; snr = 0:0.8:30; CR =4; ITERATE_NUM = 4; K = 128; % SIZE OF OFDM Symbol IF = 2; % Interpolation factor (Oversampling factor) fft_size = K*IF; % SIZE OF FFT mm=193; %when IF =1.125 =81 ;when IF =1.25 =97 ;when IF =1.5 =129; when IF = 2 =193; when IF =3 =321; when IF = 4=449 % -----------------% TRANSMITTER % -----------------% Generate 1 x 128 vector of random data points source = randsrc(1, K*(nsym-2), 0:M-1); %the data source cp_length = .25*K; % length of cyclic prefix sp = reshape(source , K , nsym-2);% s/p s = size (sp); tx = []; PAPR_Orignal = zeros(1,nsym); PAPR_RCF = zeros(ITERATE_NUM,nsym); for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); % making s/p serial_to_paralle = reshape(qpsk_mod, K , 1);% s/p ddg = dht(serial_to_paralle); % ddg = fft(serial_to_paralle); % for DFT precoding % ddg = dct(serial_to_paralle); % for DCT precoding % ddg = dst(serial_to_paralle); % for DST precoding % ddg = fwht(serial_to_paralle); % for WHT precoding

B. 47

You can use another type of precoding compnding

Appendix B

MATLAB Code

xy = [ddg(1:K/2) ; zeros(fft_size-K,1); ddg(K/2+1:K)]; ifft_data_matrix = ifft(xy,fft_size); % s/p for transmission pilot = zeros(1,length(ifft_data_matrix)); ofdm_signal = reshape(ifft_data_matrix, 1, length(ifft_data_matrix));%p/s Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); for nIter=1:ITERATE_NUM % Clipping x_tmp = ofdm_signal(Signal_Power>CR*Mean_Power); x_tmp = sqrt(CR*Mean_Power)*x_tmp./abs(x_tmp); ofdm_signal(Signal_Power>CR*Mean_Power) = x_tmp; % Filtering XX = fft(ofdm_signal,[],2); XX(K/2+(1:fft_size-K)) = zeros(1,fft_size-K); ofdm_signal = ifft(XX,[],2); % PAPR Compute Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_RCF(nIter,i) = 10*log10(Peak_Power./Mean_Power); end % make ifft to each block and add CP serial_to_paralle2 = reshape(ofdm_signal,fft_size , 1);% s/p % to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) =serial_to_paralle2(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data_cp = vertcat(cp,serial_to_paralle2); ofdm = reshape(ifft_data_cp, 1, length(ifft_data_cp));%p/s pilot = zeros(1,length(ifft_data_cp)); tx = [tx ofdm]; end figure(1) [cdf0, PAPR0] = ecdf(PAPR_Orignal); B. 48

Appendix B

MATLAB Code

[cdf1, PAPR1] = ecdf(PAPR_RCF(1,:)); [cdf2, PAPR2] = ecdf(PAPR_RCF(2,:)); [cdf3, PAPR3] = ecdf(PAPR_RCF(3,:)); [cdf4, PAPR4] = ecdf(PAPR_RCF(4,:)); %-------------------------------------------------------------------------semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r',PAPR2,1-cdf2,'-g',PAPR3,1-cdf3,'c',PAPR4,1-cdf4,'-m') legend('Orignal','One clip and filter','Two clip and filter','Three clip and filter','Four clip and filter') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; t = size (tx); Q = size(tt); figure(2) plot(real(tt)); xlabel('Time'); ylabel('Amplitude'); title('OFDM Signal');grid on; x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)) % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 6 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Path delays % Avg path

% Pass through Rayleigh channel B. 49

Appendix B

MATLAB Code

AM = h.channelFilter.alphaMatrix; g = a*AM; % Channel coefficients G(j,:) = fft(g,K); % DFT of channel coefficients % Add AWGN no rx_signal = awgn(hx ,snr(ii),'measured'); % Convert Data back to "parallel" form to perform FFT con=reshape( rx_signal, length( ifft_data_cp),1); % Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); du = [fft_data_matrix(1:K/2) ; [];fft_data_matrix(mm:fft_size)]; dg = idht(du); You can use % ddg = ifft(du); % for DFT precoding another type of % ddg = idct(du); % for DCT precoding precoding % ddg = idst(du); % for DST precoding % ddg = ifwht(du); % for WHT precoding compnding % p/s rx_serial_data = reshape(dg, 1,K); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end % r = rx(1,(K+1:length(rx)-K)); figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end ofdm_signal = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

