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Microwave and millimetre-wave design for wireless communications First Edition Chongcheawchamnan
7.5 Robust Adaptive MVDR Beamformer with Single WC Constraint
7.6 Robust LCMVBeamforming with MBWC Constraints
7.7 Geometric Interpretation
7.8 Simulation Results
7.9 Summary
References
Chapter 8: MinimumBER Adaptive Detection and Beamforming
8.1 Introduction
8.2 MBER Beamformer
8.3 MBER Simulation Results
8.4 MBER Spatial MUD in MIMO/OFDM Systems
8.5 MBER Simulation Results
8.6 Summary References
Index End User License Agreement
List of Tables
Chapter 2
Table 2.1 CP type and relevant resource block parameters in LTE ULwith SCFDMA.
Table 2.2 CP and physical RB parameters in LTE DLwith OFDM.
Table 2.3 Delay profiles for EUTRAchannel models.
Table 2.4 Typical propagation channel models used for LTE.
Table 2.5 How systems handle delay spreads.
Table 2.6 MIMO DLTxM mode.
Table 2.7 MIMO types in LTE.
Chapter 4
Table 4.1 Multiplication complexity comparison of MOEIQRD detectors.
Table 4B.1 Gram–Schmidt algorithm.
Table 4B.2 Householder transformation.
Table 4B.3 Givens rotations.
Chapter 5
Table 5.1 Complexity analysis comparison.
Table 5A.1 Summary of the robust RCG algorithm.
Chapter 6
Table 6.1 Initializations for simulated CMbased detectors.
Chapter 7
Table 7.1 MVDR RAB techniques, their notions of robustness and prior information used.
Chapter 8
Table 8.1 MBER simulation parameters in DSCDMAsystem.
Table 8.2 Computational complexity comparison.
Table 8.3 MBER simulation parameters in MIMO/OFDM system.
List of Illustrations
Chapter 1
Figure 1.1 Summary of the book.
Chapter 2
Figure 2.1 Classification of fading channels.
Figure 2.2 The relation between largescale and smallscale fading.
Figure 2.3Figure 2.3 LTE FDD frame and slot structure for normal CP.
Figure 2.4 OFDM transmitter and receiver.
Figure 2.5 Coding and modulation for transmission of data over a radio link.
Figure 2.6 Throughput of a set of coding and modulation combinations, AWGN channels assumed.
Figure 2.7 Spectral efficiency versus SNIR for baseline EUTRA.
Figure 2.8 Discrete time DSCDMASystemmodel.
Figure 2.9 I/Q code multiplexing with complex spreading circuit.
Figure 2.10 Impulse response of chip pulse shaping filter with .
Figure 2.11 Example of impulse response and frequency transfer function of a multipath channel.
Figure 2.12 Channel bandwidth and transmission bandwidth configuration for one LTE carrier.
Figure 2.13 OFDM systemblock diagram.
Figure 2.14 OFDMAsystemstructure.
Figure 2.15 MIMO system.
Figure 2.16 Massive MIMO gain.
Figure 2.17 Energy and bit allocation for a channel instance.
Figure 2.18 Plane wave incident on a ULAwith an AOAof θ.
Figure 2.19 Narrowband beamformer.
Figure 2.20 DS/CDMAsystemmodel with antenna array.
Chapter 3
Figure 3.1 The conventional single user DS/CDMAdetector: a bank of MFs.
Figure 3.2 Multiuser detection techniques.
Figure 3.3 Block diagramof PLIC structure.
Figure 3.4 MSE for MF, ZF, and MMSE detectors versus snapshot.
Figure 3.5 Average BER versus SNR for the MF, ZF, and MMSE detectors.
Figure 3.6 MSE of MOE with single constraint for three different scenarios.
Figure 3.7 Output SINR for MOE detector with equal gain channel, optimized channel, and optimumchannel.
Figure 3.8 BER for MOE detector with equal gain channel, optimized channel, and optimumchannel.
Figure 3.9 Output SINR versus SNR for MOE detector with five channel estimators and the MMSE detector with perfect power control.
Figure 3.10 Output SINR versus SNR for MOE detector with five channel estimators and the MMSE detector with 10 dB near–far effect.
Figure 3.11 Adaptive LCCMAdetector with/without whitening under equal power: (a) BER and (b) output SINR.
Figure 3.12 Adaptive LCCMAdetector with/without whitening under near–far effect: (a) BER and (b) output SINR.
Chapter 4
Figure 4.1 Block diagramof PLIC structure.
Figure 4.2 Configuration of linearly constrained adaptivearray beamformer.
Figure 4.3 Systolic array implementation and processing cells for inverse updating of the LCIQRDRLS beamforming.
Figure 4.4 Systolic array implantation for PLICMOEIQRD algorithm.
Figure 4.5 Systolic array implementation for directMOEIQRD with max/min.
Figure 4.6 Definitions of cells used in the systolic implementations.
Figure 4.7 Output SINR for IQRD-based detectors versus snapshots.
Figure 4.8 BER for IQRDbased detectors versus snapshots.
Figure 4.9 Output SINR of optimal MOEIQRD with different subspace tracking algorithms.
Figure 4.10 BER of optimal MOEIQRD with different subspace tracking algorithms.
Figure 4.11 SINR versus iterations bits for various MOE detectors at 20 dB SNR.
Figure 4.12 BER versus iterations bits for various MOE detectors at 20 dB SNR.
Figure 4.13 Output SINR of MOEIQRD w. max/min & QC for different constrained values.
Figure 4.14 BER of MOEIQRD w. max/min & QC for different constrained values.
Figure 4.15 Comparison between MOEIQRD w. max/min & VLand MOERLS w. VL.
Figure 4.16 Complexity analysis versus detector length at fixed channel length.
Figure 4.17 Complexity analysis versus channel length at fixed detector length.
Figure 4B.1 Architecture of complex Givens rotation.
Chapter 5
Figure 5.1 MOERLS w. QC (real roots).
Figure 5.2 MOERLS w. QC (imaginary roots).
Figure 5.3 Geometric interpretation for simplified VLtechnique.
Figure 5.4 SINR versus iterations for the first (ideal) initializations scenario.
Figure 5.5 BER versus iterations for the first (ideal) initializations scenario.
Figure 5.6 SINR versus iterations for the second initialization scenario.
Figure 5.7 BER versus iterations for the second initialization scenario.
Figure 5.8 SINR versus iterations for the third initialization scenario.
Figure 5.9 BER versus iterations for the third initialization scenario.
Figure 5.10 SINR versus iterations for the fourth initialization scenario.
Figure 5.11 BER versus iterations for the fourth initialization scenario.
Figure 5.12 SINR versus iterations for the fifth scenario.
Figure 5.13 BER versus iterationsfor the fifth scenario.
Figure 5.14 SINR for the MOERSD Detectors with different stepsize values.
Figure 5.15 BER for the MOERSD Detectors with different stepsize values.
Figure 5.16 SINR for the MOERSD Detectors with different alpha values.
Figure 5.17 BER for the MOERSD Detectors with different alpha values.
Figure 5.18 SINR for the MOERSD w. QC Detector with different constrained values.
Figure 5.19 BER for the MOERSD w. QC detector with different constrained values.
Figure 5.20 Multiplication complexity versus detector length.
Chapter 6
Figure 6.1 SINR versus iterations for different CMbased detectors.
Figure 6.2 BER versus iterations for different CMbased detectors.
Figure 6.3 SINR versus block iterations for first scenario.
Figure 6.4Figure 6.4 BER versus block iterations for first scenario.
Figure 6.5Figure 6.5 5 SINR versus block iterations for second scnario.
Figure 6.6 BER versus block iterations for second scenario.
Figure 6.7 Computational complexity of the developed robust detectors versus detector length.
Figure 6.8 Computational complexity versus detector length.
Figure 6.9 Computational complexity versus detector length.
Chapter 7
Figure 7.1 Geometric representation for robust Capon beamforming with ellipsoidal constraint.
Figure 7.2 Geometric interpretation of the robust WC beamformer.
Figure 7.3 Fiveelement uniformlinear array with one source and two jammers.
Figure 7.4 Output SINR versus snapshot for first scenario.
Figure 7.5 Mean squared error versus snapshots.
Figure 7.6 Signal of interest power versus snapshot.
Figure 7.7 Beampatterns of the presented beamformers.
Figure 7.8 Beampatterns of the presented beamformers.
Figure 7.9 Output SINR versus snapshot for the second scenario.
Figure 7.10 Output SINR versus noise power with 0.03π mismatch angle.
Figure 7.11 Output SINR versus noise power with 0.06π DOAmismatch.
Figure 7.12 Output SINR versus snapshot for noncoherent stationary moving scenario.
Figure 7.13Figure 7.13 Output SINR versus snapshot for coherent stationary moving scenario.
Figure 7.14 Output SINR versus snapshot for nonstationary moving scenario.
Figure 7.15 SINR versus mismatch angle.
Figure 7.16 Output SINR versus snapshot index for the first scenario.
Figure 7.17 Steady state array beampatterns versus AOI (in radians) for the first scenario.
Figure 7.18 Output SINR versus noise power for the first scenario with training data size N = 50.
