Bit level diversity combining for D-MIMO - McGill...

Bit level diversity combining for D-MIMO

Wenjing Lin

Department of Electrical & Computer Engineering

McGill UniversityMontreal, Canada

November 2011

A thesis submitted to McGill University in partial fulfillment of the requirements for the degreeof Master of Engineering (M.Eng.) in Electrical Engineering

c© 2011 Wenjing Lin

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Abstract

Multiple-Input Multiple-Output (MIMO) transmission techniques have been shown to be a pow-erful performance enhancing technology in wireless communications. However in realistic sys-tems, when increasing the number of antennas in a restricted space, the capacity gain of MIMOis limited. Furthermore, co-located MIMO (C-MIMO) systems when affected by shadowing cannot improve link quality. This motivates us to investigate Distributed MIMO (D-MIMO) system.

This work considers a bit level combining scheme, aided by bit reliability information foran uplink D-MIMO system over a composite Rayleigh-lognormal fading channel. Bit reliabilityinformation is derived based on the logarithmic likelihood ratio (LLR) and further modified forthe MIMO detection schemes: SD-ML (Sphere Decoding - Maximum Likelihood) and MMSE-OSIC (Minimum Mean Square Error - Ordered Successive Interference Cancellation). Computersimulation results demonstrate that such bit level combining scheme provides significant per-formance improvements for D-MIMO with M transmit and L receive antennas on each of its Ngeographically dispersed receive node, over conventional C-MIMO with M transmit and L receiveantennas, even in the presence of channel estimation errors or channel spatial correlation. It isfound that such a D-MIMO system provides a comparable performance to a C-MIMO systemwith M transmit and NL receive antennas, especially when space correlation becomes significant.

Furthermore, an analytical BER evaluation technique is proposed for a C-MIMO systemwith SD-ML detection over a composite Rayleigh-lognormal fading channel with and withoutspatial correlation. Numerical results show that our technique provides tight approximations forC-MIMO over space uncorrelated, space semi-correlated and space correlated channels.

We also provide a theoretical BER approximation technique for a D-MIMO system withSD-ML detection over a composite Rayleigh-lognormal fading channel with and without spatialcorrelation. Numerical results show that by optimizing two parameters, the BER approximationtechinique provides good approximation for an uncorrelated D-MIMO when the number of trans-mit antennas equals the number of receive antennas on each of its N geographically dispersedreceive node. We further notice that these optimized parameters set for an uncorrelated D-MIMOwith equal number of transmit and receive antennas on each receive node can not provide goodapproximation for an uncorrelated D-MIMO when the number of transmit antennas is less thanthe number of receive antennas on each node. These analytical results confirm the significantperformance improvement provided by D-MIMO with bit level combining.

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Sommaire

Les techniques de transmission MIMO (Multiple-Input Multiple-Output) constituent une puis-sante technologie permettant des ameliorations significatives en termes de performance dans ledomaine des communications sans fil. Cependant, en pratique, lorsque le nombre d’antennes aug-mente dans un espace relativement restreint, le gain en capacite des systemes MIMO est limite.De plus, lorsqu’ils sont affectes par l’effet d’ombrage, les systemes MIMO co-localises (C-MIMO)ne peuvent ameliorer la qualite de transmission. Ces difficultes ont motive notre investigationdes systemes D-MIMO (Distributed-MIMO).

Cette these considere une methode de combinaison au niveau du bit, utilisant l’information surla fiabilite du bit, pour le canal montant d’un systeme D-MIMO subissant des evanouissementsde Rayleigh-lognormale. L’information sur la fiabilite du bit est etablie a partir de la fonctionde vraisemblance logarithmique (LLR) et est par la suite modifiee pour differentes methodesde detection pour les sysemes MIMO, incluant SD-ML (Sphere Decoding-Maximum Likelihood)et MMSE-OSIC (Minimum Mean Square Error-Ordered Successive Interference Cancellation).Les resultats des simulations par ordinateur demontrent que comparee a un C-MIMO conven-tionnel utilisant M antennes d’emission et L antennes de reception, la methode de combinaisonau niveau du bit fournit des ameliorations de performance significatives pour le D-MIMO util-isant M antennes d’emission et L antennes de reception sur chacun des N nœuds de receptiongeographiquement disperses, et ceci meme en presence d’erreurs d’estimation de canal ou decorrelation spatiale. Il est aussi demontre qu’un tel D-MIMO fournit une performance compa-rable a celle d’un C-MIMO avec M antennes d’emission et NL antennes de reception, surtoutlorsque la correlation spatiale est significative.

De plus, une technique d’evaluation analytique de la probabilite d’error est proposee pour unsysteme C-MIMO utilisant SD-ML comme methode de detection sur un canal a evanouissementscomposites de Rayleigh-lognormale avec ou sans correlation spatiale. Les resultats numeriquesmontrent que notre technique fournit de tres bonnes approximations pour un systeme C-MIMOavec ou sans correlation spatiale.

Nous presentons egalement une technique d’approximation theorique de la probabilite d’errorpour un systeme D-MIMO utilisant SD-ML comme methode de detection sur un canal a evanouis-sements composites de Rayleigh-lognormale avec ou sans correlation spatiale. Les resultatsnumeriques montrent qu’en optimisant deux parametres, notre technique d’approximation dela probabilite d’error fournit une bonne approximation pour le D-MIMO sans correlation spa-tiale utilisant M antennes d’emission et L antennes de reception sur chacun de ses N nœuds dereception geographiquement disperses. Cependant les valeurs optimales des parametres obtenuespour un D-MIMO ou le nombre d’antennes d’emission est egal au nombre d’antennes de receptiona chacun des nœuds de reception, ne peuvent fournir de bonnes approximations lorsque le nom-bre d’antennes d’emission est inferieur au nombre d’antennes de reception sur chacun des nœudsde reception. Ces resultats analytiques confirment l’amelioration significative de performancefournie par le D-MIMO utilisant la methode de combinaison au niveau du bit.

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Acknowledgments

First and foremost, I would like to thank my supervisor, Professor Harry Leib for his encour-agement and the technical instruction throughout my studies and research. He made greatcontribution to this thesis, he proposed the bit level combining algorithm at the Fusion Centerin D-MIMO and provided scheme for theoretical BER performance analysis of D-MIMO whichare significant parts in my research.

Secondly, I would like to thank all the labmates in wireless research lab for their fruitful helpand pleasant times. In particular I want to mention Mr. Djelili Radji who gave me a lot ofhelpful advice and provided the softwares for MIMO system with square QAM modulation forMMSE-OSIC and SD-ML detection schemes respectively, which are very important resourcesto my research. I also would like to send my gratitude to Mr. Djelili Radji for translating myAbstract into French.

Thirdly, I am grateful for Professor H. Leib for his financial support from his NSERC grants.Finally, I would like to thank my parents, my sister and my friends for their encouragement

and understanding doing my whole learning process.

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Contents

1 Introduction 1

2 System and Channel Model of D-MIMO 42.1 D-MIMO System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 D-MIMO Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Small Scale Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.2.2 Large Scale Fading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

3 Combining with Reliability Information 83.1 Bit by Bit Combing at the Fusion Center . . . . . . . . . . . . . . . . . . . . . . . 83.2 Bit Reliability Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2.1 MMSE-OSIC Detection Scheme in MIMO . . . . . . . . . . . . . . . . . . 93.2.2 SD-ML Detection Scheme in MIMO . . . . . . . . . . . . . . . . . . . . . . 103.2.3 Bit Level Reliability Information . . . . . . . . . . . . . . . . . . . . . . . 14

4 Computer Simulation Results 174.1 Simulations with Two Reliability Information Scheme . . . . . . . . . . . . . . . . 174.2 Simulations with Perfect Channel Information . . . . . . . . . . . . . . . . . . . . 184.3 Impact of Channel Estimation Errors . . . . . . . . . . . . . . . . . . . . . . . . . 204.4 Impact of Channel Spatial Correlation . . . . . . . . . . . . . . . . . . . . . . . . 214.5 Comparision between a (M,N,L) D-MIMO system and a M ×NL C-MIMO system 21

5 D-MIMO Performance Analysis 535.1 Performance Analysis for C-MIMO with SD-ML Detection . . . . . . . . . . . . . 54

5.1.1 PEP in case of spatial correlation at receiver . . . . . . . . . . . . . . . . . 555.1.2 PEP in case of no spatial correlation at receiver . . . . . . . . . . . . . . . 565.1.3 BER performance for C-MIMO over space uncorrelated channels . . . . . . 575.1.4 BER performance for C-MIMO over space correlated channels . . . . . . . 68

5.2 Performance Analysis for D-MIMO with SD-ML Detection . . . . . . . . . . . . . 1065.2.1 Computation of X(n)

<C and X(n)>C . . . . . . . . . . . . . . . . . . . . . . 107

5.2.2 Computation of the characteristic function of Y(n)i . . . . . . . . . . . . . . 109

5.2.3 BER Evaluation for D-MIMO . . . . . . . . . . . . . . . . . . . . . . . . . 1105.2.4 Numerical results for Analytical BER evaluation for D-MIMO . . . . . . . 114

6 Conclusions 119

A Partial fraction expansion in integration 120A.1 ck are distinct . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120A.2 ck are same . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

Contents v

B Derivation of Transfer Function 121B.1 Transfer function for square constellation . . . . . . . . . . . . . . . . . . . . . . . 121B.2 Transfer function for 32QAM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

C Integral Evaluation with Residue Theorem 124

C.1 Computation of

∫ j∞+ǫ

−j∞+ǫ

ϕU1(z)

zdz . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

C.2 Computation of

∫ j∞−ǫ

−j∞−ǫ

ϕU2(z)

zdz . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

D Computer Simulation Overview and Guide 127

References 131

vi

List of Figures

2.1 (M,N,L) Distributed-MIMO System Model . . . . . . . . . . . . . . . . . . . . . . 5

3.1 Tree Structure for Sphere Decoding with detection layer M = 4 . . . . . . . . . . 133.2 Voronoi Regions for Gray-coded 16-QAM . . . . . . . . . . . . . . . . . . . . . . . 163.3 Bit reliability metric with transmit symbol a = 5 and k = 1 for 16QAM . . . . . 16

4.1 Construction of 32QAM from rectangular 32QAM . . . . . . . . . . . . . . . . . . 184.2 Absolute vs Absolute square for DMIMO(4,2,4) with 16QAM (f=1,g=1), where

Eb is transmit bit power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Absolute vs Absolute square for DMIMO(4,2,4) with 16QAM (f=0,g=1), where