B. 50

Appendix B

MATLAB Code

&& precoding +RCF I =1 CODE: clear all clc close % --------------% Parameters % --------------M = 4; % QPSK signal constellatio fft_size = 128; % have 128 data point fspacing=15000; fs=15000*128; Ts = 1/fs; % Sampling period of channel Fd = 0; % Max Doppler frequency shift nsym = 10; snr = 0:0.8:30; CR = 2; ITERATE_NUM = 4; % -----------------% TRANSMITTER % -----------------% Generate 1 x 128 vector of random data points source = randsrc(1, fft_size*(nsym-2), 0:M-1); %the data source cp_length = .25*fft_size; % length of cyclic prefix sp = reshape(source , fft_size , nsym-2);% s/p s = size (sp); tx = []; PAPR_Orignal = zeros(1,nsym); PAPR_RCF = zeros(ITERATE_NUM,nsym); for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); % making s/p serial_to_paralle = reshape(qpsk_mod, fft_size , 1);% s/p ddg = dht(serial_to_paralle); % ddg = fft(serial_to_paralle); % for DFT precoding % ddg = dct(serial_to_paralle); % for DCT precoding % ddg = dst(serial_to_paralle); % for DST precoding % ddg = fwht(serial_to_paralle); % for WHT precoding % to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % make ifft to each block and add CP ifft_data_matrix = ifft(ddg,fft_size); B. 51

Appendix B

MATLAB Code

% s/p for transmission pilot = zeros(1,length(ifft_data_matrix)); ofdm_signal = reshape(ifft_data_matrix, 1, length(ifft_data_matrix));%p/s Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); for nIter=1:ITERATE_NUM % Clipping x_tmp = ofdm_signal(Signal_Power>CR*Mean_Power); x_tmp = sqrt(CR*Mean_Power)*x_tmp./abs(x_tmp); ofdm_signal(Signal_Power>CR*Mean_Power) = x_tmp; % PAPR Compute Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_RCF(nIter,i) = 10*log10(Peak_Power./Mean_Power); end % make ifft to each block and add CP serial_to_paralle2 = reshape(ofdm_signal,fft_size , 1);% s/p % to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) =serial_to_paralle2(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data_cp = vertcat(cp,serial_to_paralle2); ofdm = reshape(ifft_data_cp, 1, length(ifft_data_cp));%p/s pilot = zeros(1,length(ifft_data_cp)); tx = [tx ofdm]; end figure(1) [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_RCF(1,:)); [cdf2, PAPR2] = ecdf(PAPR_RCF(2,:)); [cdf3, PAPR3] = ecdf(PAPR_RCF(3,:)); [cdf4, PAPR4] = ecdf(PAPR_RCF(4,:)); %-------------------------------------------------------------------------B. 52

Appendix B

MATLAB Code

semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r',PAPR2,1-cdf2,'-g',PAPR3,1-cdf3,'c',PAPR4,1-cdf4,'-m') legend('Orignal','One clip and filter','Two clip and filter','Three clip and filter','Four clip and filter') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; t = size (tx); Q = size(tt); x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)) % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 6 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Path delays % Avg path

Appendix B

MATLAB Code

% Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); dg = idht(fft_data_matrix); % dg = ifft(fft_data_matrix); % for DFT precoding % dg = idct(fft_data_matrix); % for DCT precoding % dg = idst(fft_data_matrix); % for DST precoding % dg = ifwht(fft_data_matrix); % for WHT precoding % p/s rx_serial_data = reshape(dg, 1,fft_size); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end % r = rx(1,(K+1:length(rx)-K)); figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end ofdm_signal = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

&& precoding + RCF( I =pilot =76 in this case )CODE: clear all clc close % --------------% Parameters % --------------M = 4; % QPSK signal constellatio fft_size = 128; % have 128 data point fspacing=15000; fs=15000*128; Ts = 1/fs; % Sampling period of channel Fd = 0; % Max Doppler frequency shift B. 54

Appendix B

MATLAB Code

nsym = 1000; snr = 0:0.8:30; CR = 1.5; ITERATE_NUM = 4; K = 76; % SIZE OF OFDM Symbol % -----------------% TRANSMITTER % -----------------% Generate 1 x 128 vector of random data points source = randsrc(1, K*(nsym-2), 0:M-1); %the data source %figure(1) cp_length = .25*K; % length of cyclic prefix sp = reshape(source , K , nsym-2);% s/p s = size (sp); tx = []; PAPR_Orignal = zeros(1,nsym); PAPR_RCF = zeros(ITERATE_NUM,nsym); for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); % making s/p serial_to_paralle = reshape(qpsk_mod, K , 1);% s/p ddg = dht(serial_to_paralle); % ddg = fft(serial_to_paralle); % for DFT precoding % ddg = dct(serial_to_paralle); % for DCT precoding % ddg = dst(serial_to_paralle); % for DST precoding % ddg = fwht(serial_to_paralle); % for WHT precoding xy = [ddg(1:K/2) ; zeros(fft_size-K,1); ddg(K/2+1:K)]; ifft_data_matrix = ifft(xy,fft_size); % s/p for transmission pilot = zeros(1,length(ifft_data_matrix)); ofdm_signal = reshape(ifft_data_matrix, 1, length(ifft_data_matrix));%p/s Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); for nIter=1:ITERATE_NUM % Clipping x_tmp = ofdm_signal(Signal_Power>CR*Mean_Power); x_tmp = sqrt(CR*Mean_Power)*x_tmp./abs(x_tmp); ofdm_signal(Signal_Power>CR*Mean_Power) = x_tmp; % Filtering XX = fft(ofdm_signal,[],2); B. 55