Figure 7.19 WC parameters of the robust MVDRWC/proposed beamformer at .
Figure 7.20 Effect of WC parameter on the output SINR of the first scenario.
Figure 7.21 WC parameters of the robust MVDRWC/proposed beamformer at .
Figure 7.22 WC parameters of the robust MVDRWC/proposed beamformer at .
Figure 7.23 WC parameters of the modified robust MVDRWC/proposed beamformer at .
Figure 7.24 Output SINR versus snapshot index for the modified robust MVDR WC/proposed beamformer with the parameters of the first scenario.
Figure 7.25 Output SINR versus snapshot index for the second scenario.
Figure 7.26 Output SINR versus snapshot index for the third scenario.
Chapter 8
Figure 8.1 MMSE versus BER cost function.
Figure 8.2 BER vs SNR for selected algorithms minimizing the BER; equal power distribution.
Figure 8.3Figure 8.3 BER vs SNR for selected algorithms minimizing the BER; desired user power 10 dB below interferers.
Figure 8.4Figure 8.4 BER vs SNR for selected algorithms minimizing the BER; equal power distribution.
Figure 8.5 BER vs SNR for selected algorithms minimizing the BER; desired user power 10 dB below interferers.
Figure 8.6 BER vs SNR for AMBER family algorithms and SDMMSE algorithm; equal power distribution.
Figure 8.7 BER vs SNR for LMBER family algorithms and LMSMMSE algorithm;
equal power distribution.
Figure 8.8 BER vs SNR for three algorithms minimizing BER cost function (LMBER, NewtonLMBER, BSMBER) and one minimizing MSE cost function (DMI); equal power distribution.
Figure 8.9 BER versus SNR for three algorithms minimizing BER cost function (LMBER, NewtonLMBER, BSMBER) and one minimizing MSE cost function (DMI); desired user power 10 dB below interferers.
Figure 8.10 BER vs SNR for AMBER family and SDMMSE algorithms; equal power distribution at SNR = 30 dB.
Figure 8.11 BER vs iterations for LMBER family and LMSMMSE algorithms; equal power distribution at SNR = 30 dB.
Figure 8.12 BER versus Iterations for three algorithms minimizing BER cost function (LMBER, NewtonLMBER, BSMBER) and one minimizing MSE cost function (DMI); equal power distribution at SNR = 15 dB.
Figure 8.13 BER versus iterations for three algorithms minimizing BER cost function (LMBER, NewtonLMBER, BSMBER) and one minimizing MSE cost function (DMI); desired user power 10 dB below interferers at SNR = 15 dB.
Figure 8.14 BER versus iterations for three algorithms minimizing BER cost function (LMBER, NewtonLMBER, BSMBER) and one minimizing MSE cost function (DMI); equal power distribution at SNR = 30 dB.
Figure 8.15 BER versus iterations for three algorithms minimizing BER cost function (LMBER, NewtonLMBER, BSMBER) and one minimizing MSE cost function (DMI); desired user power 10 dB below interferers at SNR = 30 dB.
Figure 8.16 BER versus number of users for three algorithms minimizing the BER cost function (LMBER, NewtonLMBER, BSMBER) and one minimizing the MSE cost function (DMI) with equal power distribution at SNR = 30 dB.
Figure 8.17 BER versus number of users for three algorithms minimizing BER cost function (LMBER, NewtonLMBER, BSMBER) and one minimizing MSE cost function (DMI); desired user power 10 dB below interferers at SNR = 30 dB.
Figure 8.18 BER vs iterations for AMBER family and MMSE algorithms; SUand MF as higher and lower steadystate limits, respectively; equal power distribution at SNR = 30 dB.
Figure 8.19 BER vs iterations for LMBER family and MMSE algorithms; SUand MF as higher and lower steadystate limits, respectively; equal power distribution at SNR = 30 dB.
Figure 8.20 BER vs iterations for LMBER, NewtonLMBER, BSMBER and MMSE algorithms; SUand MF as higher and lower steadystate limits, respectively; equal
power distribution at SNR = 30 dB.
Figure 8.21 BER vs SNR for AMBER family and MMSE algorithms; SUand MF as higher and lower steadystate limits, respectively; equal power distribution at SNR = 30 dB.
Figure 8.22 BER vs SNR for LMBER family and MMSE algorithms; SUand MF as higher and lower steadystate limits, respectively; equal power distribution at SNR = 30 dB.
Figure 8.23 BER vs SNR for LMBER, NewtonLMBER, BSMBER and MMSE algorithms; SUand MF as higher and lower steadystate limits, respectively; equal power distribution at SNR = 30 dB.
Simplified Robust Adaptive Detection and Beamforming for Wireless Communications
Ayman Elnashar
Emirates Integrated Telecommunications Company (EITC)
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Library of Congress Cataloging-in-Publication Data
Names: Elnashar, Ayman, author
Title: Simplified robust adaptive detection and beamforming for wireless communications / by Ayman Elnashar.
Description: First edition. | Hoboken, NJ : John Wiley & Sons, 2018. | Includes index. |
Identifiers: LCCN 2017060724 (print) | LCCN 2018007145 (ebook) | ISBN 9781118938232 (pdf) | ISBN 9781118938225 (epub) | ISBN 9781118938249 (cloth) |
Subjects: LCSH: Adaptive signal processing. | Beamforming. | Wireless communication systems.
This book is dedicated to the memory of my parents (God bless their souls). They gave me the strong foundation and unconditional love, which remains the source of motivation and is the guiding light of my life.
Also, this book is dedicated to my PhD supervisors, Prof. Said Elnoubi from Alexandria University and Prof. Hamdi Elmikati from Mansoura University. They have guided and encouraged me during my PhD thesis and inspired me to author this book.
To my dearest wife, your encouragement and patience has strengthened me always.
To my beloved children Noursin, Amira, Yousef, and Yasmina. You are the inspiration!
Finally, I acknowledge the contribution of Tamer Samir from mobily for chapter 8.
– Ayman Elnashar, PhD
About the Author
Ayman Elnashar, PhD, has 20+ years of experience in telecoms industry including 2G/3G/LTE/WiFi/IoT/5G. He was part of three major startup telecomoperators in MENA region (Orange/Egypt, Mobily/KSA, and du/UAE). Currently, he is Vice President and Head of Infrastructure Planning ICT and Cloud with the Emirates Integrated Telecommunications Co. “du”, UAE. He is the founder of the Terminal Innovation Lab and UAE 5G innovation Gate (U5GIG). Prior to this, he was Sr. Director – Wireless Networks, Terminals and IoT where he managed and directed the evolution, evaluation, and introduction of du wireless networks including LTE/LTEA, HSPA+, WiFi, NBIoT and currently working towards deploying 5G network in UAE. Prior to this, he was with Mobily, Saudi Arabia, fromJune 2005 to Jan 2008 as Head of Projects. He played key role in contributing to the success of the mobile broadband network of Mobily/KSA. FromMarch 2000 to June 2005, he was with Orange Egypt. He published 30+ papers in wireless communications arena in highly ranked journals and international conferences. He is the author of “Design, Deployment, and Performance of 4G LTE Networks: APractical Approach” published by Wiley & Sons, and “Practical Guide to LTEA, VoLTE and IoT: Paving the way towards 5G” to be published in May 2018. His research interests include practical performance analysis, planning and optimization of wireless networks (3G/4G/WiFi/IoT/5G), digital signal processing for wireless communications, multiuser detection, smart antennas, massive MIMO, and robust adaptive detection and beamforming.
About the Companion Website
This book is accompanied by a companion website:
www.wiley.com/go/elnashar49
The website include: Matlab scripts
1 Introduction
1.1 Motivation
This book presents alternative and simplified approaches for the robust adaptive detection and beamforming in wireless communications. The book adopts several systemmodels, including:
DS/CDMA, with and without antenna array
MIMOOFDM with antenna array
general smart antenna array model.
Recently developed detection and beamforming algorithms are presented and analyzed with an emphasis on robustness. In addition, simplified and efficient robust adaptive detection and beamforming techniques are developed and compared with existing techniques. The robust detectors and beamformers are implemented using wellknown algorithms including, but not limited to:
leastmeansquare
recursive leastsquares (RLS)
inverse QR decomposition RLS (IQRDRLS)
fast recursive steepest descent (RSD)
blockShanno constant modulus (BSCMA)
conjugate gradient (CG)
steepest descent (SD).
The robust detection and beamforming methods are derived fromexisting detectors/beamformers including, but not limited to: the robust minimumoutput energy (MOE) detector
partition linear interference canceller (PLIC) detector
linearly constrained constant modulus (CM) algorithm(LCCMA), linearly constrained minimumvariance (LCMV) beamforming with single constraint, minimumvariance distortionless response (MVDR) beamformer with multiple constraint block Shanno constant modulus algorithm(BSCMA) based detector/beamformer
adaptive minimumbit error rate (BER) based detectors. The adopted cost functions include the mean square error (MSE), BER, CM, MVand the
signaltonoise or signaltointerferenceplusnoise ratios (SINR/SNR). The presented robust adaptive techniques include:
quadratic inequality constraint (QIC)
diagonal loading techniques
single and multiple worstcase (WC) constraint(s)
ellipsoidal constraint
joint constraints
Detailed performance analysis in terms of MSE, SINR, BER, computational complexity, and robustness are conducted for all the presented detectors and beamformers. Practical examples based on the above systemmodels are provided to exemplify the developed detectors and beamforming algorithms. Moreover, the developed techniques are implemented using Matlab and the relevant Matlab scripts are provided to allow the readers to develop and analyze the presented algorithms. The developed algorithms will be presented in the context of DS/CDMA, MIMOOFDM, and smart antenna arrays, but they can be easily extended to other domains and other applications. Figure 1.1 provides a highlevel description of the book.