Eb is transmit bit power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284.8 DMIMO(4,2,4) and 4 × 4 CMIMO with QPSK (f=1,g=1), where Eb is transmit

bit power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.9 DMIMO(4,2,4) and 4 × 4 CMIMO with QPSK (f=1,g=0), where Eb is transmit

bit power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.10 DMIMO(4,2,4) and 4 × 4 CMIMO with QPSK (f=0,g=1), where Eb is transmit

bit power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.11 DMIMO(4,2,4) and 4× 4 CMIMO with 16QAM (f=1,g=1), where Eb is transmit







bit power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

List of Figures vii

4.18 DMIMO(4,2,4) and 4× 4 CMIMO with 64QAM (f=1,g=0), where Eb is transmitbit power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

4.19 DMIMO(4,2,4) and 4× 4 CMIMO with 64QAM (f=0,g=1), where Eb is transmitbit power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.20 Effect of channel estimation error: DMIMO(4,2,4) d(1) = d(2) = 1km and 4 × 4CMIMO d=1km with SD-ML for 16QAM, where Eb is transmit bit power . . . . . 41

4.21 Effect of channel estimation error: DMIMO(4,2,4) d(1) = 1km, d(2) = 1.2km and4 × 4 CMIMO d=1.2km with SD-ML for 16QAM, where Eb is transmit bit power 42

4.22 Effect of channel estimation error: DMIMO(4,2,4) d(1) = d(2) = 1km and 4 × 4CMIMO d=1km with SD-ML for 64QAM, where Eb is transmit bit power . . . . . 43

4.23 Effect of channel estimation error: DMIMO(4,2,4) d(1) = 1km, d(2) = 1.2km and4 × 4 CMIMO d=1.2km with SD-ML for 64QAM, where Eb is transmit bit power 44

4.24 Effect of channel estimation error: DMIMO(4,2,4) d(1) = d(2) = 1km and 4 × 4CMIMO d=1km with MMSE-OSIC for 16QAM, where Eb is transmit bit power . 45

4.25 Effect of channel estimation error: DMIMO(4,2,4) d(1) = 1km, d(2) = 1.2km and4 × 4 CMIMO d=1.2km with MMSE-OSIC for 16QAM, where Eb is transmit bitpower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4.26 Effect of channel estimation error: DMIMO(4,2,4) d(1) = d(2) = 1km and 4 × 4CMIMO d=1km with MMSE-OSIC for 64QAM, where Eb is transmit bit power . 47

4.27 Effect of channel estimation error: DMIMO(4,2,4) d(1) = 1km, d(2) = 1.2km and4 × 4 CMIMO d=1.2km with MMSE-OSIC for 64QAM, where Eb is transmit bitpower . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.28 Effect of channel spatial correlation: DMIMO(4,2,4) d(1) = d(2) = 1km and 4 × 4CMIMO d=1km with SD-ML for 16QAM, where Eb is transmit bit power . . . . . 49

4.29 Effect of channel spatial correlation: DMIMO(4,2,4) d(1) = d(2) = 1km and 4 × 4CMIMO d=1km with SD-ML for 64QAM, where Eb is transmit bit power . . . . . 50

4.30 Same space correlation for DMIMO(4,2,4) d(1) = d(2) = 1km (’D’ in legend) and4×8 CMIMO d=1km (’C’ in legend) with SD-ML for 16QAM, where Eb is transmitbit power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.31 Same space constraint at receiver of DMIMO(4,2,4) d(1) = d(2) = 1km (’D’ inlegend) and 4×8 CMIMO d=1km (’C’ in legend) with SD-ML for 16QAM, whereEb is transmit bit power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.1 Largest Euclidean distance in 32 QAM and square QAM constellation . . . . . . . 595.2 BER evaluation for uncorrelated 2×2 C-MIMO with SD-ML for QPSK. Numbers

denote the values of K in (5.39), Eb is averaged receive bit power . . . . . . . . . 625.3 BER evaluation for uncorrelated 2×2 C-MIMO with SD-ML for 16QAM. Numbers

denote the values of K in (5.39), Eb is averaged receive bit power . . . . . . . . . 625.4 BER evaluation for uncorrelated 4×4 C-MIMO with SD-ML for QPSK. Numbers

denote the values of K in (5.39), Eb is averaged receive bit power . . . . . . . . . 635.5 BER evaluation for uncorrelated 4×4 C-MIMO with SD-ML for 16QAM. Numbers

denote the values of K in (5.39), Eb is averaged receive bit power . . . . . . . . . 635.6 BER evaluation for uncorrelated 2×4 C-MIMO with SD-ML for QPSK. Numbers

denote the values of K in (5.39), Eb is averaged receive bit power . . . . . . . . . 64

List of Figures viii

5.7 BER evaluation for uncorrelated 2×4 C-MIMO with SD-ML for 16QAM. Numbersdenote the values of K in (5.39), Eb is averaged receive bit power . . . . . . . . . 64

5.8 BER evaluation for uncorrelated 2×2 C-MIMO with QPSK, Eb is averaged receivebit power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.9 BER evaluation for uncorrelated 2 × 2 C-MIMO with 16QAM, Eb is averagedreceive bit power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65





5.14 ‖R1/2T dij‖2 and dm for C-MIMO with M = 2, Q = 16, t = 0.5 . . . . . . . . . . . . 69

5.15 ‖R1/2T dij‖2 and dm for C-MIMO with M = 4, Q = 4, t = 0.5 . . . . . . . . . . . . 70

5.16 BER evaluation for 2× 2 C-MIMO with QPSK and r = 0.3. Numbers denote thevalues of K in (5.45), Eb is averaged receive bit power. . . . . . . . . . . . . . . . 78

5.17 BER evaluation using ’BER-min’ for 2 × 2 C-MIMO with QPSK and t = 0.3.Numbers denote the values of K in (5.51), Eb is averaged receive bit power. . . . . 78

5.18 BER evaluation using ’BER-avg’ for 2 × 2 C-MIMO with QPSK and t = 0.3.Numbers denote the values of K in (5.55), Eb is averaged receive bit power. . . . . 79

5.19 BER evaluation using ’BER-min’ for 2× 2 C-MIMO with QPSK and t = r = 0.3.Numbers denote the values of K in (5.56), Eb is averaged receive bit power. . . . . 79

5.20 BER evaluation using ’BER-avg’ for 2 × 2 C-MIMO with QPSK and t = r = 0.3.Numbers denote the values of K in (5.57), Eb is averaged receive bit power. . . . . 80

5.21 BER evaluation for 2 × 2 C-MIMO with 16QAM and r = 0.3. Numbers denotethe values of K in (5.45), Eb is averaged receive bit power. . . . . . . . . . . . . . 80

5.22 BER evaluation using ’BER-min’ for 2 × 2 C-MIMO with 16QAM and t = 0.3.Numbers denote the values of K in (5.51), Eb is averaged receive bit power. . . . . 81

5.23 BER evaluation using ’BER-avg’ for 2 × 2 C-MIMO with 16QAM and t = 0.3.Numbers denote the values of K in (5.55), Eb is averaged receive bit power. . . . . 81

5.24 BER evaluation using ’BER-min’ for 2×2 C-MIMO with 16QAM and t = r = 0.3.Numbers denote the values of K in (5.56), Eb is averaged receive bit power. . . . . 82

5.25 BER evaluation using ’BER-avg’ for 2×2 C-MIMO with 16QAM and t = r = 0.3.Numbers denote the values of K in (5.57), Eb is averaged receive bit power. . . . . 82





List of Figures ix


5.31 BER evaluation for 2× 4 C-MIMO with QPSK and r = 0.3. Numbers denote thevalues of K in (5.45), Eb is averaged receive bit power . . . . . . . . . . . . . . . . 85





5.36 BER evaluation for 2 × 4 C-MIMO with 16QAM and r = 0.3. Numbers denotethe values of K in (5.45), Eb is averaged receive bit power . . . . . . . . . . . . . 88





5.41 BER evaluation for 2 × 2 C-MIMO with 16QAM and r = 0.5. Numbers denotethe values of K in (5.45), Eb is averaged receive bit power. . . . . . . . . . . . . . 90










5.51 BER evaluation for 2×2 C-MIMO with QPSK and r = 0.3. Eb is averaged receivebit power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

List of Figures x

5.52 BER evaluation for 2 × 2 C-MIMO with QPSK and t = 0.3. ’Avg’ denotes BERevaluation with ’BER-avg’ scheme, ’Min’ denotes BER evaluation with ’BER-min’scheme. Eb is averaged receive bit power. . . . . . . . . . . . . . . . . . . . . . . . 96

5.53 BER evaluation for 2×2 C-MIMO with QPSK and t = r = 0.3. ’Avg’ denotes BERevaluation with ’BER-avg’ scheme, ’Min’ denotes BER evaluation with ’BER-min’scheme. Eb is averaged receive bit power. . . . . . . . . . . . . . . . . . . . . . . . 96

5.54 BER evaluation for 2 × 2 C-MIMO with 16QAM and r = 0.3. Eb is averagedreceive bit power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

5.55 BER evaluation for 2× 2 C-MIMO with 16QAM and t = 0.3. ’Avg’ denotes BERevaluation with ’BER-avg’ scheme, ’Min’ denotes BER evaluation with ’BER-min’scheme. Eb is averaged receive bit power. . . . . . . . . . . . . . . . . . . . . . . . 97

5.56 BER evaluation for 2 × 2 C-MIMO with 16QAM and t = r = 0.3. ’Avg’ de-notes BER evaluation with ’BER-avg’ scheme, ’Min’ denotes BER evaluation with’BER-min’ scheme. Eb is averaged receive bit power. . . . . . . . . . . . . . . . . 98












List of Figures xi





5.72 Numerical and simulation BER for uncorrelated (4,2,4) DMIMO d(1) = d(2) = 1kmwith SD-ML for QPSK and 16QAM. Parameter settings C and α are employedfrom Table 5.9, Eb is transmit bit power . . . . . . . . . . . . . . . . . . . . . . . 116



B.1 Structure of Dn2in one branch of constellation . . . . . . . . . . . . . . . . . . . . 121

B.2 Computation of transfer function for 32QAM . . . . . . . . . . . . . . . . . . . . . 122

C.1 Contour C1 and C2 in evaluating integral . . . . . . . . . . . . . . . . . . . . . . . 125

xii

List of Tables

4.1 The performance gain of D-MIMO over C-MIMO for SD-ML with f = 1, g = 0 atBER= 10−5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.2 The performance gain of D-MIMO over C-MIMO for MMSE-OSIC with f = 0, g =1 at BER= 10−5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