Appendix B

MATLAB Code

XX(K/2+(1:fft_size-K)) = zeros(1,fft_size-K); ofdm_signal = ifft(XX,[],2); % PAPR Compute Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_RCF(nIter,i) = 10*log10(Peak_Power./Mean_Power); end % make ifft to each block and add CP serial_to_paralle2 = reshape(ofdm_signal,fft_size , 1);% s/p % to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) =serial_to_paralle2(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data_cp = vertcat(cp,serial_to_paralle2); ofdm = reshape(ifft_data_cp, 1, length(ifft_data_cp));%p/s pilot = zeros(1,length(ifft_data_cp)); tx = [tx ofdm]; end figure(7) [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_RCF(1,:)); [cdf2, PAPR2] = ecdf(PAPR_RCF(2,:)); [cdf3, PAPR3] = ecdf(PAPR_RCF(3,:)); [cdf4, PAPR4] = ecdf(PAPR_RCF(4,:)); %-------------------------------------------------------------------------semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r',PAPR2,1-cdf2,'-g',PAPR3,1-cdf3,'c',PAPR4,1-cdf4,'-m') legend('Orignal','One clip and filter','Two clip and filter','Three clip and filter','Four clip and filter') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; t = size (tx); Q = size(tt); x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)) B. 56

Appendix B

MATLAB Code

% -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 6 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Path delays % Avg path

% --------------% RECEIVER % -----------no_of_error=[]; ratio=[]; for ii=1:length(snr) rx= []; rx_signal_matrix =reshape(tt,length( ifft_data_cp),nsym); [~, c] = size(rx_signal_matrix); for j = 2: nsym-1 hx = filter(h,rx_signal_matrix(:,j).'); % Pass through Rayleigh channel a = h.PathGains; AM = h.channelFilter.alphaMatrix; g = a*AM; % Channel coefficients G(j,:) = fft(g,K); % DFT of channel coefficients % Add AWGN no rx_signal = awgn(hx ,snr(ii),'measured'); % Convert Data back to "parallel" form to perform FFT con=reshape( rx_signal, length( ifft_data_cp),1); % Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); du = [fft_data_matrix(1:K/2) ; [];fft_data_matrix(91:fft_size)]; dg = idht(du); % ddg = ifft(du); % for DFT precoding % ddg = idct(du); % for DCT precoding % ddg = idst(du); % for DST precoding B. 57

Appendix B

MATLAB Code

% ddg = ifwht(du); % for WHT precoding

% p/s rx_serial_data = reshape(dg, 1,K); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end ofdm_signal = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

&& precoding + companding code: clear all clc close % --------------% Parameters % --------------M = 4; % QPSK signal constellatio fft_size = 128; % have 128 data point fspacing=15000; fs=15000*128; Ts = 1/fs; % Sampling period of channel Fd = 0; % Max Doppler frequency shift nsym = 100; snr = 0:0.8:30; % -----------------% TRANSMITTER % -----------------% Generate 1 x 128 vector of random data points source = randsrc(1, 128*(nsym-2), 0:M-1); %the data source cp_length = .25*128 ; % length of cyclic prefix sp = reshape(source , 128 , nsym-2);% s/p s = size (sp); B. 58

Appendix B

MATLAB Code

tx = []; for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); % making s/p serial_to_paralle = reshape(qpsk_mod, fft_size, 1); % s/p You can use ddg = dht(serial_to_paralle); another type of % ddg = fft(serial_to_paralle); % for DFT precoding precoding % ddg = dct(serial_to_paralle); % for DCT precoding % ddg = dst(serial_to_paralle); % for DST precoding compnding % ddg = fwht(serial_to_paralle); % for WHT precoding % to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % make ifft to each block and add CP ifft_data_matrix = ifft(ddg,fft_size); % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) = ifft_data_matrix(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data = vertcat(cp,ifft_data_matrix); % s/p for transmission [rows_ifft_data, cols_ifft_data]=size(ifft_data); length_ofdm_data = rows_ifft_data*cols_ifft_data; pilot = zeros(1,length_ofdm_data); ofdm_signal = reshape(ifft_data, 1, length(ifft_data));%p/s Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); A = 10; % Parameter for A-law compander V = max(abs(ofdm_signal)); compsig = compand(ofdm_signal,A,V,'A/compressor');