Recently, robust adaptive detection/beamforming has become a hot topic. Researchers seek to provide robustness against uncertainty in the direction of arrival (DOA) or the signature waveform, accuracy errors, calibration errors, small sample sizes, mutual coupling in antenna arrays, and so on. The major concern with the robust algorithms is the compromises involving robustness, complexity, and optimality. This book is aims to efficiently address this concern by presenting alternative and simplified approaches for robust adaptive detection and beamforming in wireless communications systems. The presented algorithms have low computational complexity while offering optimal or closetooptimal performance and can be practically implemented. Wireless communication applications using DS/CDMA, MIMO OFDM, and smart antenna systems are presented to demonstrate their robustness and to
Figure 1.1 Summary of the book.
compare their complexity with established techniques and optimal detectors/beamformers. The book presents and addresses current hot topics in adaptive signal processing: robustness and simplified adaptive implementation. It presents simplified approaches that add robustness to adaptive signal processing algorithms, with less computational complexity, while maintaining optimality. In addition, the presented algorithms are illustrated with practical examples and simulation results for major wireless communications systems, including DS/CDMA, MIMOOFDM, and smart antenna systems. Moreover, Matlab scripts are provided for further analysis and development. The reader can easily extend the techniques and approaches in this book to other areas and to different applications.
With the growth of mobile communication subscribers, the introduction of high datarate services, and the overall increase in user traffic, new ways are needed to increase the capacity of wireless networks. Smart antennae, MIMO and beamforming are some of the most promising technologies now being exploited to enhance the capacity of the cellular system. In wireless networks, the traditional omni and directional antennae of a basestation cause higher interference than necessary. Additionally, they are wasteful, as most transmitted signals will not be received by the target user. Adaptive antennas are a multidiscipline technology area that has exhibited growth steadily over the last four decades, primarily due to the impressive advances in the field of digital processing. Exploiting the spatial dimension using adaptive antennae promises impressive increases in systemperformance in terms of capacity, coverage, and signal quality. This will ultimately lead to increased spectral efficiency and extended coverage, especially for higherfrequency bands, such as millimetre waves (mmWave), that will be adopted for 5G evolution.
1.2 Book Overview
In Chapter 1, the mathematical models of DS/CDMAand MIMOOFDM systems are presented. These formthe foundation for the robust adaptive detection and beamforming algorithms that will be presented and/or developed in this book. DS/CDMAand OFDM are used in 3GPP 3G and 4G systems respectively. The 5G systemunder development by 3GPP will use evolved versions of MIMOOFDM. The algorithms presented in this book may fit any of these systems and may also be extended to other systems. The focus of the 3G and 4G evolutions were on mobile broadband, as a result of widespread smartphone adoption. The internet of things (IoT) evolution will lead to billions of devices being connected to the internet and this has directed the 3GPP and mobile communications industry towards narrowband technologies. 3GPP has modified the LTE systemto meet the IoT requirements by introducing NBIoT. Other proprietary technologies, such as lowpower widearea networks, have used narrowband or ultranarrowband technologies such as chirp spread spectrum. The focus of this book is not on certain technologies and readers will need to expend some effort in order to apply the detection and beamforming algorithms outlined here to specific systems. The focus of the book is the development and comparative analysis of robust adaptive detection and beamforming algorithms based on simplified systemmodels. All the results in the book are simulated using Matlab and the developed scripts are provided along with the book. The reader may need to slightly modify the scripts depending on the Matlab version. In addition, some algorithms developed by other authors are provided as part of the software package with this book for the purpose of comparative analysis.
In Chapter 3, we will provide a survey of adaptive detection algorithms based on the DS/CDMAmodel. However, the adaptive techniques that are summarized in this survey can be easily extended to MIMOOFDM and smart antenna arrays. The DS/CDMAmodel is the most complicated systemmodel, because of its need for multiuser interference cancellation and since the channel is frequency selective, as explained in Chapter 2. Despite the various advantages of the DS/CDMAsystem, it is interference limited due to multiuser interference and it cannot be easily extended to ultrabroadband systems. Aconventional DS/CDMAreceiver treats each user separately as a signal, with other users considered as noise or multiple access interference (MAI). Amajor drawback of such conventional DS/CDMAsystems is the near–far problem: degradation in performance due to the sensitivity to the power of the desired user against the power of the interference. Areliable demodulation is impossible unless tight power control algorithms are exercised. The near–far problemcan significantly reduce the capacity. Multiuser detection (MUD) algorithms can give dramatically higher capacity than conventional singleuser detection techniques. MUD considers signals fromall users, which leads to joint detection. MUD reduces interference and hence leads to a capacity increase, alleviating the near–far problem. Power control algorithms can be used but are not necessary.
Linear receiver design by minimization of some inverse filtering criterion is explained in Chapter 4. Appropriate constraints are used to avoid the trivial allzero solution. Awell known cost function for the constrained optimization problemis the variance or the power of the output signal. An MOE detector for multiuser detection is developed, based on the
constrained optimization approach. In an additive white Gaussian environment with no multipath, this detector provides a blind solution with MMSE performance. In Chapter 4, linearly constrained IQRDRLS algorithms with multiple constraints are developed and implemented for MUD in DS/CDMAsystems. As explained above, the same algorithms can be extended to MVDR beamforming algorithms. Two approaches are considered, the first with a constant constrained vector and the other with an optimized constrained vector. Three IQRD based detectors are developed as follows:
a direct formMOE detector based on the IQRD update method with fixed constraints
a MOE detector in the PLIC structure based also on the IQRDRLS algorithm
an optimal MOE algorithmbuilt using the IQRD update method and a subspace tracking algorithmfor tracking the channel vector.
The constrained vector (estimated channel vector) is obtained using the max/min approach with IQRDRLS based subspace tracking algorithms that are analyzed and tested for channel vector tracking.
The recently developed subspace tracking algorithms are tested and analyzed for channel estimation in Chapter 4. These are the fast orthogonal projection approximation subspace tracking (OPAST) algorithmand the normalized orthogonal Oja (NOOja) algorithm. In addition, a fast subspace tracking algorithmbased on the Lagrange multiplier methodology and the IQRD algorithmwill be developed and adopted for channel vector estimation and tracking. Moreover, a new strategy for combining the max/min channel estimation technique with the robust quadratic constraint technique is proposed anchored in the direct formalgorithm. Specifically, a robust MOE detector is developed, based on the max/min approach and QIC on the weight vector normto overcome noise enhancement at low SNR. Adirect formsolution is introduced for the quadratically constraint detector with a variable loading (VL) technique employed to satisfy the QIC. Thus, the IQRD algorithmacts as a core to the proposed receivers, which facilitate realtime implementation through systolic implementation. However, the same algorithms can be easily implemented using fast and robust RLSbased algorithms.
Arobust lowcomplexity blind detector is presented in Chapter 5. This is based on a recursive steepest descent (RSD) adaptive algorithmrather than the RLS algorithmand a QIC on the weight vector norm. The QIC is employed to manage the residual signal mismatch and other randomperturbations errors. In addition, the QIC will make the noise constituent in the output SINR constant and hence overcome noise enhancement at low SNR. Quadratic constraints have been used in adaptive beamforming for a variety of purposes, such as improving robustness against mismatch and modeling errors, controlling mainlobe response, and enhancing interference cancellation capability. The quadratic constraint will be analyzed along with beamforming algorithms in Chapter 7.
Analogous to the recursive conjugate gradient (RCG) algorithm, a fast RSD algorithmis developed in Chapter 5. Alowcomputational complexity recursive update equation for the gradient vector is derived. Furthermore, a variable stepsize approach is introduced for the
stepsize update of the RSD algorithmbased on an optimumstepsize calculation. The RSD algorithmis exploited to update the adaptive weight vector of the PLIC structure to suppress MAI. The same technique will be extended to MVDR beamforming in Chapter 7. Fromthis similarity, the reader can easily extend the algorithms in this book to other systems and even beyond the realmof wireless communications. Fromthis it can be seen that we have simplified the deployment of the robust techniques, such as quadratic constraints, uncertainty constraints, worstcase constraint optimization, and constrained optimization.