5.1 Transfer function for QPSK, 16QAM, 32QAM and 64QAM . . . . . . . . . . . . . 595.2 Values of K required for tight BER approximation of uncorrelated C-MIMO . . . 615.3 Values of K required for tight BER approximation of C-MIMO when space cor-

relation at receiver only, r = 0.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.4 Values of K required for tight BER approximation of C-MIMO when space cor-

relation at transmitter only with ’BER-avg’ technique, t = 0.3 . . . . . . . . . . . 765.5 Values of K required for tight BER approximation of C-MIMO when space cor-

relation at transmitter only with ’BER-min’ technique, t = 0.3 . . . . . . . . . . . 765.6 Values of K required for tight BER approximation of C-MIMO when space cor-

relation at receiver only, r = 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 775.7 Values of K required for tight BER approximation of C-MIMO when space cor-

relation at transmitter only with ’BER-avg’ technique, t = 0.5 . . . . . . . . . . . 775.8 Values of K required for tight BER approximation of C-MIMO when space cor-

relation at transmitter only with ’BER-min’ technique, t = 0.5 . . . . . . . . . . . 775.9 C and α for D-MIMO with different modulation schemes . . . . . . . . . . . . . . 115

D.1 List of C software files for MMSE-OSIC detection scheme in enclosed CD . . . . . 128D.2 List of C software files for SD-ML detection scheme in enclosed CD . . . . . . . . 129D.3 Simulation parameters of the input file . . . . . . . . . . . . . . . . . . . . . . . . 130D.4 Parameters of the output file . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

xiii

List of Acronyms

16QAM 16-point Quadrature Amplitude Modulation32QAM 16-point Quadrature Amplitude Modulation64QAM 64-point Quadrature Amplitude ModulationAWGN Additive White Gaussian NoiseBER Bit Error RateC-MIMO Co-located MIMOCSI Channel State InformationD-MIMO Distributed MIMOLLR Logarithmic Likelihood RatioMIMO Multiple-Input Multiple-OutputML Maximum LikelihoodMMSE Minimum Mean Square ErrorMS Mobile StationOSIC Ordered Successive Interference CancelationPEP Pairwise Error ProbabilityQAM Quadrature Amplitude ModulationQPSK Quadrature Phase Shift KeyingSD Sphere DecodingSER Symbol Error RateSNR Signal to Noise RatioV-BLAST Vertical Bell Labs Layered Space-TimeVSER Vector Symbol Error RateZF Zero forcing

xiv

Notational Convention

(·)T Transpose of the argument(·)H Hermitian transpose of the argument(·)∗ Complex conjugate of the argument| · | Magnitude of scaler argument‖ · ‖2 Square norm of the vector argumentb Column vector bA Matrix Aai ith column vector of matrix A

A1/2 Square root of matrix AIN N -dimension identity MatrixA Set A|A| Cardinality of set A∅ Empty setdiag(·) Block diagonal matrixtr(A) Trace of matrix AN (µ, σ2) Normal distribution with mean µ and variance σ2

CN (µ, σ2) Complex Normal distribution with mean µ and variance σ2

EX [·] Expected value over random variable XEX|Y [·] Conditional expectation of X given event YfX(x) Probability density function (PDF) of random variable XfX|Y (x) Conditional probability density function (PDF) of random variable X

given random variable YFX(x) Cumulative distribution function (CDF) of random variable XFX|Y (x) Conditional cumulative distribution function (CDF) of random variable X

given random variable YPr[X] Probability of event XPr[X|Y ] Conditional probability of event X given event Y(

MN

)Binomial coefficient

ℜ(·) Real part of a complex numberℑ(·) Imaginary part of a complex number

Q(·) Q-function with Q(x) =1√2π

∫ ∞

x

e−u2

2 du

1

Chapter 1: Introduction

Multiple-Input Multiple-Output (MIMO) transmission techniques constitute a powerful per-formance enhancing technology in wireless communications. MIMO techniques provide threemain gains: spatial multiplexing, spatial diversity and array gains. Spatial multiplexing in-creases capacity for no additional bandwidth expenditure, spatial diversity mitigates fading andincreases robustness to interference, and array gains provides an increased average signal-to-noiseratio [1].

A wireless communication architecture for MIMO known as V-BLAST has been proposed byFoschini, which has demonstrated to achieve spectral efficiencies of 20 - 40 bps/Hz at averagesignal-to-noise ratio (SNR) ranging from 24 to 34 dB [2]. In such architecture, a single datastream is demultiplexed into Nt substreams, each substream is then encoded into symbols andfed to its respective transmitter; at the receiver, each receive antenna receives the signal radiatedfrom all Nt transmit antennas. Several conventional detection algorithms for V-BLAST havebeen established. The optimal receiver, implementing maximum likelihood detection (MLD),is realized by searching for the transmit symbol vecotr over all possible candidates. Due toits exponential complexity increases with the number of transmit antennas, it is impracticalto be implemented in MIMO system [1]. Hence, suboptimal receivers with ordered successiveinterference cacellation such as ZF-OSIC and MMSE-OSIC [2] [3] attracted significant interestfor practical applications. These suboptimal receivers achieve lower complexity at a significantperformance degradation in high SNR when compared to the ML receiver [1]. The sphere decoder(SD) can reach ML performance with low decoding complexity. The principle of the SD algorithmis to search the closest lattice point to the received signal within a sphere of radius R centeredat the received signal [4]. In [5], a Complex SD by using complex Cholesky factorization of thechannel matrix was introduced. Since the complex SD avoids decoupling the complex system intoa real-valued system, it has a speed advantage over the real-valued SD. An improved complexSD algorithm to reduce the complexity of the SD was presented in [6]. In this new algorithm, asystem ordering criterion proposed in [7] was applied to maximize the probability that the firstfeasible solution is optimal, a minimum mean square error criterion was used in the selection ofcandidates, and a ”best-first” search strategy was applied to find new feasible solutions with asfew number of steps as possible.

Several works consider the performance analysis of MIMO system. In [8], the performanceof maximum likelihood detection (MLD) over Rayleigh fading channels with imperfect channelstate information (CSI) was analyzed, and a tight union bound on the probability of symbolerror rate (SER) was presented. The performance analysis of SD over spatially correlated MIMOchannels, by using Monte-Carlo simulation, was presented in [9], showing that its performancedegrades when the spatial correlation in the channel increases. Furthermore, [9] mentioned thatsince SD achieves equivalent performance to MLD, the performance of pairwise error probability(PEP) can be used to determine the performance of SD and the average PEP can be obtainedby averaging over the channel realization. A simplified approximation of vector symbol errorrate (VSER) for a given channel in high SNR was proposed in [10]. This approximation utilizedthe idea of ’nearest neighbor’ which is based on the geometrical relation between lattice points

1 Introduction 2

and was varified by simulation over three kinds of channels: unitary channel, dense channel andsparse channel. Due to the difficulty of analyzing the performance of MIMO with optimal orderingsymbol-by-symbol detection algorithm, [11] provided the theoretical analysis on the performanceof VSER for ZF-SIC with fixed order over Rayleigh fading channel. A closed-form analyticalexpression for average BER of a 2 ×NR system when the optimal ordering is implemented wasderived in [12]. In [13], the BER performances of ZF-SIC and MMSE-SIC for the NT -th layerwithout optimal ordering have been analyzed. The SER performance of linear MMSE detectionfor a small number of transmit and receive antennas was analyzed in [14].

Due to the increased capacity needs, MIMO techniques are anticipated to be widely employedin future wireless networks. However in realistic systems, increasing the number of antennas ina restricted space reduces the inter-antenna spacing, leading to an increase in spatial correlationthat could limit the capacity gain of a traditional co-located MIMO (C-MIMO) [15]. Furthermore,C-MIMO when affected by shadowing can not improve link quality [15]. These reasons motivateus to investigate Distributed MIMO (D-MIMO) scheme, which can potentially remedy some ofthe problems associated with conventional C-MIMO systems. The main feature of D-MIMO isthat multiple antennas at one end of a communication link are distributed among multiple widely-separated radio Ports (i.e. base stations) [15]. In [15] [16] [17], a composite fading channel modelthat addresses the small-scale fast fading and large-scale fading for D-MIMO was presented. Itshows that the composite fading channel from MS to a radio Port is a product of a large-scalefading coefficient and a small-scale fading matrix.

Channel capacity for D-MIMO systems has attracted considerable research interest. In [16],the influence of macroscopic diversity on the capacity of D-MIMO with a composite channelmodel by using Monte-Carlo simulation has been presented and simulation results showed thatmultiple ports are essential to achieve high channel capacity both for the uplink and the downlink.Capacity loss due to spatial correlation and shadowing has been analyzed in [15] to demonstratethe advantage of D-MIMO versus C-MIMO. Mean spectral efficiency (MSE) and mean outagespectral efficiency (MOSE) metrics for D-MIMO and C-MIMO have been derived in [18] by apply-ing Gaussian approximation to the distribution of the mutual information (MI). Analytical andsimulation results show that due to macro-diversity gain, D-MIMO has significant larger MOSEthan C-MIMO. In [19], the authors proved that MI of D-MIMO is asymptotically equivalent toa Gaussian random variable and derive a closed-form approximation of the outage probabilityfor D-MIMO systems.

The BER performance of an uplink distributed-antenna/direct-sequence code-division multiple-access wireless communication system (DA/DS-CDMA) over composite lognormal shadowingslow-fading and Nakagami-m fast-fading channels is analyzed and evaluated in [20], where onlyone transmit antenna per user is considered and both correlation-based single-user detector(SUD) and MMSE-assisted multiuser detector (MUD) are employed. Simulation results for BERperformance of a downlink distributed antenna system (DAS) with only one receive antennaare presented in [21], where both SD and zero forcing (ZF) detection schemes are considered.However, to the author’s best knowledge, there is no research work which deals with the BERperformance analysis of a generalized D-MIMO system [15] [16] in literatures.

In this thesis, we propose a bit level combining scheme which combines the detected bitsfrom all Ports, aided by use of bit reliability information. By using such a bit combining schemeat the Fusion Center, bit detection for D-MIMO system is achieved with a complexity suitablefor practical applications. We derive the bit reliability information by applying logarithmic

1 Introduction 3

likelihood ratios (LLR) [22] [23] [24] and using the max-log approximation [25]. We present BERperformance for both C-MIMO and D-MIMO over a composite Rayleigh-lognormal channel byusing Monte-Carlo simulations for SD-ML and MMSE-OSIC detection. Furthermore, we providetheoretical BER performance analysis for C-MIMO and D-MIMO with SD-ML detection over acomposite Rayleigh-lognormal channel respectively, with consideration of spatial correlation andno spatial correlation.

The contributions of this thesis are:

• Proposed bit level combining scheme with bit reliability information at the Fusion Centerin D-MIMO and derived bit reliability information based on logarithmic likelihood ratio.

• Extended software development to D-MIMO with bit combining scheme and performedcomputer simulations to show BER performance of a 4×4 C-MIMO and BER performanceof a (4,2,4) D-MIMO over a composite Rayleigh-lognormal channel by considering MMSE-OSIC and SD-ML detection schemes for QPSK, 16QAM, 32QAM and 64QAM.

