Performance Analysis of Convolutional Coding in CDMA Communication Systems by Alejandro Iturri

Journal of Vectorial Relativity

JVR 4 (2009) 4 1-11

Performance Analysis of Convolutional Coding in CDMA Communication Systems G Leija-Hernández 1, M Badaoui 2 Y A Iturri-Hinojosa 3

A performance analysis of convolutional coding on wide band CDMA communications systems is presented. The two main performance parameters used for the analysis are: bit energy to spectral density rate, Eb/No, and the frame error rate, Pf(e). Some convolutional encoders are studied, and appropriate values for the performance parameters are suggested. These values correspond to the K stage codec register of a digital communication system.

RESUMEN:

KEYWORDS: Pf(e), Eb/No, CDMA.

INTRODUCTION

Finding solutions to optimize data traffic in CDMA systems is of extreme importance. This can be achieved by analyzing different performance parameters, capable of diagnosing the channel’s traffic functions to ensure the best flow in the transmission of information during communication. Two important parameters are the probability of frame error, Pf(e), and the bit energy to spectral noise density ratio, Eb/No. By finding convenient values for the Eb/No ratio for an appropriate Pf(e) of 0.01, the proper functionality of the CDMA system with convolutional codification can be guaranteed [1].

II. II.1

SYSTEM DESCRIPTION

Digital communication systems

Digital input data is usually binary, although they can be coded for any alphabet of q ≥ 2 symbols. Data reaches a symbol velocity for each Ts seconds and is stored in a registry until a block of K symbols is filled. Afterwards, this block is presented to the coding channel as one of the M possible messages, referenced as H1, H2, …, HM where M = qK and q is the alphabet’s dimension. The combination of the encoder and the modulator maps the set of the M messages, {HM}, to the set of the M signals of finite energy, {xm(t)}, of finite duration T = KTs. Figure 1 shows the block diagram of a digital communication system on the transmitter’s side.

1, 2, 3

ESIME-Zacatenco/IPN, Área. de Diciembre, 2009 E-mail: aiturri@ipn.mx

1,3

Ing. en Comunic. y Electrón, Ing. Eléctrica. Anexo Edif. 5, Col. Lindavista CP 07738 México DF

G Leija-Hernández et al.: Análisis de Desempeño de la Codif. Convolucional en Sist. de Com. CDMA

Diciembre, 2009

Fig. 1 Block diagram of a digital communication system transmitter [1]

II.2

Frame error rate

The frame error rate is a historic registry of the real functionality of the system with regards to errors. For example, if a system has a probability of frame error of 10-3, it means that in the past there has been one erroneous frame for each 1000 transmitted frames. The frame error rate is first calculated and compared with the expected error probability to evaluate the system’s performance [2,5,9]. Any symmetric channel without memory depends on interference and fading, and is independent from symbol to symbol. For this reason there is a convolutional binary code with constraint length K that depends on the code rate, R, given in bits/symbol. Considering that the frame contains L-bits, the probability error is defined by [6]: Pf <

L 2 − Kro / r , 2 1 − 2 − ( ro / r − 1 )

(

)

r < ro

(1)

where r0 is an exclusive statistic function of the memoryless channel, often considered a practical limit of the code rate r of the binary convolutional code, given by:

r0 = 1 − log 2 (1 + Z )

(2)

Being Z the performance parameter for the choice of decoder to be used [6]. The convolutional codes are generated by the continuous execution of logical operations on a limited sequence of m bits contained in a message. They are appropriate for use in channels with a lot of noise (high probability of error). A encoder with memory uses convolutional codes. The convolutional encoder accepts b binary symbols at the input and produces n binary symbols at the output. The coding rate is defined by r = b/n, and typical values of b and n may vary from 1 to 8, and r may vary from ½ to 7/8, see Fig. 2.

JVR 4 (2009) 4 1-9

©Journal of Vectorial Relativity

G Leija-Hernández et al.: Análisis de Desempeño de la Codif. Convolucional en Sist. de Com. CDMA

Diciembre, 2009

Fig. 2 Process of a Convolutional Encoder (2,1,9) for r=1/2 and (3,1,9) for r=1/3

In consequence, each frame of k-bits produces an output frame of n-bits. The redundancy foreseen at the output is because of nb. Figure 2 shows a convolutional encoder (2,1,9), and the reason for these values is to maintain the probability of error below 10-2 and to obtain a coding rate of 1/2 and of 1/3. A reduced value of the coding velocity r points to a high degree of redundancy; this brings a more effective error control at the cost of incrementing the coded signal’s bandwidth [7]. To obtain an effective convolutional code with constraint length K and rate r, the following expression is suggested [6].