Signal_Power = abs(compsig.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal1(i) = 10*log10(Peak_Power./Mean_Power); tx = [tx compsig]; B. 59

You can use another type of compnding compnding

Appendix B

MATLAB Code

end figure(1) [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_Orignal1); %-------------------------------------------------------------------------semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; t = size (tx); Q = size(tt); x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)); disp('PAPR of original signal in dB'); disp(papr); % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 6 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Path delays % Avg path power gains

Appendix B

MATLAB Code

rx_signal = awgn(hx ,snr(ii),'measured'); V = max(abs(rx_signal)); compsig = compand(rx_signal,A, V,'A/expander'); % Convert Data back to "parallel" form to perform FFT con=reshape( compsig, length(ifft_data),1);

You can use another type of compnding compnding

You can use another type of precoding compnding

% p/s rx_serial_data = reshape(dg, 1,fft_size); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end x = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

&& RCF + companding CODE: clear all clc close % --------------% Parameters B. 61

Appendix B

MATLAB Code

% --------------M = 4; % QPSK signal constellatio fft_size = 128; % have 128 data point fspacing=15000; fs=15000*128; Ts = 1/fs; % Sampling period of channel Fd = 0; % Max Doppler frequency shift nsym = 1000; snr = 0:0.8:30; CR = 4; ITERATE_NUM = 4; K = 128; % SIZE OF OFDM Symbol IF = 2; % Interpolation factor (Oversampling factor) fft_size = K*IF; % SIZE OF FFT % -----------------% TRANSMITTER % -----------------% Generate 1 x 128 vector of random data points source = randsrc(1, K*(nsym-2), 0:M-1); %the data source cp_length = .25*K; % length of cyclic prefix sp = reshape(source , K , nsym-2);% s/p s = size (sp); tx = []; PAPR_Orignal = zeros(1,nsym); PAPR_RCF = zeros(ITERATE_NUM,nsym); for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); % making s/p serial_to_paralle = reshape(qpsk_mod, K , 1);% s/p xy = [serial_to_paralle(1:K/2) ; zeros(fft_size-K,1); serial_to_paralle(K/2+1:K)]; ifft_data_matrix = ifft(xy,fft_size); % s/p for transmission pilot = zeros(1,length(ifft_data_matrix)); ofdm_signal = reshape(ifft_data_matrix, 1, length(ifft_data_matrix));%p/s Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); for nIter=1:ITERATE_NUM % Clipping x_tmp = ofdm_signal(Signal_Power>CR*Mean_Power); B. 62

Appendix B

MATLAB Code

x_tmp = sqrt(CR*Mean_Power)*x_tmp./abs(x_tmp); ofdm_signal(Signal_Power>CR*Mean_Power) = x_tmp; % Filtering XX = fft(ofdm_signal,[],2); XX(K/2+(1:fft_size-K)) = zeros(1,fft_size-K); ofdm_signal = ifft(XX,[],2); % PAPR Compute Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_RCF(nIter,i) = 10*log10(Peak_Power./Mean_Power); end % make ifft to each block and add CP serial_to_paralle2 = reshape(ofdm_signal,fft_size , 1);% s/p % to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) =serial_to_paralle2(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data_cp = vertcat(cp,serial_to_paralle2); ofdm = reshape(ifft_data_cp, 1, length(ifft_data_cp));%p/s pilot = zeros(1,length(ifft_data_cp)); k= 5; k1 =1; y = 1; x = k1.*tanh((((abs(ofdm).*k).^(y)))).* sign(ofdm); Signal_Power = abs(x.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal1(i) = 10*log10(Peak_Power./Mean_Power);

tx = [tx x]; end figure(1) [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_RCF(1,:)); [cdf2, PAPR2] = ecdf(PAPR_RCF(2,:)); [cdf3, PAPR3] = ecdf(PAPR_RCF(3,:)); [cdf4, PAPR4] = ecdf(PAPR_RCF(4,:)); B. 63

You can use another type of compnding compnding

Appendix B

MATLAB Code

[cdf5, PAPR5] = ecdf(PAPR_Orignal1); %-------------------------------------------------------------------------semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r',PAPR2,1-cdf2,'-g',PAPR3,1-cdf3,'c',PAPR4,1-cdf4,'-m',PAPR5,1-cdf5,'-k') legend('Orignal','One clip and filter','Two clip and filter','Three clip and filter','Four clip and filter','tanh') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; t = size (tx); Q = size(tt); x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)) % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 6 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Path delays % Avg path