• Proposed theoretical BER evaluation scheme for C-MIMO with SD-ML detection over acomposite Rayleigh-lognormal channel with consideration of both spatial correlation andno spatial correlation.

• Proposed theoretical BER evaluation scheme for D-MIMO with SD-ML detection over acomposite Rayleigh-lognormal channel with consideration of both spatial correlation andno spatial correlation.

This thesis is organized as follows. Chapter 2 describes the system and channel model ofD-MIMO. In Chapter 3, the bit level combining scheme is presented and bit reliability informa-tion is derived. Furthermore, two detection schemes, MMSE-OSIC and SD-ML, are reviewed.Computer simulation results considering both perfect CSI and imperfect CSI as well as chan-nel spatial correlation are presented in chapter 4. In chapter 5, BER performance analysis forboth C-MIMO and D-MIMO over a composite fading channel with SD-ML detection as wellas corresponding numerical results are presented. Discussions and conclusions are presented inchapter 6. Appendix A illustrates partial fraction expension of a rational function. In AppendixB, we present the derivation of transfer function for square QAM constellaton and 32QAM. Theapplication of Residue theorem to evaluate integrals is illstrated in Appendix C. An overviewand user guide to the source code used to generate the simulations are included in Appendix D,and an attached CD contains all C source codes.

4

Chapter 2: System and Channel Model of D-MIMO

2.1 D-MIMO System Model

A D-MIMO system consisting of one transmit node, or Mobile Station (MS), and N geograph-ically dispersed receive nodes, or Base Stations (Ports) is illustrated in Fig. 2.1. The MStransmitter employs M co-located antennas, and each Port receiver has L co-located antennas(L ≥ M). Such system can be referred to as an (M,N,L) D-MIMO system [16]. Let Portn be thenth Port, and the channel from the MS to Portn be characterized by an L×M matrix H(n)(d(n))which is a product of a large-scale fading coefficient hsh,n(d

(n)) and a small-scale fading matrix

H(n)SSF , hence H(n)(d(n)) = hsh,n(d

(n)) · H(n)SSF , where d(n) is the distance between MS and Portn.

A single data stream, or transmitted bit vector b, has components bk ∈ {0, 1}, k = 1, . . . ,MQ0

where Q0 = log2Q and Q is the symbol constellation size. The data stream b is demultiplexedinto M substreams, each mapped into symbol si, i = 1, . . . ,M and fed to its respective transmitantenna [2], forming the transmitted symbol vector s of dimension M. The detected bit vector at

Portn is b(n)

, and is of dimension MQ0. Furthermore, w(n) is a bit-level reliability information

vector at Portn, of dimension MQ0. Each port passes b(n)

and w(n) to the Fusion Center (FC)

where the output bit vectorˆb of dimension MQ0 is produced based on a bit level combining

scheme we proposed in Chapter 3. We assume the FC is part of the core network, and hence thelinks from the ports are capacity limited. For example in LTE networks, the interface betweenthe base station and the core network is referred to as the S1 interface. It is usually carriedeither over a high-speed copper or fiber cable, or alternatively over a high-speed microwave link,which are, for example, based on Ethernet [26].

We assume that the wireless channel is static during a symbol duration, and experiencesflat-fading. Furthermore, we also assume perfect CSI is available at all radio port receivers. Theoverall uplink received signal can be written as [16] [18]:

r = H(d)s + n (2.1)

where r = [r1, . . . , rNL]T is the received signal vector at all ports, s = [s1, . . . , sM ]T is the trans-mitted symbol vector with unit transmit power satisfying E(sHs) = 1. Assuming the transmit

power is equally divided among transmit antennas, E(ssH) = σ2sIM =

1

MIM . Furthermore,

n = [n1, . . . , nNL]T is the complex additive white Gaussian noise vector with E[n] = 0 and co-variance matrix E[nnH ] = σ2

nINL. H(d) is an NL×M composite channel matrix with influenceof small-scale fading as well as large-scale fading and d is an N × 1 distance vector with the nthelement indicats the distance between MS and Portn.

2 System and Channel Model of D-MIMO 5

MSb s

1

1

1

1

L

L

L

M

H(1)(d(1))

H(2)(d(2))

H(N)(d(N))

Port1

Port2

PortN

b(1)

b(2)

b(N)

w(1)

w(2)

w(N)

ˆb

Fusion

Center

Fig. 2.1 (M,N,L) Distributed-MIMO System Model

2.2 D-MIMO Channel Model

The D-MIMO channel matrix H(d) is parameterized by the distance vector d = [d(1), d(2), . . . ,d(N)]T with d(n) indicates the distances from MS to Portn. For the uplink, H(d) consists of Nindependent L × M subchannel matrices for each port [16]:

H(d) =[

H(1)(d(1)),H(2)(d(2)), . . . ,H(N)(d(N))]T

(2.2)

with H(n)(d(n)) = [h(n)1 (d(n)),h

(n)2 (d(n)), . . . ,h

(n)M (d(n))] and h(n)

m (d(n)) = [h(n)1m(d(n)), h

(n)2m(d(n)),

. . . , h(n)Lm(d(n))]T , where h

(n)lm (d(n)) is the complex channel gain from the mth mobile antenna to

the lth antenna of Portn.The composite fading channel matrix H(d) that models both small-scale fading and large-

scale fading, can be expressed as [17]:

H(d) = HSH(d)HSSF (2.3)

where HSSF is an NL×M matrix representing small-scale fading, and HSH(d) is an NL×NLmatrix representing large-scale fading.

2.2.1 Small Scale Fading

In [17], a model for small-scale fading with spatial correlation at the transmitter and receiverwas considered. The overall fading correlation can be partitioned into transmit fading correlationand receive fading correlation. The small-scale fading channel matrix HSSF can be expressed as:

HSSF = R1/2R HwR

1/2T (2.4)

where Hw is a NL × M random matrix with i.i.d. CN (0, 1) entries. The M × M transmitcorrelation matrix, according to the exponential correlation model and uniform linear arrays


employed at MS, is given by [27]:

RT =

1 t∗ . . . (t(M−1)2)∗

t 1 . . . (t(M−2)2)∗

......

. . ....

t(M−1)2

t(M−2)2 . . . 1

(2.5)

where t ∈ [0, 1] is fading correlation coefficient of neighboring transmit antennas, and (·)∗ denotescomplex conjugation. Since RT is Hermitian, it is diagonalizable by a unitary matrix U: RT =UDU−1 where D is a diagonal matrix consisting of the eigenvalues of RT , and hence:

R1/2T = UD1/2U−1 (2.6)

Same definition of matrix square root applies to receive correlation matrix RR.For D-MIMO receiver, we assume independent small-scale fading at different ports due to

large spacing between them, with small-scale fading within each port being correlated due toinsufficient spacing between antennas in a port. According to this assumption, the NL × NLreceive correlation matrix RR can be expressed as:

RR = diag(R(1)R ,R

(2)R , . . . ,R

(N)R ) (2.7)

where diag(·) denotes a block diagonal matrix, having main diagonal blocks square matrices,

R(1)R ,R

(2)R , . . . ,R

(N)R . The receive correlation matrix at Portn, R

(n)R , also follows the exponential

correlation model,

R(n)R =

1 (rn)∗ . . . (r(L−1)2

n )∗

rn 1 . . . (r(L−2)2

n )∗

......

. . ....

r(L−1)2

n r(L−2)2

n . . . 1

(2.8)

where rn is the fading correlation coefficient of neighboring receive antennas at Portn.From (2.4), (2.7), we get the equivalent small-scale fading matrix:

HSSF =

(R(1)R )1/2

(R(2)R )1/2

. . .

(R(N)R )1/2

H(1)w

H(2)w...

H(N)w

· (RT )1/2

=

(R(1)R )1/2H(1)

w (RT )1/2

(R(2)R )1/2H(2)

w (RT )1/2

...

(R(N)R )1/2H(N)

w (RT )1/2

(2.9)

where R(n)R is an L × L receive correlation matrix of Portn, H(n)

w is an L × M matrix with

i.i.d. CN (0, 1) entries, RT is an M × M transmit correlation matrix of MS and H(n)SSF =

(R(n)R )1/2H(n)

w (RT )1/2 is an L×M subchannel small scale fading matrix between MS and Portn,n = 1, 2, . . . , N .


2.2.2 Large Scale Fading

Large-scale fading is the effect of combined path loss and shadowing. Due to large spacingbetween Ports, we assume that shadow fading between the N base station ports and the MS areindependent, and antennas at the same port are experiencing same shadowing effect [15] [16].

Models for combined path loss and shadowing can be represented as a power decrease versusdistance along with the random attenuation due to shadowing. From [28], [29] the received signal

power at the nth base station P(n)R is given in terms of the transmitted signal power PT by:

P(n)R =

A

(d(n)

d0)τ

· φn · PT (2.10)

where A is the path gain at reference distance d0, d(n) is the distance between MS and Portn,

τ is the path-loss exponent that strongly depends on the base station antenna height and theterrain category [28] with typical values 3.7 − 6.5 in Urban macrocells [29]. Furthermore, φn isa log-normal shadow fading variable of Portn, with 10 log10 φn ∼ N (0, σ2

φdB). Empirical studies

support σφdBranging from 4dB to 13dB [29] for outdoor channels.

At 1.9GHz, A is close to the free-space path gain at d0 = 100m [28], and from [29]:

A =

(λ

4πd0

)2

(2.11)

where λ =c

fcis the wavelength in meters, c is the speed of light and fc is the carrier frequency.

Combining (2.10)-(2.11), we have:

P(n)R =

(c

fc·4·π·d0

)2

(d(n)

d0

)τ · φn · PT =

(c

fc · 4 · π

)2

· d(τ−2)0 · φn

[d(n)]τ· PT (2.12)

Define κ =

(c

fc · 4 · π

)2

· d(τ−2)0 , we have:

hsh,n(d(n)) =

√

κ · φn

[d(n)]τ(2.13)

where hsh,n(d(n)) represents the channel coefficient affected by shadowing and path loss. This

model applies to base station antenna heights from 10 to 80m, BS-to-MS distances from 0.1 to8 km [28]. Finally the channel matrix of large-scale fading is:

HSH = diag[hsh,1(d(1)) · IL, hsh,2(d

(2)) · IL, · · · , hsh,N(d(N)) · IL] (2.14)

where IL is an L-dimension identity matrix.From (2.2), (2.3), (2.9), (2.14), we can get the composite fading channel matrix H(d) as:

H(d) =[

H(1)(d(1)),H(2)(d(2)), . . . ,H(N)(d(N))]T

(2.15)

where

H(n)(d(n)) = hsh,n(d(n))[(R

(n)R )1/2H(n)

w (RT )1/2] = hsh,n(d(n))H

(n)SSF (2.16)

is the L × M subchannel matrix associated with the link from MS to Portn, n = 1, 2, . . . , N .