The drawbacks of diagonal loading techniques are tackled in Chapter 5. An alternative way of robust adaptive detection based on the RSD adaptive algorithmis presented. This involves an accurate technique for precisely computing the diagonal loading level without approximation or eigendecomposition. We combined the QIC with the RSD algorithmto produce a robust recursive implementation with O(N2) complexity. Anew optimal VLtechnique is developed and integrated into the RSD adaptive algorithm. In addition, the diagonal loading termis optimally computed, with O(N) complexity, using a simple quadratic equation. Geometrical interpretations of the scaled projection (SP) and VLtechniques, along with RLS and RSD algorithms, are illustrated and analyzed. The performance of the robust detectors is compared with traditional detectors and the former are shown to be more accurate and more robust against signal mismatch and randomperturbations. Finally, the presented approach can be reformulated to handle an uncertainty constraint – imposed on the signature waveformin MUD, or on a steering vector in beamforming – such as the ellipsoidal constraint. It can also be exploited with any of the robust approaches to produce a simple recursive implementation.
In Chapter 6, the quadratic inequality constraint is imposed on the weight vector normof the LCCMAand BSCMAalgorithms in order to enhance their performance. The weight norm constraint will control the gradient vector norm, meaning that there is no need to check the gradient vector normincrease in BSCMA. Additionally, the iteration inside the block can continue without affecting algorithmstability due to the weight vector normconstraint. We will investigate the effect of adding a quadratic inequality constraint on the LCCMAand BSCMA algorithms. The proposed VLtechnique in Chapter 5 is exploited to estimate the optimum diagonal loading value. The LCCMAand BSCMAalgorithms are used to update the adaptive vector of the PLIC structure. The PLIC structure with multiple constraints is employed to identify the MAI and hence help in avoiding interference capture. Moreover, the different forms of BSCMAalgorithms – the blockconjugate gradient CMAalgorithm(BCGCMA) and block gradient descent constant modulus algorithm(BGDCMA) – are investigated as well. The resistance of BSCMAbased algorithms against the near–far effect is discussed and evaluated.
In Chapter 7, we will present four approaches for robust adaptive beamforming as follows:
Improved recursive realization for robust LCMV beamforming We first develop an improved recursive realization for robust LCMVbeamforming. This includes an ellipsoidal uncertainty constraint on the steering vector. The robust recursive implementation presented here is based on a combination of the ellipsoidal constraint formulation and the variable diagonal loading technique demonstrated in Chapter 5. As a
consequence, an accurate technique for computing the diagonal loading level without eigendecomposition or SOCP is developed. The geometrical interpretation of the diagonal loading technique is demonstrated and compared with eigendecomposition approach. Note that this approach adopts a spherical constraint on the steering vector to optimize the beamformer output power. Unfortunately, the adaptive beamformer developed here is apt to noise enhancement at low SNR and an additional constraint is required to bolster the ellipsoidal constraint.
Joint constraint approach for a joint robustness beamformer The second approach is the development of a joint constraint approach for a joint robustness beamformer. Ajoint constraint approach is presented for joint robustness against steering vector mismatch and unstationarity of interferers. An alternative approach involves imposing an ellipsoidal uncertainty constraint and a quadratic constraint on the steering vector and the beamformer weights, respectively. We introduce a new simple approach to get the corresponding diagonal loading value. The quadratic constraint is invoked as a cooperative constraint to overcome noise enhancement at low SNR. The performance of the robust adaptive schemes developed and other robust approaches are demonstrated in scenarios with steering vector mismatch and several moving jammers.
Beamformer with a single WC constraint In the third approach, a robust MVDR beamformer with a single WC constraint is implemented using an iterative gradient minimization algorithm. This involves a simple technique to estimate the Lagrange multiplier instead of a Newtonlike algorithm. This algorithmexhibits several merits, including simplicity, low computational load, and no need for either samplematrix inversion or eigendecomposition. Ageometric interpretation of the robust MVDR beamformer is demonstrated to supplement the theoretical analysis.
LCMV beamformer with MBWC constraints In the last approach, a robust LCMV beamformer with multiplebeamWC (MBWC) constraints is developed using a novel multipleWC constraints formulation. The optimization problementails solving a set of nonlinear equations. As a consequence, a Newtonlike method is mandatory to solve the systemof nonlinear equations, which yields a vector of Lagrange multipliers. The Lagrange method is used to give the solution.
The traditional MMSE detector is the most popular technique for beamforming. An adaptive implementation of the MMSE can be achieved by minimizing the MSE between the desired output and the actual array output. The LCMVand MVDR beamformers in Chapter 7 are different forms of MMSE detectors. For a practical communication system, it is the BER or block BER, not the MSE performance, that really matter. Ideally, the systemdesign should be based directly on minimizing the BER rather than the MSE. For application in singleuser channel equalization, multiuser detection, and beamforming, it has been shown that the MMSE solution can, in certain situations, be distinctly inferior to the minimumBER (MBER) solution. However, the BER cost function is not a linear function of the detector or the beamformer, making it difficult to minimize. Several adaptive MBER beamformer/detectors implementations are developed in the literature.
It must be stated here that the cost function of the MMSE criterion has a circular shape. This means that we have one global minimum. Hence, convergence can be easily achieved. In contrast, the cost function of the BER is highly nonlinear. This means that during minimization steps we may converge to a local minimum. The MMSE and MBER solutions lead to very different detector weight vectors. Clearly, the MBER design is more intelligent in utilizing the detector's resources. However, special attention is mandatory with the minimization algorithm in order to avoid convergence to a local minimum, and hence the algorithmdiverging rather than converging.
Beamforming is a key technology in smart antenna systems, and can increase capacity and coverage and mitigate multipath propagation in mobile radio communication systems. The most popular criterion for linear beamforming is MMSE. However, the MSE cost function is not optimal in terms of the bit error probability performance of the system. In Chapter 8, a class of adaptive beamforming algorithms using direct minimization of the BER cost function is presented. Unfortunately, the popular least minimumBER stochastic beamforming algorithm suffers fromlow convergence speeds. Gradient Newton algorithms are presented as an alternative. These speed up the convergence rate and enhance performance but only at the expense of complexity. In Chapter 8, a block processing objective function for the MBER is formulated, and a nonlinear optimization strategy that produces the socalled ‘blockShanno MBER’ is developed. Acomplete consideration of the complexity calculations of the proposed algorithmis given. Simulation scenarios are carried out in a multipath Rayleighfading DS CDMAsystemto explore the performance of the proposed algorithm. Simulation results show that the proposed algorithmoffers good performance in terms of convergence speed, steady state performance, and even systemcapacity, compared to other MBER and MSEbased algorithms.
Finally, we will extend the adaptive filtering algorithms using the concept of spatial multiuser detection in a MIMOOFDM systemmodel rather than beamforming in a DSCDMAmodel. As stated above, a fundamental goal in any digital communications systemis to directly minimize the BER. Wiener solution based algorithms indirectly minimize the BER by optimizing other cost functions (SNR, SINR, or MSE), which may result in suboptimal BER performance.
2
Wireless System Models
2.1 Introduction
In this chapter, mathematical models of DS/CDMAand orthogonal frequency division multiplexing (OFDM) systems are presented. These formthe foundation of the robust adaptive detection algorithms that will be presented in this book. DS/CDMAand OFDM are used in 3GPP 3G and 4G systems respectively. The 5G systems under development by 3GPP will use evolved versions of OFDM. The algorithms presented in this book suit any of these systems and may also be extended to other systems.
The focuses of 3G and 4G were on mobile broadband thanks to smartphone adoption. The internet of things (IoT), with billions of devices expected to be connected to the internet, has directed the 3GPP and industry towards narrowband technologies. 3GPP has modified the LTE systemto meet IoT requirements by introducing NBIoT. Other proprietary technologies in unlicensed bands, known as lowpower wide area (LPWA) networks, have used narrowband or ultranarrowband approaches such as SigFox. Other LPWAnetworks have adopted a wideband CDMAapproach based on chirp spread spectrum(CSS). One example is LoRa. The focus of this book is not on particular commercial technologies, and readers will require some effort in order to match the detection and beamforming algorithms developed in this book to specific practical systems. The focus of the book is on the development and comparative analysis of robust adaptive detection and beamforming algorithms based on simplified system models. All results in the book are simulated using Matlab and the scripts used are provided along with the book. The reader may need to slightly modify the scripts depending on the Matlab version. Also, some algorithms developed by other authors are provided as part of the software package with this book for the purpose of comparative analysis.
Due to the complex nature of CDMAsystems, there have been many different formulations of the DS/CDMAmodel. In this book, we consider a general DSCDMAsystemmodel, which account for user asynchronism, multipath propagation, and frequencyselective fading propagation channels. The link fromthe base station to the mobile station is referred to as the “downlink” and is typically characterized by synchronous data transmission. More challenging for demodulation is the “uplink” frommobile to base station, where different user transmissions are typically asynchronous and of widely disparate power levels. Reliable modulation might also require mitigation of multipath interference, especially in wideband CDMA(WCDMA) schemes where multipath effects can be significant. The robust adaptive receivers/detectors presented and/or developed in this book are suitable for both uplink and downlink scenarios. Important performance measures are formulated for assessment and comparative analysis. DS/CDMAwas adopted in 3G/HSPA+ mobile communication systems and IEEE 802.15.4. The main reasons for using CDMAtechniques are as follows:
resistance to unintended or intended jamming/interference sharing of a single channel among multiple users reduced signal/backgroundnoise level hampering interception determination of relative timing between transmitter and receiver robustness against multipath propagation and frequency selective channels.