Pf <

L T (Z ) 2

(3)

T(Z) is the code generating function, and Z is obtained from (2):

Z = 2 1− R 0 − 1

(4)

Now, a wireless personal communication system with CDMA access technique, that uses convolutional coding with M-ary orthogonal signaling is considered [8]. Let b bits flowing through the input of the register, to select n successive M-ary signals generating a coding rate of R=b/n bits/symbol. The frame error probability of the L-bits, for a constraint length register of bK bits, is given by [6]:

Pf <

L 2

⎛ 2b −1 ⎞ 2 − bKRo / R ⎜⎜ ⎟⎟ − b [( Ro / R )− 1 ] ⎝ b ⎠1− 2

(5)

Using expression (5) we can find the frame error probability with respect to α = R0/R. In disperse spectrum modulation there exists a coding redundancy. Consequently, no limitation is imposed on the code rate. The convolutional code is convolutioned over another, i.e., redundancy bits are added to a block of data at the input.

JVR 4 (2009) 4 1-9

©Journal of Vectorial Relativity

G Leija-Hernández et al.: Análisis de Desempeño de la Codif. Convolucional en Sist. de Com. CDMA

Diciembre, 2009

Fig. 3 Probability of frame error for K = 9 [6]

As one can see in Fig. 3, for α = 0.6 dB there is a corresponding value of Pf = 0.01. In other words, one erroneous frame for each 100 frames transmitted. A probability of frame error below 10-2 is considered as an acceptable level for the operation of the vocoder over voice traffic [6]. The vocoder is in charge of generating the traffic channel of CDMA and coding voice in PCM frames [4]. II.3

Frame Error Probability

Frame error probability, Pf(e), is the theoretic or mathematic expectation of a given system with regards to the frame error rate. The probability of error is a function of the carrier to noise ratio, or being more specific, the average of the bit energy and noise power density ratio. The probability of error is dependent on the possible conditions of coding that are used (M-ary signaling) [5]. Now, using equation (5) we can find the frame error probability in terms of α . For example, for values of bK = 9, b = 1, 2, y 3, where bK is the registry of input frame displacement of the convolutional code. Fig. 4 shows the response to these values.

JVR 4 (2009) 4 1-9

©Journal of Vectorial Relativity

G Leija-Hernández et al.: Análisis de Desempeño de la Codif. Convolucional en Sist. de Com. CDMA

Diciembre, 2009

1 b=1 b=2 b=3

0.1

0.01

0.001 0.5

1.5

2.5

ro/r

Fig. 4 Frame error rate in terms of

The values of Pf obtained from Fig. 4, represent an erroneous frame for a given number of frames transmitted in a CDMA system. For example, for the relation R0/R =2 dB we can obtain a frame probability of 1/0.0186 that represents an erroneous frame for each 52 transmitted frames. Table 1 presents the number of frames received with one failed frame. Table 1 Number of Received Frames with one Failed Frame

II.4

α =Ro/R

For b=1

For b=2

For b=3

0.5

1/0.236=4

1/0.193=5

1/0.217=4

1/0.0186=52

1/0.0137=72

1/0.0169=59

2.5

1/0.0036=277

1/0.00345=289

1/0.00426=234

1/0.0008=1250

1/0.00106=940

Performance of Convolutional Codes

The operation of convolutional codes depends mainly on multipath fading, which is produced when a direct signal is partially cancelled out by reflection on earth, water, or intermediate objects between transmitter and receiver. This fading reduces signal intensity in 20 dB or more. Any wireless CDMA system must include a margin for fading in its calculations of system gain. Fading is a reduction of signal intensity below its normal level [2,3]. Through the development of the formula m/R/α the optimal performance of convolutional codes using orthogonal signals interlaced in multipath environment will be found. For this case of convolutional coding with M-ary orthogonal signaling: JVR 4 (2009) 4 1-9

©Journal of Vectorial Relativity

G Leija-Hernández et al.: Análisis de Desempeño de la Codif. Convolucional en Sist. de Com. CDMA

Diciembre, 2009

Ro = − log 2 z

(6)

where [6]:

⎡ 1+ S ⎤ Z =⎢ 2⎥ ⎣ (1 + S / 2) ⎦

(7)

From (6) and (7) : ⎤ ⎡ 1+ S Ro = − log 2 ⎢ 2 ⎥ ⎣ (1 + S / 2 ) ⎦

(8)

where: α = Ro/R

(9)

After a series of operations with these expressions, the following second grade equation is obtained [10].

2−Ro/m 2 −Ro/m S + 2 −1 S + 2−Ro/m −1 =0 4

(

) (

)

(10)

Using equation (10), the performance of convolutional codes is obtained, see Fig. 5. It describes the behavior of the signal received by m trajectories. Similarly, it provides necessary information to determine if the signal will be optimal or not when being received. For example, for Eb/N0/α = 6.684 dB, α = 2dB. The value of α = 2dB, is there because Pf must be below 0.01 for the correct operation of the vocoder. 8.5

(Eb/No)/Alpha, dB

7.5

6.5 -4

Fig. 5.