8

Chapter 3: Combining with Reliability Information

After receiving the detected bits from all Ports, the Fusion Center makes the final bit detectionby using bit combining aided with bit reliability information. In this chapter, we present analgorithm for bit combing and derive the bit reliability information. Furthermore, two MIMOdetection schemes, MMSE-OSIC and SD-ML are reviewed in this chapter.

3.1 Bit by Bit Combing at the Fusion Center

Let b(n)k be the kth detected bit at Portn, k = 1, . . . ,MQ0. Let p

(n)k be the probability that b

(n)k

is in error, assume p(n)k ≤ 1

2. Let b ∈ {0, 1}, bk ∈ {0, 1} be the kth transmitted bit. Define

dH(b(n)k , b) =

{

1 b(n)k 6= b

0 b(n)k = b

as the Hamming distance between b(n)k and b. Then:

P[

b(1)k , . . . , b

(N)k |bk = b

]

=N∏

n=1

P[

b(n)k |bk = b

]

=

N∏

n=1

[

p(n)k

]dH(b(n)k ,b)

·[

1 − p(n)k

]1−dH (b(n)k ,b)

=

N∏

n=1

[

1 − p(n)k

]

·N∏

n=1

[

p(n)k

1 − p(n)k

]dH(b(n)k ,b)

(3.1)

where the first equality is because b(1)k , . . . , b

(N)k are independent, the second equality is based on:

P[

b(n)k |bk = b

]

=

{

p(n)k b

(n)k 6= b

1 − p(n)k b

(n)k = b

=[

p(n)k

]dH(b(n)k ,b)

·[

1 − p(n)k

]1−dH(b(n)k ,b)

(3.2)

Taking the logarithm of (3.1), we have:

logP[

b(1)k , . . . , b

(N)k |bk = b

]

=N∑

n=1

log[

1 − p(n)k

]

−D[

{b(1)k , . . . , b(N)k }, b

]

(3.3)where

D[

{b(1)k , . . . , b(N)k }, b

]

=N∑

n=1

dH(b(n)k , b) · log

[

1 − p(n)k

p(n)k

]

. (3.4)

Since p(n)k ≤ 1

2, we have D

[

{b(1)k , . . . , b(N)k }, b

]

≥ 0.

After receiving the detected bits from all of the N ports, the Fusion Center will make the final

bit detectionˆbk corresponding to each transmitted bit bk by implementing the Maximum Likeli-

hood decision rule,ˆbk = arg max

b∈{0,1}P[

b(1)k , . . . , b

(N)k |bk = b

]

= arg minb∈{0,1}

D[

{b(1)k , . . . , b(N)k }, b

]

.

When there is a tie, thenˆbk = 1 with probability 1/2 and

ˆbk = 0 with probability 1/2. With

equal a-prior probabilities for bk, this will result in minimizing the probability of error.

Define ∆[

b(1)k , . . . , b

(N)k

]

= D[

{b(1)k , . . . , b(N)k }, 1

]

− D[

{b(1)k , . . . , b(N)k }, 0

]

. The decision rule

3 Combining with Reliability Information 9

will be: ∆[

b(1)k , . . . , b

(N)k

]

< 0,ˆbk = 1

> 0,ˆbk = 0

= 0,ˆbk = 1 with probability 1/2 andˆbk = 0 with probability 1/2

. From (3.4), we have:

∆[

b(1)k , . . . , b

(N)k

]

=

N∑

n=1

[

dH(b(n)k , 1) − dH(b

(n)k , 0)

]

log

[

1 − p(n)k

p(n)k

]

=

N∑

n=1

(−1)b(n)k log

[

1 − p(n)k

p(n)k

]

(3.5)Since p

(n)k are not available, (3.5) can not be used as is. Recognizing that log

[

1−p(n)k

p(n)k

]

can be

considered as reliability indication, we suggest the following class of decision rules:

∆[

b(1)k , . . . , b

(N)k

]

< 0,ˆbk = 1

> 0,ˆbk = 0

= 0,ˆbk = 1 with probability 1/2 andˆbk = 0 with probability 1/2

(3.6)

where∆[

b(1)k , . . . , b

(N)k

]

=

N∑

n=1

(−1)b(n)k w

(n)k (3.7)

and w(n)k are reliability informations with w

(n)k ≥ 0. The larger is w

(n)k the more reliable is b

(n)k .

3.2 Bit Reliability Information

Bit reliability information, depends on the type of MIMO detection employed at the Ports. Inthis work, we consider MMSE-OSIC as well as Sphere Decoding.

3.2.1 MMSE-OSIC Detection Scheme in MIMO

Compared with linear MMSE (Minimum Mean-Square Error) detection scheme in MIMO, thenon-linear detection scheme MMSE-OSIC improves the overall system performance by imple-menting successive interference cancellation (SIC) with optimized ordering (choosing the bestSNR at each iteration in the detection process [2]). Let the ordered set P = {p1, p2, . . . , pM}indicate the order in which an element of s is detected; hence pm corresponds to the indexof the symbol which has the highest SNR during the mth iteration. At the mth iteration,m = 0, 1, . . . ,M − 1, MMSE-OSIC detection is performed in three steps [3]:

Firstly: Calculate the MMSE filter, GM−m, at the mth iteration: GM−m = HM−mQM−m,where HM−m is a L×(M−m) matrix derived from H by removing columns p1, . . . , pm, QM−m =[

HHM−mHM−m +

1

γIM−m

]−1

with γ =σ2

s

σ2n

and IM−m is an (M −m)-dimension identity matrix.

Applying this MMSE filter to rm, the receive vector of dimension L derived by removing theeffects of all former m detected symbols as well as the effects of their corresponding channelsfrom r, we have the symbol estimate vector s with M −m elements at the mth iteration:

s = GHM−mrm =

s1

s2...sM−m

=

qHM−m,1H

HM−mrm

qHM−m,2H

HM−mrm

...qH

M−m,M−mHHM−mrm

, (3.8)


where si is the ith element of s and qM−m,j is the j th column vector of QM−m. During theinitialization stage (m = 0), we have HM = H and r0 = r.

Secondly: The element of s with the highest SNR is detected. In [3], this element isassociated with the smallest diagonal entry of QM−m: pm+1 = arg min

iqM−m,ii, where qM−m,ii are

the diagonal elements of matrix QM−m. We have the detected symbol:

spm+1 = Q[spm+1 ] = Q[qHM−m,pm+1

HHM−mrm] (3.9)

where Q[·] indicates the quantization procedure according to the signal constellation.Thirdly: We cancel the detected symbol spm+1 as well as its corresponding channel from the

receive vector rm, resulting in a new received vector for symbol detection of next (m+1 )th layer:

rm+1 = rm − spm+1hpm+1 (3.10)

Steps 1-3 are then performed for the (m+1 )th iteration with modified HM−(m+1), QM−(m+1)

and rm+1, till all the symbols sp1 , . . . , spMare detected.

The algorithm for V-BLAST with MMSE-OSIC [3] can be presented as:

Initialization (m=0):r0 = r, HM = H, QM = [HH

MHM + αIM ]−1

p1 = arg miniqM,ii, sp1 = qH

M,p1HH

Mr0, sp1 = Q[sp1 ]

Recursion: (m = 1, 2, . . . ,M − 1)Step 1. rm = rm−1 − spmhpm

Step 2. Determine HM−m by removing the pmth column from HM−m+1

Step 3. QM−m = [HHM−mHM−m + αIM−m]−1

Step 4. pm+1 = arg miniqM−m,ii

Step 5. spm+1 = qHM−m,pm+1

HHM−mrm

Step 6. spm+1 = Q[spm+1]

where α =σ2

n

σ2s

=1

γ. At Step 6. of each iteration, we consider spm+1 as the symbol estimate

corresponding to the pm+1th layer.

3.2.2 SD-ML Detection Scheme in MIMO

The Maximum Likelihood (ML) detection in V-BLAST MIMO system is: sML = arg mins∈Λ

‖r −Hs‖2, where Λ is a set of all possible transmit signal vector s, having the size QM . Because ofthis exponential complexity, ML is not realizable in practical systems with high modulation sizeQ and large number of transmit antennas M .

A new complex sphere decoding (SD) algorithm for signal detection in V-BLAST MIMOsystems has been introduced in [4]. The principle of SD is to search the closest lattice point tothe received signal r within a sphere of radius R centered at r,

‖r− Hs‖2 ≤ R2. (3.11)

With a good choice of initial sphere radius R, the size for the set of possible s will be reduced sothat SD can achieve approximate ML performance with significantly lower complexity.

Solving (3.11) is equivalent to solving the inequality:∥∥∥∥∥r −

M∑

i=1

hisi

∥∥∥∥∥

2

≤ R2 ⇒L∑

j=1

∣∣∣∣∣rj −

M∑

i=1

hjisi

∣∣∣∣∣

2

≤ R2. (3.12)


Due to the general structure of H, (3.12) is an inequality with summation of L terms, and eachterm consists of M unknown parameters si, it is shown to be a nondeterministic polynomial-time(NP)-hard problem [4]. According to [5], ‖r − Hs‖2 can be decomposed as:

‖r −Hs‖2 = (s− s)HHHH(s− s) + rH(I − H(HHH)−1HH

)r (3.13)

where s is the unconstrained ML estimate of s: s = (HHH)−1HHr. Therefore, the SD detectionis equivalent to:

(s − s)HHHH(s− s) ≤ R2 (3.14)

Since Gramian matrix HHH is Hermitian and positive-definite, and applying Cholesky factor-ization, we have HHH = UHU, where U is an upper triangular M ×M matrix with positivediagonal entries. (Compared with QR factorization which needs M3 flops, Cholesky factorizationneeds only M3/3 flops [30]). Hence (3.14) can be written as:

(s − s)HUHU(s − s) = ‖U(s − s)‖2 =

∥∥∥∥∥∥∥∥∥

u11 u12 · · · u1M

0 u22 · · · u2M...

.... . .