Appendix B

MATLAB Code

xx = abs((atanh(abs(rx_signal)./(k))).^(1/y))./((k1).^(1/y)) .* sign(rx_signal); % Convert Data back to "parallel" form to perform FFT con=reshape(xx , length(ifft_data_cp),1); % Remove CP con(1:cp_length,:)=[]; % Perform FFT

You can use another type of compnding compnding

% FFT fft_data_matrix = fft(con,fft_size); du = [fft_data_matrix(1:K/2) ; [];fft_data_matrix((fft_size/2+K/2)+1:fft_size)]; % p/s rx_serial_data = reshape(du, 1,K); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end ofdm_signal = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

&& RFC + companding CODE: clear all clc close % --------------% Parameters % --------------M = 4; % QPSK signal constellatio fft_size = 128; % have 128 data point fspacing=15000; fs=15000*128; Ts = 1/fs; % Sampling period of channel Fd = 0; % Max Doppler frequency shift B. 65

Appendix B nsym = 1000; snr = 0:0.8:30; CR = 3; ITERATE_NUM = 4; K = 128; IF = 2; factor) fft_size = K*IF; % SIZE OF FFT d =.8;

MATLAB Code

% SIZE OF OFDM Symbol % Interpolation factor (Oversampling

% -----------------% TRANSMITTER % -----------------% Generate 1 x 128 vector of random data points source = randsrc(1, K*(nsym-2), 0:M-1); %the data source cp_length = .25*K; % length of cyclic prefix sp = reshape(source , K , nsym-2);% s/p s = size (sp); tx = []; PAPR_Orignal = zeros(1,nsym); PAPR_RCF = zeros(ITERATE_NUM,nsym); for i=2:nsym-1 % QPSK modulation (mapping) qpsk_mod = pskmod(sp(:,i-1), M); % making s/p serial_to_paralle = reshape(qpsk_mod, K , 1);% s/p xy = [serial_to_paralle(1:K/2) ; zeros(fft_size-K,1); serial_to_paralle(K/2+1:K)]; ifft_data_matrix = ifft(xy,fft_size); % s/p for transmission pilot = zeros(1,length(ifft_data_matrix)); ofdm_signal = reshape(ifft_data_matrix, 1, length(ifft_data_matrix));%p/s Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal(i) = 10*log10(Peak_Power./Mean_Power); for nIter=1:ITERATE_NUM

% Filtering XX = fft(ofdm_signal,[],2); XX(K/2+(1:fft_size-K)) = zeros(1,fft_size-K); ofdm_signal = ifft(XX,[],2); % Clipping x_tmp = ofdm_signal(Signal_Power>CR*Mean_Power); x_tmp = sqrt(CR*Mean_Power)*x_tmp./abs(x_tmp); B. 66

Appendix B

MATLAB Code

ofdm_signal(Signal_Power>CR*Mean_Power) = x_tmp; % PAPR Compute Signal_Power = abs(ofdm_signal.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_RCF(nIter,i) = 10*log10(Peak_Power./Mean_Power); end % make ifft to each block and add CP serial_to_paralle2 = reshape(ofdm_signal,fft_size , 1);% s/p % to know the start and end of cp cp_start = fft_size-cp_length; cp_end = fft_size; % Compute and append Cyclic Prefix for j=1:cp_length, cp(j,1) =serial_to_paralle2(j+cp_start,1); end % Append the CP to the existing block to create the actual OFDM block ifft_data_cp = vertcat(cp,serial_to_paralle2); ofdm = reshape(ifft_data_cp, 1, length(ifft_data_cp));%p/s pilot = zeros(1,length(ifft_data_cp));

a =var(abs(ofdm))+ mean(abs(ofdm)); bb =cos(-((abs(ofdm)))./std(ofdm)); b =exp(-((abs(ofdm)))./std(ofdm)); c =(1-b).^2; e =(c).^(d/2); E1 =( a./mean(e)).^(d/2); f= (E1.*(1-bb)).^(d/2); h= sign(ofdm).*f;

You can use another type of compnding compnding

Signal_Power = abs(h.^2); Peak_Power = max(Signal_Power,[],2); Mean_Power = mean(Signal_Power,2); PAPR_Orignal1(i) = 10*log10(Peak_Power./Mean_Power); tx = [tx h]; end figure(1) [cdf0, PAPR0] = ecdf(PAPR_Orignal); [cdf1, PAPR1] = ecdf(PAPR_RCF(1,:)); [cdf2, PAPR2] = ecdf(PAPR_RCF(2,:)); [cdf3, PAPR3] = ecdf(PAPR_RCF(3,:)); [cdf4, PAPR4] = ecdf(PAPR_RCF(4,:)); [cdf5, PAPR5] = ecdf(PAPR_Orignal1); B. 67