Second and thirdgeneration mobile systems are based on either TDMAor CDMA technologies. Although these technologies can theoretically be extended to nextgeneration wireless broadband systems, practical implementation issues and complexities limit their adoption. On the other hand, OFDM offers an easier solution and simple implementation. However, OFDM is not without its issues.
Multipath signal propagation makes the channel response time dispersive; the amount of signal dispersion depends on the environment of operation. For example, the channel dispersion is about 5 µs in typical urban areas and 15–20 µs in rural and hilly terrain. The factor that affects the receiver is the number of resolvable channel taps over the channel dispersion interval. In a TDMAsystem, it is the ratio of the channel dispersion to signal symbol time. However, in a CDMAsystem, it is the number of channel taps with strong energy at chiptime resolution over the channel dispersion period. The channel time dispersion is viewed as frequency selective or nonselective in the frequency domain. Afrequency nonselective channel means the signal, over its entire bandwidth, will have the same effect as due to the multipath channel. This is also called a flat fading channel. In the time domain, the channel is not dispersive relative to its symbol time, and hence there is no intersymbol interference (ISI). In the frequencyselective channel, the signal will have independent effects over its bandwidth due to the channel, and it is time dispersive relative to its symbol time.
In narrowband TDMAsystems, such as GSM, multipath propagation makes the channel frequency nonselective or less selective, making the receiver less complex. Extending TDMAtechniques to broadband systems makes the receiver complexity unmanageable, as the channel becomes highly frequency selective. More specifically, GSM is a 200 kHz channel TDMAsystem, of 270.833 kHz symbol rate with either binary GMSK or 8PSK modulations. The baseband signal uses partial response signalling, which spreads the symbol to three symbol periods. For a typical urban case with about 5 µs channel dispersion, the received signal can have a signal dispersion of about 5 symbol periods, including its partial response signalling. Therefore, a typical GSM receiver requires a 16state MLSE (maximum likelihood sequence estimation) equalizer for Gaussian minimumshift keying (GMSK) modulation and an 8 or 64state DFSE (decision feedback sequence equalizer) for an 8 PSK EDGE signal. Suppose we want to scale up this technique to a wideband or broadband systemby factor of 100; that is, a 20 MHz channel bandwidth like LTE systemwith 27.0833 MHz symbol rate. For the same amount of channel dispersion, the received symbol will be spread over 200 symbol periods, which means a very frequencyselective channel. The receiver with an equalizer for 200 channel taps will be impractical to implement due to the very high complexity.
Similarly, WCDMAcan also be extended to broadband systems, but its complexity increases, as it requires a larger number of rake receiver fingers. Complexity of a rake receiver, and often its gain, are based on the number of rake fingers the receiver can process. Atypical WCDMA rake receiver requires about 5–8 rake fingers for a typical urban channel with channel spread of 5 µs. More advanced receivers, such as generalized rake receivers (Grake), require even more fingers, placed around the desired signal, and often called “interference fingers”. Extending WCDMAto a 20 MHz broadband systemwill require higher chip rates, meaning that it can resolve channel taps with finer resolution. This results in more fingers for the rake receiver with strong signal energy. Therefore, extension of WCDMA/HSPA+ systems to a 20 MHz broadband systemrequires expansion, by a similar factor, of the number of fingers in the rake receiver, and thus an increase in its detection complexity especially at the mobile side. In addition, the receiver design will be further complicated if we need to add multipleinput, multipleoutput (MIMO) on top of such complicated receivers; the gain of MIMO will be minimized due to the frequency selective nature and the high number of detector taps. This is the main reason behind the delay in deploying MIMO with 3G/HSPA+ mobile systems while MIMO 2 × 2 has been used with 4G/LTE OFDM based systems fromday one and currently MIMO 4 × 4 and MIMO 8 × 8 systems are in development. Moreover, 5G evolution will take MIMO and beamforming to the next level by introducing evolved node B (eNB), with 64 antenna elements. The 3GPP has introduced other ways of extending HSPA+ systems to broadband, based on multicarrier HSPA. Most commercial 3G networks adopt DCHSPA+ and have evolved to DC/2C/4CHSPA+ by combining 2, 3 and 4 carriers. The three/four carriers provide 63 Mbps/84 Mbps in the DL, respectively. However, it is still challenging to go beyond the dual carrier approach due to the network optimization complexity with each carrier considered as one cell. In addition, DC is mainly deployed in the downlink and its adoption in the uplink is still limited. More particularly, the multicarrier approach in the ULis still not widely deployed. With 3CHSPA+ modems, dual carriers in the ULwill be supported. The complexity in deploying multicarrier HSPA+ systems has been a motivation to develop and deploy the LTE system.
OFDM has become a most favored technique for broadband wireless systems due to the susceptibility to signal spread under multipath conditions. OFDM can also be viewed as a multicarrier narrowband system, where the whole systembandwidth is split into multiple smaller subcarriers with simultaneous transmission. Simultaneous data transmission and reception over these subcarriers are handled almost independently. Each subcarrier is usually narrow enough that the multipath channel response is flat over the individual subcarrier frequency range; that is, there is a frequency nonselective fading channel. Therefore, the OFDM symbol time is much larger than the typical channel dispersion. Hence OFDM is inherently susceptible to channel dispersion due to multipath propagation.
The fading channels are illustrated in Figure 2.1. The fading can be mainly classified into two different types: largescale fading and smallscale fading. Largescale fading occurs as the mobile moves over a large distance, say of the order of the cell size [52]. It is caused by the path loss of the signal as a function of distance, and shadowing by large objects such as buildings, intervening terrain, and vegetation. Shadowing is a slowfading process,
characterized by variation of median path loss between the transmitter and receiver in fixed locations. In other words, largescale fading is characterized by average path loss and shadowing. On the other hand, smallscale fading involves the rapid variation of signal levels due to the constructive and destructive interference of multiple signal paths (multipaths) when the mobile station moves over short distances. Depending on the relative extent of a multipath, frequency selectivity of a channel is characterized (say, as frequencyselective or frequency flat) for smallscaling fading. Meanwhile, depending on the time variation in a channel due to mobile speed (characterized by the Doppler spread), shorttermfading can be classified as either fast fading or slow fading. Largescale fading and timevariance fading are addressed in the link budget of the system. The path loss and shadowing identify the cell radius in the mobile system. Multipath fading is addressed in the receiver design and the multiple access technique deployed. The link budget usually assumes slowfading channels. For fastfading channels, such as with users inside a highspeed train, a fastfading compensation factor is added to the link budget [52].
2.1
The relationship between largescale fading and smallscale fading is illustrated in Figure 2.2 [52]. Largescale fading is manifested by the mean path loss, which decreases with distance, and shadowing, which varies along the mean path loss. The received signal strength may be different at two different places, even at the same distance froma transmitter, due to shadowing caused by obstacles on the path. Furthermore, the scattering components incur smallscale fading, which ultimately gives a shorttermvariation of the signal, which has already experienced shadowing. Adetailed practical link budget for an LTE systemis provided in the literature [53]. This illustrates how all of these fading factors are addressed in the link budget.
Figure
Classification of fading channels.
Figure 2.2 The relation between largescale and smallscale fading. In traditional systems (TDMA or CDMAbased systems), symbol detection is on the samples at either the symbol or chip rate, and the carriertointerference level only matters at the sampling points. However, OFDM symbol detection requires that the entire symbol duration be free of interference fromprevious symbols, thus preventing ISI. Even though the OFDM symbol duration is much larger than the channel dispersion, even a small amount of channel dispersion causes some spilling of each OFDM symbol to the next symbol, thus causing some ISI. However, this ISI spillover is limited to only the initial part of the neighboring symbol. Hence this ISI spillover at the beginning of each symbol can easily be tackled by adding a cyclic prefix (CP) to each transmit symbol. This prefix extends each symbol by duplicating a portion of the signal at the symbol ends. The prefix is removed at the receiver. The amount of symbol extension – that is, the length of the CPs – is a systemdesign parameter, and is based on the expected signal dispersion in the environment of system operation. For example, the LTE systemuses an OFDM symbol of 66 µs plus 5 µs of CP. This means it is susceptible to a maximumsignal dispersion of 5 µs due to multipath channel propagation.
The LTE FDD frame structure is shown in Figure 2.3 [53] for normal CP. Each LTE FDD radio frame is long and consists of 20 slots of length , numbered from0 to 19. The number of ULSCFDMAsymbols in a slot depends on the CP length configured by the higher layer parameter (Table 2.1). The number of DLOFDM symbols in a slot depends on the CP length and the subcarrier spacing configured (Table 2.2). The CP length NCP,l that is to be used is provided in Table 2.2. Note that different OFDM symbols within a slot in some cases have different CP lengths.