...0 0 · · · uMM

s1 − s1

s2 − s2...

sM − sM

∥∥∥∥∥∥∥∥∥

2

≤ R2 (3.15)

Using the property of upper triangular matrix U, (3.15) implies the set of conditions:

M∑

k=m

∣∣∣∣∣

M∑

j=k

ukj(sj − sj)

∣∣∣∣∣

2

≤ R2 m = 1, . . . ,M (3.16)

Consider above conditions in the order from M to 1, we successively obtain the set of candidatesfor sm, symbol of mth layer, with back-substitution of given symbols sM , . . . , sm+1. Followingsare the detail processes:

Starting from layer m = M and only considering the bottom term of the Euclidean norm

in (3.15), the corresponding condition in (3.16) is: |sM − sM |2 6R2

u2MM

. A complex disk for

the M th layer DiskM is formed, with center sM = sM and radius RM = R2

u2MM

. Let A be the

set of whole constellation points. By searching the constellation points within the DiskM , wehave the set of candidates for sM : SM,cand = {a : a ∈ A and a within DiskM} and let thecandidate which is closest to the center sM be the temporary symbol detection of the M th layer:sM = arg min

a′∈SM,cand

|a′ − sM |2.Go to the upper layer m = M − 1 and consider the bottom two terms of the Euclidean norm

in (3.15), the implied condition in (3.16) will be:

M∑

k=M−1

∣∣∣∣∣

M∑

j=k

ukj(sj − sj)

∣∣∣∣∣

2

≤ R2

=> |uM−1,M−1(sM−1 − sM−1) + uM−1,M(sM − sM)|2 + u2M,M |sM − sM |2 6 R2

=>

∣∣∣∣sM−1 − sM−1 +

uM−1,M

uM−1,M−1(sM − sM)

∣∣∣∣

2

61

u2M−1,M−1

[R2 − u2M,M |sM − sM |2]

=> |sM−1 − sM−1|2 6 RM−1 (3.17)

With back-substitution of sM , we have DiskM−1, a complex disk for the (M-1 )th layer, with cen-


ter sM−1 = sM−1−uM−1,M

uM−1,M−1(sM−sM) and radius RM−1 =

1

u2M−1,M−1

[R2−u2M,M |sM−sM |2]. Sim-

ilarly, we get the set of the candidates for sM−1: SM−1,cand = {a : a ∈ A and a within DiskM−1}and the temporary symbol detection of the (M-1 )th layer: sM−1 = arg min

a′∈SM−1,cand

|a′ − sM−1|2.In general, at layer m = i, considering the bottom M − i+ 1 terms of the Euclidean norm in

(3.15), the corresponding condition in (3.16) becomes:

M∑

k=i

∣∣∣∣∣

M∑

j=k

ukj(sj − sj)

∣∣∣∣∣

2

≤ R2

=>

∣∣∣∣∣uii(si − si) +

M∑

j=i+1

uij(sj − sj)

∣∣∣∣∣

2

+

∣∣∣∣∣ui+1,i+1(si+1 − si+1) +

M∑

j=i+2

ui+1,j(sj − sj)

∣∣∣∣∣

2

+ · · ·+ u2MM |sM − sM |2 6 R2

=> u2ii|si − si|2 + u2

i+1,i+1|si+1 − si+1|2 + · · ·+ u2MM |sM − sM |2 6 R2

=> |si − si|2 ≤1

u2ii

[

R2 −M∑

j=i+1

(ujj|sj − sj |)2

]

=> |si − si|2 ≤ Ri (3.18)

where sp = sp −M∑

q=p+1

upq

upp(sq − sq) p = i, i+ 1, . . . ,M (3.19)

andRi =

1

u2ii

[

R2 −M∑

j=i+1

(ujj|sj − sj |)2

]

(3.20)

with back-substitution of previously determined sj , j = M,M − 1, . . . , i + 1, a complex diskDiski of radius Ri (3.20) centered at si (3.19) is specified. The set of candidates for si isSi,cand = {a : a ∈ A and a within Diski} and the temporary symbol detection of ith layer issi = arg min

a′∈Si,cand

|a′ − si|2.The process is continued to (i−1)th layer and so on. At last, two possible things will happen:

1. The leaf node is reached (s1 is found), a feasible solution s is obtained. The new radius iscalculated: Rnew = ‖U(s − s)‖2, satisfying Rnew ≤ R2. Then re-do the search for betters with Rnew. In [6], a new algorithm of ’best-first’ search strategy was proposed to reduceSD complexity. In this new algorithm, the search resumes with the path that is closest tocompletion rather than restarting the search from the root node.

2. There is no leaf node reached at all, which means the initial radius R is too small so thatno feasible s can be found within the hypersphere. In such case, the initial radius R shouldbe increased and the search should be restart from root node again.

The center of a complex disk at ith layer, si of (3.19), is the symbol estimate of ith layer.Such SD-ML detection process can be performed by using a tree structure with M layers

as illustrated in Fig. 3.1. SD-ML starts searching from root of the tree and works down alongthe branch till the leaf node is reached. The nodes on the ith layer stands for Si,cand, the setof the candidates for si, which can be obtained from the bound (3.18) with back-substitution ofpreviously detected symbols. The nodes on the branch from the root to leaf with the smallestradius are chosen to be the symbol vector solution s.


root node

leaf nodeM=4

(first layer)m = M − 3

m = M − 2

m = M − 1

m = M

Fig. 3.1 Tree Structure for Sphere Decoding with detection layer M = 4

The algorithm of SD [6] consists of the following steps with index i from 0 to M − 1:Step 1. Set the initial radius R, set flag = 0, compute unconstrained ML estimate s.Step 2. Initialize i = M , D = 0 and the candidate sets Sk,cand = ∅, k=0,. . . ,M-1. Set symboldetection vector s = 0 and symbol estimate vector s = 0. If flag = 1, set R = 1.2R.Step 3. Set i = i − 1, compute the symbol estimate si using (3.19) and obtain the candidateset for si within the range defined by (3.18).Step 4. If candidates for symbol si are found, goto step 7. Otherwise, goto step 5.Step 5. If the candidate sets for all symbols are nonempty Sk,cand 6= ∅, k = 0, . . . ,M − 1, gotostep 12. Otherwise, goto step 6.Step 6. If no solution has been found yet, set flag = 1 and goto step 2. Otherwise, output the”current-best” solution.Step 7. For each candidate of si found in Step 3, compute the corresponding distances δ =|(candidate of si)− si|2. Denote the smallest δ as δi, arrange these δ in increasing order and storethem in candidate set Si,cand (excluding δi). Set si equal to the candidate associated with δi.Step 8. Set D = D + u2

iiδi. If D < R, goto step 10. Otherwise, goto step 9.Step 9. If the candidate sets for all symbols are nonempty Sk,cand 6= ∅, k = 0, . . . ,M − 1, gotostep 12. Otherwise, goto step 6.Step 10. If i > 0, goto Step 3.Step 11. If i = 0, update the ’current-best’ solution with s, update the ’current-best’ symbolestimate with s and let R = D. If the candidate sets for all symbols are nonempty Sk,cand 6= ∅,k = 0, . . . ,M − 1, goto step 12. Otherwise, output the ”current-best” solution.Step 12. Select the nonempty candidate set closest to the last tree level and set i equal tothis level. Select the candidate with the smallest distance δ in set Si,cand, set δi equal to δcorresponding to this candidate and update si equal to this candidate, remove the first elementin set Si,cand and goto Step 8.

Compared with ML detector, which needs to perform exhaust search among all possibletransmitted symbol vector of size QM , the SD algorithm achieves approximate ML performancewith much lower complexity by only searching the points that lie within a hypersphere of radiusR around r. With a good choise of R, the roughly complexity of the algorithm is O(M3) [5].


3.2.3 Bit Level Reliability Information

Consider a signal space constellation composed of the set of signal points A = {a1, . . . , aQ} anda mapping of Q0 information bits b1, . . . , bQ0 into the constellation points A where Q0 = log2Q.

Here, we consider quadrature amplitude modulation (QAM) with Gray mapping. Define A(0)i as

the set of constellation points for which bi = 0. Similarly, define A(1)i as the set of constellation

points for which bi = 1. Then for i = 1, 2, . . . , Q0, we have: A(0)i

⋂A(1)i = ∅ and A(0)

i

⋃A(1)i = A,

with |A(0)i | = |A(1)

i | = Q/2, where | · | denotes the cardinality of the set.Let bk be the kth transmitted bit and s be the symbol estimate. The a posteriori probability

ratio for the kth bit is: Λk =Pr{bk = 1|s}Pr{bk = 0|s} . According to Bayes Rule:

Pr{bk|s}Pr{s}Pr{bk}

=

Pr{s|bk}. With equally likely bk, we have: Λk =Pr{s|bk = 1}Pr{s|bk = 0} .

The reliability metrics (soft output) for bk are calculated in the form of logarithmic likelihoodratios (LLR) [22] [23]:

Lk = log Λk = logPr{s|bk = 1}Pr{s|bk = 0} = log

∑

a∈A(1)kPr{s|sk = a}

∑

a∈A(0)kPr{s|sk = a} . (3.21)

For an AWGN channel, (3.21) will be: Lk = log

∑

a∈A(1)k

exp(−γ|s− a|2)∑

a∈A(0)k

exp(−γ|s− a|2) [23][24], where γ is

the average SNR per symbol.

Applying the so-called max-log approximation [25]: log∑

j

exp(αj) ≈ maxj

(αj), the logarith-

mic likelihood ratio can be simplified as [23][24]:

Lk = γ

[

mina∈A

(0)k

{|s− a|2} − mina∈A

(1)k

{|s− a|2}]

(3.22)

In an M × L V-BLAST MIMO system, the symbol covariance matrix at the receiver can be

calculated as: E[(Hs)(Hs)H

]=

M∑

i=1

hihHi · σ2

si, since E(sis

∗j) =

{σ2

sii = j

0 i 6= j. Let ‖hi‖2 · σ2

si

be the receive power of symbol si. Since the transmit power is equally divided among transmit

antennas σsi=PT

Mand the AWGN noise on each channel has the same power σ2

n, the average

receive SNR for symbol si will be:γi ∝ ‖hi‖2 (3.23)

and it is the same for all bits associated with the same symbol.Due to the positive requirement of the reliability information, based on (3.22), we could

take

∣∣∣∣∣min

a∈A(0)k

{|s− a|2} − mina∈A

(1)k

{|s− a|2}∣∣∣∣∣

as the reliability weight for the kth bit. Let ak,j =

arg mina∈A

(j)k

|s− a| with j = 0, 1. We have:∣∣∣∣∣min

a∈A(0)k

{|s− a|2} − mina∈A

(1)k

{|s− a|2}∣∣∣∣∣=∣∣|s− ak,0|2 − |s− ak,1|2

∣∣

=∣∣|s− ak,0| − |s− ak,1|

∣∣ · (|s− ak,0| + |s− ak,1|) = |lk,0 − lk,1| · (lk,0 + lk,1) (3.24)


where lk,j = |s− ak,j| indicating the distance between s and ak,j. The larger the value of (3.24),the more reliable the detection of the kth bit is.