Appendix B

MATLAB Code

%-------------------------------------------------------------------------semilogy(PAPR0,1-cdf0,'-b',PAPR1,1-cdf1,'-r',PAPR2,1-cdf2,'-g',PAPR3,1-cdf3,'c',PAPR4,1-cdf4,'-m',PAPR5,1-cdf5,'-k') legend('Orignal','One clip and filter','Two clip and filter','Three clip and filter','Four clip and filter','COS') xlabel('PAPR0 [dB]'); ylabel('CCDF (Pr[PAPR>PAPR0])'); tt =[ pilot tx pilot]; t = size (tx); Q = size(tt); x_abs=abs(tt); papr=10*log(max(x_abs.^2)/mean(x_abs.^2)) % -----------% CHANNEL % -----------% Create Rayleigh fading channel object. % Frequency selective channel with 4 taps tau = [0 .2e-9 .5e-9 1.6e-9 2.3e-9 5e-9]; pdb = [0.189 0.379 0.239 0.095 0.061 0.037]; power gains h = rayleighchan(Ts, Fd,tau,pdb); h.StoreHistory = 0; h.StorePathGains = 1; h.ResetBeforeFiltering = 1;

% Path delays % Avg path

Appendix B

MATLAB Code

a =abs(rx_signal); c =a.^(2/d); aa =var(abs(rx_signal))+ mean(abs(rx_signal)); b =exp(-((abs(rx_signal)).^2)./var(rx_signal)); cc =(1-b).^2; e =(cc).^(d/2); E =( aa./mean(e)).^(d/2); dd=acos(1-(c)./E); v =std(rx_signal); s =(-v.*dd); ff=abs(s); rr =sign(rx_signal).*ff;

You can use another type of compnding compnding

% Convert Data back to "parallel" form to perform FFT con=reshape( rr , length(ifft_data_cp),1); % Remove CP con(1:cp_length,:)=[]; % Perform FFT % FFT fft_data_matrix = fft(con,fft_size); du = [fft_data_matrix(1:K/2) ; [];fft_data_matrix((fft_size/2+K/2)+1:fft_size)]; % p/s rx_serial_data = reshape(du, 1,K); fftrx = rx_serial_data./G(j,:); % Demodulate the data qpsk_dem_data = pskdemod(fftrx,M); rx =[rx qpsk_dem_data]; end % r = rx(1,(K+1:length(rx)-K)); figure(2) [no_of_error(ii),ratio(ii)]=biterr(source , rx); % error rate calculation end ofdm_signal = size (source); v = size(rx); semilogy(snr,ratio,'--*r','linewidth',1); hold on; axis([0 30 10^-4 1]) legend('simulated') grid on xlabel('snr'); ylabel('BER') title('Bit error probability curve for qpsk using OFDM');