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middle, while the other half must be squared and straight and set with an empty space below, in order that it may hold as does an arch, the wall on the external face appearing worked with vertical joints. [116] Do not let the stones of the said frieze rest on the architrave, but let a finger’s breadth be between them; in this way, making an arch, the frieze comes to support itself and does not burden the architrave. Afterwards make on the inside, for filling up the said frieze, a flat arch of bricks as high as the frieze, that stretches from die to die above the columns. Then make a piece of cornice as wide as the die[117] above the columns, which has the joints in front like those of the frieze, and within let the said cornice be keyed like the blocks of the frieze, care being taken to make the cornice, as the frieze, in three pieces, of which the two at the sides hold from within the middle piece of the cornice above the die of the frieze,[118] and mind that the middle piece of the cornice, C, C, slips down into the sinkings so as to span the void, and unites the two pieces at the sides so as to lock them in the form of an arch. In this fashion everyone can see that the frieze sustains itself, as does the cornice, which rests almost entirely on the arch of bricks.[119] Thus one thing helping another, it comes about that the architrave does not sustain any but its own weight, nor is there danger of its ever being broken by too heavy a load. Because experience shows this method to be the most sure, I have wished to make particular mention of it, for the convenience and benefit of all; especially as I know that when the frieze and the cornice were put above the architrave as was the practice of the ancients, the latter broke in course of time, possibly on account of an earthquake or other accident, the arch of discharge which was introduced above the cornice not being sufficient to preserve it. But throwing the arches above the cornices made in this form, and linking them together with iron, as usual,[120] secures the whole from every danger and makes the building endure eternally.
Returning to the matter in hand, let us explain then that this fashion of work may be used by itself alone, or can be employed in the second floor from the ground level, above the Rustic Order, or it can be put higher up above another variety of Order such as Ionic, Corinthian or Composite, in the manner shown by the ancients in the Colosseum in Rome, in which arrangement they used skill and judgement. The Romans, having triumphed not only over the Greeks but over the whole world, put the Composite Order at the top, of
which Order the Tuscans have composed many varieties. They placed it above all, as superior in force, grace, and beauty, and as more striking than the others, to be a crown to the building; for to be adorned with beautiful members gives to the work an honourable completion and leaves nothing more to be desired.
§ 23. The proportions and parts of the Doric Order.
To return to the Doric Order, I may state that the column is made seven heads in height. Its pedestal must be a little less than a square and a half in height and a square in width,[121] then above are placed its mouldings and beneath its base with torus and two fillets, as Vitruvius directs. The base and capital are of equal height, reckoning the capital from the astragal upwards. The cornice with the frieze and architrave attached projects over every column, with those grooved features, usually called triglyphs, which have square spaces[122] interposed between the projections, within which are the skulls of oxen, or trophies, or masks, or shields, or other fancies. The architrave, jutting out, binds these projections with a fillet, and under the fillet are little strips square in section, at the foot of each of which are six drops, called by the ancients ‘guttae’ (goccie). If the column in the Doric order is to be seen fluted, there must be twenty hollow facets instead of flutes,[123] and nothing between the flutes but the sharp arris. Of this sort of work there is an example in Rome at the Forum Boarium which is most rich;[124] and of another sort are the mouldings and other members in the theatre of Marcellus, where to-day is the Piazza Montanara, in which work there are no bases (to the Doric columns) and those bases which are visible are Corinthian. It is thought that the ancients did not make bases, but instead placed there a pedestal of the same size as the base would have been. This is to be met with in Rome by the prison of the Tullianum where also are capitals richer in members than others which appear in the Doric Order.[125] Of this same order Antonio da San Gallo has made the inner court of the Casa Farnese in the Campo di Fiore at Rome, which is highly decorated and beautiful; thus one sees continually ancient and modern temples and palaces in this style, which for stability and assemblage of the stones have held together better and lasted longer than all other edifices.
F . 6. Drawing by Giuliano da San Gallo of a portion of the Basilica Aemilia in the Roman Forum, that survived to the time of Vasari.
§ 24. The Ionic Order.
The Ionic Order, more slender than the Doric, was made by the ancients in imitation of persons who stand mid-way between the fragile and the robust; a proof of this is its adoption in works dedicated to Apollo, Diana, and Bacchus, and sometimes to Venus. The pedestal which sustains the column is one and a half squares high and one wide, and the mouldings, above and below, are in accordance with this Order. Its column measures in height eight times the head, and its base is double with two tori, as described by Vitruvius in the third chapter of his third book. Its capital with its volutes or scrolls or spirals, as anyone may call them, should be well turned, as one sees in the theatre of Marcellus in Rome, above the Doric Order; and its cornice adorned with modillions and with
F . 7. Roman Doric cap, with stucco finish, at S. Nicola in Carcere, Rome.
dentils, and its frieze slightly convex (pulvinated). Should it be desired to flute the columns, there must be twenty-four flutes, but divided in such a manner as to leave between each two of them a flat piece that measures the fourth part of the flute. This order has in itself the most beautiful lightness and grace and is consequently adopted by modern architects.
§ 25. The Corinthian Order.
The Corinthian style was invariably a favourite among the Romans, who delighted in it so greatly that they chose this Order for their most elaborate and most prized buildings to remain as a memorial of themselves; as is seen in the Temple at Tivoli above the Teverone, in the remains of the temple of Peace,[126] in the arch of Pola, and in that of the harbour of Ancona; but much more beautiful is the Pantheon, that is the Ritonda of Rome. This Order is the richest and most decorated of all the Orders spoken of above. The pedestal that supports the column is measured in the following way; a square and two thirds wide (high)[127] and the mouldings above and below in proportion, according to Vitruvius[128]: the height of the column nine heads with base and capital, which last shall be in height the diameter of the column at the foot, and its base half of the said thickness. This base the ancients used to carve in various ways. Let the ornament of the capital be fashioned with its tendrils and its leaves, as Vitruvius directs in the fourth book, where he records that this capital has been taken from the tomb of a Corinthian girl. Then follow its proper architrave, frieze and cornice measured as he describes, all carved with the modillions and ovolos and other sorts
of carving under the drip. The friezes of this Order may be carved with leafage, or again they may be plain, or adorned with letters of bronze let into marble, as those on the portico of the Ritonda. There are twenty-six flutes in the Corinthian columns, although sometimes also there are fewer, and the fourth part of the width of each flute remains flat between every two, as is evident in many ancient works and in modern works copied from the ancients.
§ 26. The Composite Order.
The Composite Order, although Vitruvius has not made mention of it—having taken account of none others than the Doric, Ionic, Corinthian, and Tuscan, and holding those artists lawless, who, taking from all four Orders, constructed out of them bodies that represented to him monsters rather than men—the Composite Order has nevertheless been much used by the Romans and in imitation of them by the moderns. I shall therefore proceed, to the end that all may have notice of it, to explain and give the proportions of buildings in this Order also, for I am convinced of this, that if the Greeks and Romans created these first four Orders and reduced them to a general rule and measure, there may have been those who have done the same for the Composite Order, forming of it things much more graceful than ever did the ancients.
As an example of the truth of this I quote the works of Michelagnolo Buonarroti in the Sacristy and Library of San Lorenzo in Florence, where the doors, niches, bases, columns, capitals, mouldings, consoles and indeed all the details, have received from him something of the new and of the Composite Order, and nevertheless are wonderful, not to say beautiful. The same merit in even greater measure is exhibited by the said Michelagnolo in the second story of the Court of the Casa Farnese[129] and again in the cornice which supports on the exterior the roof of that palace. He who wishes to see in this manner of work the proof of this man’s excellence—of truly celestial origin—in art and design of various kinds, let him consider that which he has accomplished in the fabric of St. Peter’s in compacting together the body of that edifice and in making so many sorts of various and novel ornaments, such beautiful profiles of mouldings, so many different niches and
numerous other things, all invented by him and treated differently from the custom of the ancients. Therefore no one can deny that this new Composite Order, which through Michelagnolo has attained to such perfection, may be worthily compared with the others. In truth, the worth and capacity of this truly excellent sculptor, painter, and architect have worked miracles wherever he has put forth his hand. Besides all the other things that are clear as daylight, he has rectified sites which were out of the straight and reduced to perfection many buildings and other objects of the worst form, covering with lovely and fanciful decoration the defects of nature and art.[130] In our days certain vulgar architects, not considering these things judiciously and not imitating them, have worked presumptuously and without design almost as if by chance, without observing ornament, art, or any order. All their things are monstrous and worse than the German.
Returning now to our subject, it has become usual for this manner of work to be called by some the ‘Composite,’ by others the ‘Latin,’ and by others again the ‘Italic’ Order. The measure of the height of this column must be ten heads, the base the half of the diameter of the column, measured in the same way as the Corinthian column, as we see in the arch of Titus Vespasianus in Rome. And he who wishes to make flutes in this column can do so, following the plan of the Ionian or Corinthian column—or in any way that pleases him who adopts this style of architecture, which is a mixture of all the Orders. The capitals may be made like those of the Corinthian except that the echinus moulding of the capital must be larger and the volutes or tendrils somewhat larger, as we see in the above mentioned arch. The architrave must be three quarters of the thickness of the column and the rest of the frieze supplied with modillions, and the cornice equal to the architrave, because the projection gives the cornice an increase of size, as one sees in the uppermost story of the Roman Colosseum; and in the said modillions grooves can be cut after the manner of triglyphs, and there can be other carving according to the taste of the architect; the pedestal on which the column rests must be two squares high, with the mouldings just as he pleases.
§ 27. Of Terminal Figures.