Consider the Voronoi region associated with signal point aq, denoted as V(aq), defined as theset of all points in the signal space that are closer, in terms of Euclidean distance, to aq than toany other signal point ak, k 6= q. Fig. 3.2 illustrates example of the Voronoi regions for 16QAM.

Suppose a is a transmit symbol, then a symbol estimate can be modeled as: s = a + n. Inhigh SNR condition, the noise n is small, symbol estimate s is likely to be in the Voronoi regionassociated with transmit symbol a, s ∈ V(a), then a is the signal point closest to s and eitherak,0 or ak,1 is a. Assume for example, that the k -th bit of symbol a is 0, as in Fig.3.3. s willbe close to a, so |s− a| is very small indicating that lk,0 is very small and lk,1 is large, see Fig.3.3(a). In low SNR condition, the noise n is large, symbol estimate s will be far away fromtransmit symbol a and will fall in the Voronoi region associated with another signal point a′,s ∈ V(a′). Considering that errors happen in the nearest neighbor will play a dominant role inthe performance, we assume that a′ is the signal point neibouring to transmit symbol a, and swill be closer to symbol a′ than to symbol a, implying that lk,0 is large. In this case lk,1 is morelikely to be large since it can be small only if s is very close to a′, see Fig. 3.3(b).

If we take |lk,0 − lk,1| · (lk,0 + lk,1), then (lk,0 + lk,1) will increase the bit reliability even when sis far from symbol a, reducing the quality of bit reliability information. Hence we only considerthe absolute difference part in (3.24), and define the reliability weight for kth bit of ith layersymbol si as: ∣

∣∣∣∣min

a∈A(0)k

{|si − a|} − mina∈A

(1)k

{|si − a|}∣∣∣∣∣. (3.25)

Simulation results in Chapter 4. (Fig. 4.2 ∼ Fig. 4.7) confirm the improvement of using (3.25)as reliability information.

With (3.23)(3.25), we define the kth bit reliability information corresponding to symbol si atPortn in D-MIMO as:

w(n)k,i = (‖h(n)

i ‖2)f ·∣∣∣∣∣min

a∈A(0)k

{|s(n)i − a|} − min

a∈A(1)k

{|s(n)i − a|}

∣∣∣∣∣

g

(3.26)

where s(n)i is the ith layer symbol estimate at Portn, i = 1, . . . ,M , k = 1, . . . , Q0 and n =

1, . . . , N . Since bit reliability information is composed of two factors, symbol level SNR andbit level reliability weight. By using parameters f and g, which can assume the values of 0,1,it is easier to find which factor has more significant influence on the performance of D-MIMOcorresponding to different MIMO detections, MMSE-OSIC and SD-ML.

The reliability information vector at Portn is:

w(n) = [w(n)1,1 , . . . , w

(n)Q0,1, w

(n)1,2 , . . . , w

(n)Q0,2, . . . , w

(n)1,M , . . . , w

(n)Q0,M ]T (3.27)

Applying the reliability information w(n)k , the kth element of w(n), to the bit level combing

scheme (3.6) and (3.7), final bit detection of D-MIMO,ˆbk, is obtained on the Fusion Center.

In Chapter 4, we present Monte-Carlo simulation results for the BER performance of D-MIMOimplemented with such bit level combing scheme.


V (0)

V (1)

V (2)

V (3)

V (4)

V (5)

V (6)

V (7)

V (8)

V (9)

V (10)

V (11)

V (12)

V (13)

V (14)

V (15)

Fig. 3.2 Voronoi Regions for Gray-coded 16-QAM

0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

s

a

lk,0lk,1

(a) Case of small noise

0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

s

a a′

lk,0lk,1

(b) Case of large noise

Fig. 3.3 Bit reliability metric with transmit symbol a = 5 and k = 1 for 16QAM

17

Chapter 4: Computer Simulation Results

We present computer simulation results for bit error rate (BER) performance of a (4,2,4)

D-MIMO system and a 4×4 C-MIMO system. BER is defined as: BER=1

MQ0.

MQ0∑

k=1

Pr(ˆbk 6= bk)

and it is plotted versusEb

No=

σ2s

Q0σ2n

where Eb is transmit bit power. Monte-Carlo simulations

are used to obtain Pr(ˆbk 6= bk).

Two detection schemes are considered: MMSE-OSIC [3] and SD-ML [5] [6]. The consideredmodulation schemes include QPSK, 16QAM and 64QAM, all with Gray code mapping havingthe property that two adjacent symbols differ in only one single bit [29] [31]. We consider also32QAM with quasi-Gray code mapping where some symbols adjacent may not differ only in onebit. The construction of 32QAM is done by first forming a 4×8 rectangular 32QAM constellationwith Gray mapping. Then the last columns of symbols on the far left and far right are movedto the top and bottom [32]. This construction is shown in Fig 4.1 where arrows indicate movingdirections.

For each channel use, a new transmitted symbol vector of dimension M is randomly generatedand each symbol consists of Q0 binary bits. The composite fading channel (2.15) varies randomlyand independently from one use to another. The small scale fading channel matrix (2.9) isconsidered. In case of no space correlation at the transmitter and the reciever, each element ofthe small scale fading channel matrix is generated as i.i.d complex Gaussian random variablewith zero mean and unit variance. The large scale fading channel coefficient (2.13) is a functionof d(n) with parameters: fc = 1.9GHz, d0 = 100m, τ = 4 and σφdB

= 4. In addition, at eachSNR, the performance is averaged over at least 1 million channel realizations and a minimumof 500 frame errors are accumulated. A frame error is defined as the event ˆs 6= s where ˆs is thedetected symbol vector at the Fusion Center.

4.1 Simulations with Two Reliability Information Scheme

In this section, we present simulation results to show the effect of the reliability informationscheme on BER performance of D-MIMO. Two reliability information schemes are considered:reliability information with absolute value (3.25) and reliability information with absolute squarevalue (3.24). The considered modulation schemes are 16QAM, 32QAM and 64QAM. For eachmodulation scheme, parameter settings for the reliability information with f = 1, g = 1 andf = 0, g = 1 are considered. Two types of D-MIMO systems are considered: a (4,2,4) D-MIMOwith d(1) = d(2) = 1km and a (4,2,4) D-MIMO with d(1) = 1km, d(2) = 1.2km.

From Fig.4.2 ∼ Fig.4.7 we see that for all modulation schemes, the performance correspondingto reliability information based on absolute value is better than that based on absolute squarevalue. For SD-ML, the performance gain with reliability parameter settings f = 1, g = 1 (Fig4.2, Fig 4.4, Fig 4.6) is larger than that with f = 0, g = 1 (Fig 4.3, Fig 4.5, Fig 4.7), while forMMSE-OSIC, the performance gain with reliability parameter settings f = 0, g = 1 (Fig 4.3, Fig

4 Computer Simulation Results 18

0000000000

00001

00001

00010

00010

00011

00011

00100

00101

00110

00111

01000

01001

01010

01011

01100

01101

01110

01111

1000010000

10001

10001

10010

10010

10011

10011

10100

10101

10110

10111

11000

11001

11010

11011

11100

11101

11110

11111

Fig. 4.1 Construction of 32QAM from rectangular 32QAM

4.5, Fig 4.7) is larger than that with f = 1, g = 1 (Fig 4.2, Fig 4.4, Fig 4.6). At BER= 10−5,for SD-ML with reliability parameter settings f = 1, g = 1, the performance gain of reliabilityinformation with absolute value over that with absolute square value is 0.8dB, 1.0dB, 1.2dBfor 16QAM, 32QAM and 64QAM respectively, and for MMSE-OSIC with reliability parametersettings f = 0, g = 1, the performance gain of reliability information with absolute value over thatwith absolute square value is around 6.4dB, 5.3dB and 5.8dB for 16QAM, 32QAM and 64QAMrespectively. This varifies that our improvement of the reliability information scheme proposedin Chapter 3 provides gains, and this improvement has more significant effect on MMSE-OSICdetection scheme.

4.2 Simulations with Perfect Channel Information

In this Section, we present simulation results of BER performance for uncorrelated D-MIMO andC-MIMO with perfect CSI. Fig 4.8, Fig 4.11, Fig 4.14, Fig 4.17 show the BER performance withreliability parameter settings f = 1, g = 1 for QPSK, 16QAM, 32QAM and 64QAM respectively.Fig 4.9, Fig 4.12, Fig 4.15, Fig 4.18 show the BER performance with reliability parameter settingsf = 1, g = 0 for QPSK, 16QAM, 32QAM and 64QAM respectively. Fig 4.10, Fig 4.13, Fig 4.16,Fig 4.19 show the BER performance with reliability parameter settings f = 0, g = 1 for QPSK,16QAM, 32QAM and 64QAM respectively. In all schemes, we present BER performance forthe following two kinds of D-MIMO systems: a (4,2,4) D-MIMO with d(1) = d(2) = 1km and a(4,2,4) D-MIMO with d(1) = 1km, d(2) = 1.2km, we also present results for two kinds of C-MIMOsystems: a 4 × 4 C-MIMO with d = 1km and a 4 × 4 C-MIMO with d = 1.2km.

We see that reliability parameter settings f and g play an important role on the performanceof D-MIMO with different detection schemes. According to Fig 4.9, Fig 4.12, Fig 4.15, Fig4.18, we see that for SD-ML with reliability parameter settings f = 1, g = 0, D-MIMO withd(1) = 1km, d(2) = 1.2km not only outperforms C-MIMO with d = 1.2km but also outperforms


C-MIMO with d = 1km; according to Fig 4.10, Fig 4.13, Fig 4.16, Fig 4.19, we see that forSD-ML with reliability parameter settings f = 0, g = 1, the performance of D-MIMO withd(1) = 1km, d(2) = 1.2km is better than that of C-MIMO with d = 1.2km, but it is worse thanthat of C-MIMO with d = 1km. Furthermore, for all modulation schemes, the performancegains of D-MIMO with d(1) = d(2) = 1km over C-MIMO with d = 1km for SD-ML detection withf = 1, g = 0 are much larger than that with f = 0, g = 1. So for D-MIMO with SD-ML, theperformances with reliability parameter setting f = 1, g = 0 are much better than those withreliability parameter settings f = 0, g = 1. We also see that for all modulation schemes, theperformances with reliability parameter setting f = 1, g = 0 (Fig 4.9, Fig 4.12, Fig 4.15, Fig4.18) are similar to those with reliability parameter settings f = 1, g = 1 (Fig 4.8, Fig 4.11, Fig4.14, Fig 4.17). This implies that for D-MIMO with SD-ML detection in all modulation schemes,the effect of symbol level SNR has more significant effect on the performance.