B. 69

‫الخالصة‬ ‫ٔيعاػفح ذمغٍى انرشدد انًرؼايذ (‪َ ْٕ )OFDM‬ظاو ذشيٍض يرؼذد انُالم فؼال ‪ٔ,‬انزي أصثخ‬ ‫يؤخشا شؼثٍا َغثٍا فً كم يٍ انُظى االذصاالخ انغهكٍح ٔانَلعهكٍح نهثٍاَاخ انٕعائػ انًرؼذدج‬ ‫َمم‪ًٌ OFDM .‬كٍ اعرخذايّ فً صًٍى انُظى انًؼشٔفح يصم خػّ اشرشان سلًً غٍش يرًاشم‬ ‫(‪ )ADSL‬اإلَرشَد‪ٔ ،‬انرهفضٌٌٕ انشلًً ‪ /‬انثس اإلراػً‪ٔ ،‬انشثكح انًذهٍح انَلعهكٍح (‪،)LANs‬‬ ‫يششٔع يرطٕس غٌٕم األيذ (‪.)LTE‬‬ ‫اسذفاع َغثح انمذسج انؼظًى إنى انًؼذل )‪ ْٕ (PAPR‬انؼٍة انشئٍغً ل‪ ،OFDM‬يًا ٌؤدي إنى‬ ‫اَخفاض انكفاءج فً اعرَٓلن انمذسج ٔتانرانً ٌؼشلم ذُفٍز ‪.OFDM‬يشكهح ال‪ ًْ PAPR‬أكصش‬ ‫أًٍْح فً اإلسعال ألٌ كفاءج يعخى انمذسج أيش تانغ األًٍْح خاصح فً انًذطح انًرُمهح ألٌ‬ ‫انثطاسٌح نذٌٓا غالح يذذٔدج‪.‬‬ ‫اسذفاع ‪ PAPR‬ذذذز َرٍجح نرمهثاخ كثٍشج فً إشاسج انذايهح نم‪ْٔ OFDM‬زا االيش ٌرطهة‬ ‫دسجح ػانٍح يٍ انخطٍح فً يعخى انمذسج انؼانً (‪ .)HPA‬يعخًاخ انمذسج انؼانٍح ػُذيا ذكٌٕ‬ ‫خطٍح تشكم كثٍش ذصثخ غانٍح انصًٍ‪ ،‬ظخًح تُغثح ذصم انىٓ٘‪ ٪‬يٍ دجى االسعال ٔصؼٕتح‬ ‫انرصٍُغ‪.‬‬ ‫يٍ أجم انذذ يٍ ‪،PAPR‬نمذ ذى الرشاح ػذج ذمٍُاخ فً ْزِ األغشٔدح‪ ،‬أال لذ ذى الرشح ػهى‬ ‫ذكشاس ذصفٍح َطاق انرشدد ٔ لصّ (‪ٔ )RFC‬تانًماسَح يغ انطشٌمح انًرٕفشج ًْٔ ذكشاس انمص ٔ‬ ‫ذصفٍح َطاق انرشدد (‪ RFCٔ .)RCF‬أفعم ‪ RCF‬فً األداء ٔخصٕصا ػُذيا كُد ٕ ‪،I‬‬ ‫ػهى انشغى يٍ أٌ نذٌٓى َفظ انرؼمٍذ ٔانركهفح‪.‬‬ ‫انطشٌمح انًمرشدح نٍغد فمػ ذؼًم ػهى ذذغٍٍ ‪ٔ PAPR‬اًَا أٌعا ذذغٍ يؼذل خطأ انشاسج‬ ‫)‪ .(BER‬أفعم َرٍجح فً ْزِ انطشٌمح نم‪ ًْ BER‬ػُذ ‪ ،CR = 4 ٔ I = 4‬دٍس فً دٍس‬ ‫ذذغُد َغثح اإلشاسج إنى انعجٍجغ )‪ (SNR‬ػُذيا )‬ ‫( ‪ BER‬تًمذاس (ٔٓ‪٘.6ٙ‬‬ ‫دٌغٍثم)‪ٔ ،‬ذذغٍ انذانح انًكًهح نهرٕصٌغ انرشاكًً(‪ )CCDF‬نم ‪ PAPR‬تًمذاس (٘‪ٗ.66‬‬ ‫دٌغٍثم)‪ٔ ،‬ذذغٍ ‪ PAPR‬تًمذاس (‪ ٔٔ.ٗٔ66‬دٌغٍثم)‪.‬‬ ‫أفعم ذذغٍ ٔادذج فً ‪ CCDF of PAPR ٔ PAPR‬تذٍس ال ذرذْٕس ‪ ْٕ BER‬فً ‪ٔ I = 4‬‬ ‫‪ٔ .CR = 1.75‬يمذاسانرذغٍ فً ‪ ٔ8.ٕ681( = PAPR‬دٌغٍثم)‪(=CCDF of PAPR ،‬‬ ‫‪ 8.0187‬دٌغٍثم)‪ٔ ،‬ذرذغٍ )‬ ‫(‪ SNR at BER‬يمذاس = (ٔٓٔ‪ ٓ.ٙ‬دٌغٍثم)‪.‬‬ ‫تاإلظافح إنى (‪ٔ )RFC‬لذ ذى الرشاح عرح إَٔاع جذٌذج يٍ ‪ٔٔ companding‬يماسَرٓا يغ لإٌَ‬ ‫‪ ٔ -μ‬لإٌَ‪ .- compandings A‬كم ْزِ األعانٍة انًمرشدح نٓا أداء أفعم يٍ لإٌَ ‪ٔ μ‬‬ ‫لإٌَ ‪ٔ ، compandings A‬أفعم ‪ companding‬يمرشح ْٕ انًطهك األعً )‪)AEXP‬‬ ‫ٔافعم يمذاسذذغٍ فً ‪ ًْ CCDF of PAPR ٔ PAPR‬ػُذيا ذكٌٕ ‪ٔ .ٔ.ٔ =d‬يمذاس‬ ‫انرذغٍ فً ‪ ٔ6.ٙٗ1ٕ( = PAPR‬دٌغٍثم)‪ 7.2405( = CCDF of PAPRٔ ،‬دٌغٍثم)‪،‬‬ ‫تًٍُا )‬ ‫(‪ SNR at BER‬ذرذْٕسيمذاس= (‪ ٖ.ٗٔ8ٙ-‬دٌغٍثم)‪.‬‬ ‫ٔذى اعرخذاو خًغح إَٔاع يٍ لثم انرشيٍض(‪ )precoding‬فً ْزِ االغشٔدح ٔيٍ شى يماسَرٓا يغ‬ ‫تؼعٓا انثؼط‪ .‬أفعم َٕع يٍ ‪ precoding‬فً ذمهٍم ‪ ْٕ BERٔ PAPR‬ذذٌٕم فٕسًٌ‬ ‫انًرمطغ)‪ (DFT‬تًٍُا اعٕء َٕع يماسَح يغ انثمٍح ْٕذذٌٕم ٔانش ْاداياسد (‪.(WHT‬‬ ‫‪B. 70‬‬