The ancients were accustomed to use for doors or sepulchres or other kinds of enrichment, various sorts of terminal figures instead of columns, here a figure which has a basket on the head for capital, there a figure down to the waist, the rest, towards the base, a cone or a tree trunk; in the same way they made virgins, chubby infants, satyrs, and other sorts of monsters or grotesque objects, just as it suited them, and according as the ideas occurred to them so the works were put into operation.
§ 28. German Work (the Gothic Style).
We come at last to another sort of work called German, which both in ornament and in proportion is very different from the ancient and the modern. Nor is it adopted now by the best architects but is avoided by them as monstrous and barbarous, and lacking everything that can be called order. Nay it should rather be called confusion and disorder. In their buildings, which are so numerous that they sickened the world, doorways are ornamented with columns which are slender and twisted like a screw, and cannot have the strength to sustain a weight, however light it may be. Also on all the façades, and wherever else there is enrichment, they built a malediction of little niches one above the other, with no end of pinnacles and points and leaves, so that, not to speak of the whole erection seeming insecure, it appears impossible that the parts should not topple over at any moment. Indeed they have more the appearance of being made of paper than of stone or marble. In these works they made endless projections and breaks and corbellings and flourishes that throw their works all out of proportion; and often, with one thing being put above another, they reach such a height that the top of a door touches the roof. This manner was the invention of the Goths, for, after they had ruined the ancient buildings, and killed the architects in the wars, those who were left constructed the buildings in this style.[131] They turned the arches with pointed segments, and filled all Italy with these abominations of buildings, so in order not to have any more of them their style has been totally abandoned.
May God protect every country from such ideas and style of buildings! They are such deformities in comparison with the beauty
of our buildings that they are not worthy that I should talk more about them, and therefore let us pass on to speak of the vaults.
CHAPTER IV.
On forming Vaults in Concrete, to be impressed with Enrichment: when the Centerings are to be removed, and how to mix the Plaster.
§ 29. The Construction of enriched Stucco Vaults.
When walls have reached the point where the arches of brick or light stone or tufa have to spring, it is necessary to turn a centering with planks in a close circle, over the framework of struts or boarding. The planks are fitted together according to the form of the vault, or in the shape of a boat, and this centering for the vaults must be fixed with strong props in whatever mode you wish, so that the material above does not strain it by its weight; and afterwards every crevice, in the middle, in the corners, and everywhere, must be firmly stopped up with clay so that when the concrete is spread the mixture shall not filter through. This finished, above that surface of boards they make caissons of wood, which are to be worked contrariwise, with projections where a hollow is wanted; in the same way let the mouldings and details that we wish to make be worked by opposites, so that when the material is cast, it may come, where (the mould is) hollow, in relief; where in relief, hollow, and thus similarly must all the members of the mouldings be arranged. Whether the vault is to be smooth or enriched, it is equally necessary to have shapes of wood, which mould the desired forms in clay; with this clay also are made the square panels for such decoration, and these are joined the one to the other on the flat or by mouldings or enriched bands, which can be made to follow the line of this centering. Having finished covering it all with enrichments of clay, formed in intaglio and fitted together, as was said above, one must then take lime, with pozzolana earth or sand riddled finely, mixed liquid and mostly lime, and of that lay evenly a coating over all, till every mould is full. Afterwards, above this coating make the vault with bricks, raising or lowering them according as the vault turns, and continually adding till the arch be closed. This done, it must all be left to set and get firm, till the work be dry and solid.[132] Then when the props are removed and
the vault is left free, the clay is easily taken away and all the work remains modelled and worked as if done in stucco, and those parts that have not come out well are gone over with stucco till they are complete. In this manner have been executed all the works in the ancient edifices, which had afterwards stucco enrichment upon them. This the moderns have done to-day in the vaults of St. Peter’s, and many other masters throughout Italy have done the same.
§ 30. Stucco made with Marble Dust.
Now let us show how the stucco is mixed.[133] Chips of marble are pounded in a stone mortar; no other lime is used for this stucco save white lime made either of marble chips or of travertine; instead of sand the pounded marble is taken and is sifted finely and kneaded with the lime, in the proportion of two thirds lime to one third pounded marble. The stucco is made coarser or finer, according as one wishes to work coarsely or finely. Enough now of stuccoes because the rest will be said later, when I shall treat of them in connection with Sculpture. Before passing to this subject, we shall speak briefly of fountains which are made for walls and of their various ornaments.
CHAPTER V.
How Rustic Fountains are made with Stalactites and Incrustations from water, and how Cockle shells and Conglomerations of vitrified stone are built into the Stucco.
§ 31. Grottoes and Fountains of ‘Rocaille’ work.
The fountains which the ancients made for their palaces, gardens, and other places, were of different kinds; some stood alone, with basins and vases of different sorts, others were attached to the walls, and bore niches with masks, figures, or ornaments suggesting the sea; others again for use in hot baths, were simpler and plainer, and finally others resembled woodland springs that rise naturally in the groves; while those which the moderns have made and continue to make are also of different kinds. The moderns, always varying them, have added to the inventions of the ancients, compositions of Tuscan work,[134] covered with stalactites from petrified waters, which hang down resembling roots, formed in the lapse of time of congelations of such waters as are hard and are charged with sediment. These exist not only at Tivoli, where the river Teverone petrifies the branches of trees, and all objects that come in contact with it, turning them into gum-like exudations and stalactites; but also at the lake Piè di Lupo, [135] where the stalactites are very large; and in Tuscany at the river Elsa,[136] whose water makes them clear so that they look like marble, glass, or artificial crystals. But the most beautiful and curious of all are found behind Monte Morello[137] also in Tuscany, eight miles from Florence. Of this sort Duke Cosimo has had made in his garden at Olmo near Castello[138] the rustic ornaments of the fountains executed by the sculptor Tribolo. These stalactites removed from where nature has produced them are introduced into work done by the artificer and fixed with iron bars, with branches soldered with lead or in some other way, or they are grafted into the stones so as to hang suspended. They are fixed on to the Tuscan work in such a way as to leave it here and there exposed to view. Then by adjusting leaden tubes hidden between these stalactites, and distributing holes among them, jets of water are made to pour out, when a key at the
entrance of the conduit is turned; and thus are arranged pipes for water and various jets through which the water rains down among the incrustations of these stalactites, and in falling sounds sweet to the ear and is beautiful to the eye.
There is also another kind of grotto, of a more rustic fashion, imitating sylvan fountains in the following way. Some take spongelike stones and joining them together sow grass over them, thus, with an order which appears disorder and wild, the grottoes are rendered very natural and real. Others make smoother and more polished grottoes of stucco, in which are mingled both stones and stucco, and while the stucco is fresh they insert, in bands and compartments, knobs or bosses, cockle shells, sea snails, tortoise shells, shells large and small, some showing the outside and some the reverse: and of these they make flower vases and festoons, in which the cockle shells represent the leaves, and other varieties of shells the fruit;[139] and to these they add shells of turtles, as is seen in the vineyard at the foot of Monte Mario that Pope Clement VII, when still Cardinal, had made by the advice of Giovanni da Udine.[140]
Again a rustic and very beautiful mosaic in many colours is made by using little bits of old bricks that have been too much baked, and pieces of glass which has run owing to the pans of glass bursting in an overheated furnace. The work is done by sticking these bits into the stucco on the wall as was said above, and arranging between them corals and other spoils from the sea, things in themselves full of grace and beauty. Thus are made animals and figures, covered with the shells already mentioned as well as with coloured pastes in various pieces arranged in rustic fashion, very quaint to look upon. There have been many fountains of this kind recently set up at Rome, which by their charm have incited the minds of countless persons to be lovers of such work. Another kind of ornament entirely rustic is also used now-a-days for fountains, and is applied in the following manner. First the skeleton of the figure or any other object desired is made and plastered over with mortar or stucco, then the exterior is covered in the fashion of mosaic, with pieces of white or coloured marble, according to the object designed, or else with certain little many coloured pebbles: and these when carefully worked have a long life. The stucco with which they build up and work these things is the same that we have before described, and when once set it holds them
securely on the walls. To such fountains pavements are made of sling-stones, that is, round and flat river pebbles, set on edge and in ripples as water goes, with excellent effect. Others, for the finer fountains, make pavements with little tiles of terra cotta in various divisions and glazed in the fire, as in clay vases painted in various colours and with painted ornaments and leafage; but this sort of pavement is more suitable for hot-air chambers and baths than for fountains.[141]
INTERIOR OF GROTTO IN BOBOLI GARDENS, FLORENCE
Showing an unfinished statue ascribed to Michelangelo
P IV
CHAPTER VI.
On the manner of making Pavements of Tesselated Work.
§ 32. Mosaic Pavements.