-2

2 4 (m/R)/Alpha, dB

Performance of convolutional codes

Since Eb/No is the bit energy to spectral noise density ratio, in other words, the relationship between the energy contained in a single information bit and noise power in a bandwidth normalized to 1Hz [10]. We get Eb=No = 8.684 dB if α = 2dB. JVR 4 (2009) 4 1-9

©Journal of Vectorial Relativity

G Leija-Hernández et al.: Análisis de Desempeño de la Codif. Convolucional en Sist. de Com. CDMA

Diciembre, 2009

The following table summarizes some relationships in these performance parameters. Table 2. Performance Parameters of m=qK multipaths to be combined at the receiver

Eb/No/α 6.68 dB 6.85 dB 7.24 dB 7.78 dB 8.44 dB

III.

(m/R)/α 2 4 6 8 10

Eb/No 8.68 dB 8.85 dB 9.24 dB 9.78 dB 10.44 dB

RESULT ANALYSIS

The bit energy to spectral noise density response, Eb/No, is shown in Fig. 6 [10]. 9.5 9.4 9.3

Eb/No (dB)

9.2 9.1 9 8.9 8.8 8.7 8.6 8.5 1

3 4 (Eb/No)/Alpha, dB

Fig. 6 Bit energy to spectral noise density ratio as a function of Eb/No/α

The following Table summarizes some relationships. The frame error probability, Pf(e), and the bit energy to spectral noise density ratio, Eb/No, are measured at band base signal level at the receiver. Table 3. Some values of Eb/No

Eb/No (dB)

Eb/No/α(dB)

8.684 8.855 9.245 9.787 10.44

6.684 6.855 7.245 7.787 8.440

The coding gain is defined as the decrease of Eb/No obtained when some sort of coding is used.

JVR 4 (2009) 4 1-9

G Leija-Hernández et al.: Análisis de Desempeño de la Codif. Convolucional en Sist. de Com. CDMA

Diciembre, 2009

Fig. 7 shows the response for the Frame Error Probability with respect to Eb/No for some constrained lengths bK of the convolutional encoder with b=1. The coding gain is increased as Eb/No increases. An 8.99dB gain is achieved for the typical value of Pf=0.001, the minimum value required, for bK=9. For example, for a frame error probability of 0.01, the value of Eb/No for bK=12 must be greater than or equal to 8.16 dB. 0

-1

-2

bK=6 bK=9 bK=12 -3

6.5

7.5

8 8.5 Eb/No, dB

9.5

Fig. 7 Frame error probability in terms of Eb/No for bK=5,9 and 12

IV.

CONCLUSIONS

A performance analysis of convolutional codes for high bandwidth CDMA communication systems is presented. The results obtained for the frame error probability in terms of the bit energy to spectral noise density ratio for some constrained lengths of the convolutional encoder K=5,9 and 12 are also shown. The encoders with large storage registries reach lower Eb/No ratios with respect to encoders with smaller storage registries, for the same required frame error probability. For example, between the registries K=9 and 12, the difference of Eb/No ratio is approximately 0.8 dB, for a frame error probability of 0.01. This difference increases as the length of the registry decreases. In the same manner, the behavioral curve of the relationship Eb/No with Eb/No/α is shown. REFERENCES [1]

Andrew J. Viterbi, Jim K. Omura, Principles of Digital Communication and Coding, Mc. Graw Hill Series in Electrical Engineering. 1979.

[2]

Wayne Tomasi, Sistemas de

[3]

Samuel C. Yang, CDMA RF System Engineering, Artech House, 1998.

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Comunicaciones Electrónicas Prentice Hall, cuarta edición 2003.

G Leija-Hernández et al.: Análisis de Desempeño de la Codif. Convolucional en Sist. de Com. CDMA

Diciembre, 2009

[4]

Diplomado en Telecomunicaciones. Módulo I Telefonía, Alcatel.

[5]

Roger L. Freeman, Ingeniería de Sistemas de Telecomunicaciones (Diseño de Redes Digitales y Analógicas), Limusa Noriega.

[6]

Andrew J. Viterbi, Ephraim Zehavi “Performance of Power Controlled Wideband Terrestrial Digital Communication”, 41, No. 4. 1993 IEEE.

[7]

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[8]

Qiang Shen, Power Assignment in CDMA Personal Systems With Integrated Voice/data Traffic, 1996.

[9]

Andy Bateman, Digital Communications Design of the Real World, México: Marcombo, 2003.

[10] L. Alejandro Iturri, Luis Maldonado Alcalá, Gabriela Leija Hernández, “Parámetros de Desempeño en Sistemas de Comunicaciones CDMA de Banda Ancha”, 2do. Congreso Internacional de Sistemas Computacionales y Electrónicos, Septiembre 2-5, 2008, México D.F.

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