‫كًا ذى الرشاح أستؼح إَٔاع جذٌذج يٍ ذمٍُاخ ْجٍُح نهذذ يٍ ‪ْ .PAPR‬زِ انطشق ًْ‪:‬‬ ‫ٔ‪ RCF .‬يغ ‪ ،WHT) precodings‬ذذٌٕم جٍة انرًاو انًرمطغ)‪ ، (DCT‬ذذٌٕم جٍة‬ ‫انًرمطغ (‪ ٔ ،)DST‬ذذٌٕم ْاسذهً انًرمطغ(‪.))DHT‬‬ ‫ٕ‪ RCF . .‬يغ ‪( compandings‬نجًٍغ إَاع ال‪ compandings‬انًمرشدح‪ ،‬انمإٌَ‪μ-‬‬ ‫ٔانمإٌَ‪.)compandings A-‬‬ ‫ٖ‪ RFC . .‬يغ ‪( compandings‬نجًٍغ إَاع ال‪ compandings‬انًمرشدح‪ ،‬انمإٌَ‪μ-‬‬ ‫ٔانمإٌَ‪)compandings A-‬‬ ‫ٗ‪ٔ .‬أخٍشا ‪ ،.)DHTٔ ،DST ،DCT ،WHT) precodings‬يغ ‪( compandings‬نجًٍغ‬ ‫إَاع ال‪ compandings‬انًمرشدح‪ ،‬انمإٌَ‪ٔ μ-‬انمإٌَ‪)compandings A-‬‬ ‫أفعم دانح ًْ (‪ RFC‬يغ ‪ )AEXP‬ألَٓا ذؼًم ػهى ذذغٍٍ كم يٍ ‪CCDF of ٔ ،PAPR‬‬ ‫‪ٔ .BERٔ ، PAPR‬افعم لًٍح نرذغٍ ‪ ًْ CCDF of PAPR, PAPR‬ػُذ ‪ٔ ٓ.ٙ = d‬‬ ‫‪ .CR = 4‬يمذاسانرذغٍ فً ‪ ٕٔ.ٓ٘ٓ1( = PAPR‬دٌغٍثم)‪8.7178(= CCDF of PAPR,‬‬ ‫دٌغٍثم) )‬ ‫(‪ ٓ.ٓٔٔٙ( = SNR at BER‬دٌغٍثم)‪.‬‬ ‫ٔ‪ DHT‬يغ ظم ذًاو انجزسي (‪ )tanhR‬نٓا َرائج جٍذج دٍس ذرذغٍ كم يٍ ‪PAPR‬‬ ‫ٔ‪ CCDF of PAPR‬تًٍُا ال ذرذْٕس ‪ BER‬يمذاس كثٍش‪.‬‬ ‫أفعم لًٍح نرذغٍ ‪ ًْ CCDF of PAPR, PAPR‬ػُذ ‪.DHTٔ ٓ.8 = y ،ٔ٘ = k‬‬ ‫يمذاسانرذغٍ فً ‪ ٕٕ.66ٔٔ( = PAPR‬دٌغٍثم)‪ 8.9691(=CCDF of PAPR ،‬دٌغٍثم)‬ ‫تًٍُا )‬ ‫(‪ SNR at BER‬ذرذْٕس يمذاس= (‪ ٔ.ٔ8ٕ8-‬دٌغٍثم)‪.‬‬ ‫ذى يذاكاج كم انطشق تاعرخذاو ياذَلب‪.‬‬

‫‪B. 71‬‬

‫نظرة عامة ‪ :‬تقلٌل نسبة القدرة العظمى الى المعدل‬ ‫فً نظام مضاعفة تقسم التردد المتعامد باستخدام‬ ‫طرق بعض الطرق الجدٌدة‬ ‫(مع ماتالب كود)‬ ‫تأليف‬ ‫زٌنب سعد هادي الهاشمً‬ ‫خرٌجة كلٌة الهندسة قسم االكترونٌك واالتصاالت‬ ‫جامعة بغداد‬

‫‪1436‬‬

‫‪2015‬‬

‫نظرة عامة‪ :‬تقلٌل نسبة القدرة العظمى الى المعدل‬ ‫فً نظام مضاعفة تقسم التردد المتعامد باستخدام‬ ‫بعض الطرق الجدٌدة (مع ماتالب كود)‬ ‫زٌنب سعد هادي الهاشمً‬