There are no possible devices in any department that the ancients did not find out or at any rate try very hard to discover,—devices I mean that bring delight and refreshment to the eyes of men. They invented then, among other beautiful things, stone pavements diversified with various blendings of porphyry, serpentine, and granite, with round and square or other divisions, whence they went on to conceive the fabrication of ornamental bands, leafage, and other sorts of designs and figures. Therefore to prepare the work the better to receive such treatment, they cut the marble into little pieces, so that these being small they could be turned about for the background and the field, in round schemes or lines straight or twisted, as came most conveniently. From the joining together of these pieces they called the work mosaic,[142] and used it in the pavements of many of their buildings, as we still see in the Baths of Caracalla in Rome and in other places, where the mosaic is made with little squares of marble, that form leaves, masks, and other fancies, while the background for these is composed of squares of white marble and other small squares of black. The work was set about in the following manner. First was spread a layer of fresh stucco of lime and marble dust thick enough to hold firmly in itself the pieces fitting into each other, so that when set they could be polished smooth on the top; these in the drying make an admirably compacted concrete, which is not hurt by the wear of footsteps nor by water. Therefore this work having come into the highest estimation, clever people set themselves to study it further, as it is always easy to add something valuable to an invention already found out. So they made the marble mosaics finer, and of these, laid pavements both for baths and for hot rooms, and with the most subtle mastery and diligence they delicately fashioned various fishes in them, and imitated painting with many colours suitable for that work, and with many different sorts of marbles, introducing also
among these some pieces cut into little mosaic squares of the bones of fishes which have a lustrous surface.[143] And so life-like did they make the fishes, that water placed above them, veiling them a little, even though clear, made them appear actually alive in the pavements; as is seen in Parione in Rome, in the house of Messer Egidio and Fabio Sasso.[144]
§ 33. Pictorial Mosaics for Walls, etc.
Therefore, this mosaic work appearing to them a picture, capable of resisting to all eternity water, wind, and sunshine, and because they considered such work much more effective far off than near, the ancients disposed it so as to decorate vaults and walls, where such things had to be seen at a distance, for at a distance one would not perceive the pieces of mosaic which when near are easily distinguished. Then because the mosaics were lustrous and withstood water and damp, it was thought that such work might be made of glass, and so it was done, and producing hereby the most beautiful effect they adorned their temples and other places with it, as we still see in our own days at Rome in the Temple of Bacchus[145] and elsewhere.[146] Just as from marble mosaics are derived those which we now call in our time glass mosaics, so from the mosaic of glass we have passed on to egg-shell mosaic,[147] and from this to the mosaic in which figures and groups in light and shade are formed entirely of tesserae, though the effect is like painting; this we shall describe in its own place in the chapters on that art.[148]
CHAPTER VII.
How one is to recognize if a Building have good Proportions, and of what Members it should generally be composed.
§ 34. The principles of Planning and Design.
But since talking of particular things would make me turn aside too much from my purpose, I leave this minute consideration to the writers on architecture, and shall only say in general how good buildings can be recognized, and what is requisite to their form to secure both utility and beauty. Suppose then one comes to an edifice and wishes to see whether it has been planned by an excellent architect and how much ability he has shown, also whether the architect has known how to accommodate himself to the site, as well as to the wishes of him who ordered the structure to be built, one must consider the following questions. First, whether he who has raised it from the foundation has thought if the spot were a suitable one and capable of receiving buildings of that style and extent, and (granted that the site is suitable) how the building should be divided into rooms, and how the enrichment on the walls be disposed in view of the nature of the site which may be extensive or confined, elevated or low-lying. One must consider also whether the edifice has been tastefully arranged and in convenient proportion, and whether there has been furnished and distributed the proper kind and number of columns, windows, doors, and junctions of wall-faces, both within and without, in the given height and thickness of the walls; in short whether every detail is suitable in and for its own place. It is necessary that there should be distributed throughout the building, rooms which have their proper arrangement of doors, windows, passages, secret staircases, anterooms, lavatories, cabinets, and that no mistakes be apparent therein. For example there should be a large hall, a small portico or lesser apartments, which being members of the edifice, must necessarily, even as members of the human body, be equally arranged and distributed according to the style and complexity of the buildings; just as there are temples round, or octagonal, or six sided, or square, or in the form of a cross, and also
various Orders, according to the position and rank of the person who has the buildings constructed, for when designed by a skilful hand these exhibit very happily the excellence of the workman and the spirit of the author of the fabric.
§ 35. An ideal Palace.
To make the matter clearer, let us here imagine a palace,[149] and this will give us light on other buildings, so that we may be able to recognize, when we see them, whether they are well fashioned or no. First, then, if we consider the principal front, we shall see it raised from the ground either above a range of outside stairs or basement walls, so that standing thus freely the building should seem to rise with grandeur from the ground, while the kitchens and cellars under ground are more clearly lighted and of greater elevation. This also greatly protects the edifice from earthquakes and other accidents of fortune. Then it must represent the body of a man in the whole and similarly in the parts; and as it has to fear wind, water, and other natural forces it should be drained with sewers, that must be all in connection with a central conduit that carries away all the filth and smells that might generate sickness. In its first aspect the façade demands beauty and grandeur, and should be divided as is the face of a man. The door must be low down and in the middle, as in the head the mouth of the man, through which passes every sort of food; the windows for the eyes, one on this side, one on that, observing always parity, that there be as much ornament, and as many arches, columns, pilasters, niches, jutting windows, or any other sort of enrichment, on this side as on that; regard being had to the proportions and Orders already explained, whether Doric, Ionic, Corinthian, or Tuscan. The cornice which supports the roof must be made proportionate to the façade according to its size, that rainwater may not drench the façade and him who is seated at the street front. The projection must be in proportion to the height and breadth of the façade. Entering within, let the first vestibule have a great amplitude, and let it be arranged to join fittingly with the entrance corridor, through which everything passes; let it be free and wide, so that the press of horses or of crowds on foot, that often congregate there, shall not do themselves any hurt in the entrance on fête days
or on other brilliant occasions. The courtyard, representing the trunk, should be square and equal, or else a square and a half, like all the parts of the body, and within there should be doors and wellarranged apartments with beautiful decoration. The public staircase needs to be convenient and easy to ascend, of spacious width and ample height, but only in accordance with the proportion of the other parts. Besides all this, the staircases should be adorned or copiously furnished with lights, and, at least over every landing-place where there are turns, should have windows or other apertures. In short, the staircases demand an air of magnificence in every part, seeing that many people see the stairs and not the rest of the house. It may be said that they are the arms and legs of the body, therefore as the arms are at the sides of a man so ought the stairs to be in the wings of the edifice. Nor shall I omit to say that the height of the risers ought to be one fifth of a braccio at least,[150] and every tread two thirds wide,[151] that is, as has been said, in the stairs of public buildings and in others in proportion; because when they are steep neither children nor old people can go up them, and they make the legs ache. This feature is most difficult to place in buildings, and notwithstanding that it is the most frequented and most common, it often happens that in order to save the rooms the stairs are spoiled. It is also necessary that the reception rooms and other apartments downstairs should form one common hall for the summer, with chambers to accommodate many persons, while upstairs the parlours and saloons and the various apartments should all open into the largest one. In the same manner should be arranged the kitchens and other places, because if there were not this order and if the whole composition were broken up, one thing high, another low, this great and that small, it would represent lame men, halt, distorted, and maimed. Such works would merit only blame, and no praise whatever. When there are decorated wall-faces either external or internal, the compositions must follow the rules of the Orders in the matter of the columns, so that the shafts of the columns be not too long nor slender, not over thick nor short, but that the dignity of the several Orders be always observed. Nor should a heavy capital or base be connected with a slender column, but in proportion to the body must be the members, that they may have an elegant and beautiful appearance and design. All these things are best appreciated by a correct eye, which, if it have discrimination, can
hold the true compasses and estimate exact measurements, because by it alone shall be awarded praise or blame. And this is enough to have said in a general sense of architecture, because to speak of it in any other way is not matter for this place.
NOTES ON ‘INTRODUCTION’ TO ARCHITECTURE
PORPHYRY AND PORPHYRY QUARRIES.
[See § 2, Of Porphyry, ante, p. 26.]
Porphyry, which is mineralogically described as consisting of crystals of plagioclase felspar in a purple felspathic paste, is a very hard stone of beautiful colour susceptible of a high polish. ‘No material,’ it has been said, ‘can approach it, either in colour, fineness of grain, hardness or toughness. When used alone its colour is always grand; and in combination with any other coloured material, although displaying its nature conspicuously, it is always harmonious’ (Transactions, Royal Institute of British Architects, 1887, p. 48). Though obtained, as Vasari knew, from Egypt, it was not known to the dynastic Egyptians, but was exploited with avidity by the Romans of the later imperial period. The earliest mention of it seems to be in Pliny, Hist. Nat., XXXVI, 11, under the name ‘porphyrites’ and statues in the material were according to this author sent for the first time to Rome from Egypt in the reign of Claudius. The new material was however not approved of, and for some time was by no means in fashion. It was not indeed till the age of the Antonines that as Helbig remarks ‘the preference for costly and rare varieties of stone, without reference to their adaptability for sculpture, began to spread.’ After this epoch, the taste for porphyry and other such strongly marked or else intractable materials grew till it became a passion, and the Byzantine emperors carried on the tradition of its use inherited by them from the later days of paganism. The material was quarried in the mountains known as Djebel Duchan near the coast of the Red Sea, almost opposite the southern point of the peninsula of Sinai, and the Romans carried the blocks a distance of nearly 100 miles to Koptos on the Nile whence they were transported down stream to Alexandria, where Mr Brindley thinks there would be reserve dépôts where lapidaries and artists resided, a source of supply for the large quantities used by Constantine. The same authority estimates that there must be about