According to Fig 4.10, Fig 4.13, Fig 4.16, Fig 4.19, we see that for MMSE-OSIC with reliabilityparameter settings f = 0, g = 1, D-MIMO with d(1) = 1km, d(2) = 1.2km has significant betterperformance than both C-MIMO with d = 1.2km and C-MIMO with d = 1km; according toFig 4.9, Fig 4.12, Fig 4.15, Fig 4.18, we see that for MMSE-OSIC with reliability parametersettings f = 1, g = 0, D-MIMO with d(1) = 1km, d(2) = 1.2km still provide some improvementover C-MIMO with d = 1.2km, but it doesn’t outperform C-MIMO with d = 1km significantly.Furthermore, the performance gains of D-MIMO with d(1) = d(2) = 1km over C-MIMO withd = 1km for MMSE-OSIC detection with f = 0, g = 1 are much larger than that with f =1, g = 0. So for D-MIMO with MMSE-OSIC, the performances with reliability parameter settingf = 0, g = 1 are much better than those with reliability parameter settings f = 1, g = 0. Wealso see that for all modulation schemes, the performances with reliability parameter settingsf = 0, g = 1 (Fig 4.10, Fig 4.13, Fig 4.16, Fig 4.19) are a bit better than those with reliability

parameter settings f = 1, g = 1 (Fig 4.8, Fig 4.11, Fig 4.14, Fig 4.17) in the region of highEb

No

.

This implies that for D-MIMO with MMSE-OSIC detection the bit level reliability weight hasmore significant effect on the performance.

At BER= 10−5, the performance gains of a (4,2,4) D-MIMO over a 4 × 4 C-MIMO for SD-ML detection with reliability parameter settings f = 1, g = 0 (Fig 4.9, Fig 4.12, Fig 4.15, Fig4.18) corresponding to different modulation schemes are presented in Table 4.1. We see that theperformance gain for QPSK is a bit higher compared with the other three modulation schemes,and the performance gains for 16QAM, 32QAM and 64QAM are almost same which indicatesthat the performance improvment for D-MIMO with SD-ML is maintained as modulation sizeincreases. At BER= 10−5, the performance gains of a (4,2,4) D-MIMO over a 4 × 4 C-MIMOfor MMSE-OSIC with reliability parameter settings f = 0, g = 1 (Fig 4.10, Fig 4.13, Fig 4.16,Fig 4.19) corresponding to different modulation schemes are presented in Table 4.2. We see thatthe performance gain for QPSK is smaller compared with the other three modulation schemes.The performance gain for 16QAM is larger than that for 32QAM, and the performance gain for32QAM is larger than that for 64QAM. This indicates that the performance improvement forD-MIMO with MMSE-OSIC decreases when modulation size increases.

According to Table 4.1 and Table 4.2, we see that for both detection schemes, the performanceimprovement of D-MIMO with d(1) = 1km, d(2) = 1.2km over C-MIMO with d = 1.2km is greaterthan that of D-MIMO with d(1) = d(2) = 1km over C-MIMO with d = 1km, and the performanceimprovement of D-MIMO with d(1) = 1km, d(2) = 1.2km over C-MIMO with d = 1km is less


QPSK 16QAM 32QAM 64QAM

D-MIMO d(1) = d(2) = 1kmvs. 5.2dB 4.7dB 4.5dB 4.5dB

C-MIMO d = 1km

D-MIMO d(1) = 1km d(2) = 1.2kmvs. 3.6dB 3.1dB 3.1dB 2.9dB

C-MIMO d = 1km


C-MIMO d = 1.2km

Table 4.1 The performance gain of D-MIMO over C-MIMO for SD-ML withf = 1, g = 0 at BER= 10−5

QPSK 16QAM 32QAM 64QAM

D-MIMO d(1) = d(2) = 1kmvs. 5.1dB 6.7dB 6.5dB 5.7dB

C-MIMO d = 1km


C-MIMO d = 1km


C-MIMO d = 1.2km

Table 4.2 The performance gain of D-MIMO over C-MIMO for MMSE-OSICwith f = 0, g = 1 at BER= 10−5

than that of D-MIMO with d(1) = d(2) = 1km over C-MIMO with d = 1km. Furthermore, for16QAM, 32QAM and 64QAM, the performance improvement of D-MIMO over C-MIMO forMMSE-OSIC with f = 0, g = 1 is larger than that for SD-ML with f = 1, g = 0.

4.3 Impact of Channel Estimation Errors

In this section, we consider the effect of channel estimation errors on the performance of D-

MIMO. Assume an L×M channel estimation matrix H(n)

, associated with the link between MSand Portn, as modeled in [33]:

H(n)

= H(n) + H(n)

(4.1)

where H(n) is the true channel matrix from MS to Portn, H(n)

is an L×M error matrix associatedwith Portn which is independent of H(n) and with i.i.d. CN (0, γ2σ2

n) entries, and γ2 is a constantindicating the relative error power with respect to the channel noise.

We consider two kinds of D-MIMO systems: a (4,2,4) D-MIMO with d(1) = d(2) = 1kmand a (4,2,4) D-MIMO with d(1) = 1km, d(2) = 1.2km, we also consider two kinds of C-MIMOsystems: a 4×4 C-MIMO with d = 1km and a 4×4 C-MIMO with d = 1.2km. Two modulationschemes are considered: 16QAM and 64QAM, both with Gray mapping. Two detection schemesare considered: SD-ML with reliability parameter settings f = 1, g = 0 and MMSE-OSIC with


reliability parameter settings f = 0, g = 1.Fig 4.20 ∼ Fig 4.27 present the BER performance with different values of γ2 for SD-ML and

MMSE-OSIC respectively. We observe that for all schemes, the performance improvement of D-MIMO over C-MIMO is maintained. The channel estimation error results in some degradationsin performance, which increases with γ2, and the degradations with a given channel estimationerror level γ2 are almost same for all modulation schemes. With SD-ML (Fig 4.20 ∼ Fig 4.23),at BER=10−5, the degradations are about 3dB for γ2 = 1 and about 10.3dB for γ2 = 10. WithMMSE-OSIC (Fig 4.24 ∼ Fig 4.27), at BER=10−5, the degradations are about 3dB for γ2 = 1and about 13dB for γ2 = 10.

4.4 Impact of Channel Spatial Correlation

In this section, we investigate the impact of channel spatial correlation on the performance ofD-MIMO. We consider a (4,2,4) D-MIMO system with d(1) = d(2) = 1km and a 4 × 4 C-MIMOwith d = 1km. The detection scheme is SD-ML with f = 1, g = 0. Three situations forspatial correlation are considered: spatial correlation at transmitter only, spatial correlation atreceiver only and spatial correlation at both transmitter and receiver. The channel model forspatial correlation employed is decribed in Chapter 2 (2.16), with R

(n)R and RT indicating the

receive correlation matrix at Portn (2.8) and transmit correlation matrix (2.5) respectively. Twomodulation schemes are considered: 16QAM and 64QAM, both with Gray mapping. We foundthat for both modulation schemes, performance with spatial correlation at receiver only is veryclose to that with spatial correlation at transmitter only, hence we do not include the resultswith spatial correlation at receiver only.

Fig. 4.28 ∼ Fig. 4.29 present the BER performance with correlation coefficient of 0.6 for16QAM and 64QAM respectively. We see that for both 16QAM and 64QAM, performanceimprovement of D-MIMO over C-MIMO is maintained. At BER=10−5, the improvement isaround 4.6dB for no space correlation, 4.2dB for space correlation at transmitter only and 3.8dBfor space correlation at both transmitter and receiver. We also observe that for both 16QAM and64QAM channel spatial correlation results in some degradations in performance, that increasesas space correlation increases.

4.5 Comparision between a (M,N,L) D-MIMO system and a M ×NL

C-MIMO system

In this section, we present simulation results for a (4,2,4) D-MIMO system with d(1) = d(2) = 1kmand a 4 × 8 C-MIMO system with d = 1km. Hence both systems have the same number oftransmit and receive antennas. We consider 16QAM with Gray code mapping and SD-MLdetection with f = 1, g = 0. The correlation coefficient in (2.5) and (2.8) in a given wave-field[34] is approximated as,

r(d) ≈ exp(−23Λ2d2) (4.2)

where Λ is the angular spread according to [34], d is the distance in wavelengths between twoantennas [27]. Assume uniform linear arrays at receiver with equidistant antenna spacing. Thenfor a 4× 8 C-MIMO system,using (4.2), we have when doubling the number of receive antennas


in the same space,r′ = exp(−23Λ2 · (d2/4)) = r

1/4. (4.3)

Fig(4.30) presents the BER performance for a (4,2,4) D-MIMO and a 4 × 8 C-MIMO withand without space correlation, where the same correlations at transmitter and receiver for bothD-MIMO and C-MIMO are considered. We see that the performance of an uncorrelated 4×8 C-MIMO is better than an uncorrelated (4,2,4) D-MIMO and the difference is 2.1dB at BER= 10−5.Performance for both C-MIMO and D-MIMO becomes worse as spatial correlation increases.When there is same spatial correlation at both transmitter and receiver, the performance ofC-MIMO is better than that of D-MIMO and the difference increases as spatial correlationincreases. At BER= 10−5, the difference is 2.5dB when t = r = 0.3, 2.92dB when t = r = 0.5 and3.75dB when t = r = 0.8. In this case, the detection scheme for a 4 × 8 C-MIMO is equivalentto implement a joint symbol detection of eight receive antennas from two base stations, whichneeds perfect channel state information (CSI) of two base stations together. According to thesystem assumption in Chapter 2, each base station only has its own perfect CSI. Hence, in orderto get perfect CSI for both base stations, connections between two basestations are requiredto exchange their CSI. However, such operations lead to large amount of information transferand increase the complexity and load of receiver. Hence, for case of same space correlation atreceiver, although the performance of a M × NL C-MIMO is better than a (M,N,L) D-MIMO,more complexity and heavier load are required at receiver of C-MIMO.

Fig(4.31) presents the BER performance for a (4,2,4) D-MIMO and a 4×8 C-MIMO with andwithout space correlation, where the same space restrictions at the receiver of C-MIMO and theradio ports of D-MIMO are considered. In this case, the detection scheme for a 4 × 8 C-MIMOis equivalent to implement a joint symbol detection of eight receiver antennas amounted on onebase station, which has the same space restriction to that of D-MIMO. Hence, space correlationat receiver is higher for a 4 × 8 C-MIMO (4.3) than for a (4,2,4) D-MIMO. With same spacecorrelation at transmitter for both 4 × 8 C-MIMO and (4,2,4) D-MIMO, the performance ofC-MIMO is a bit better than that of D-MIMO and the difference decreases as spatial correlationincreases. At BER= 10−5, the difference between C-MIMO with t = 0.3, r = 0.74 and D-MIMO with t = r = 0.3 is 0.43dB, the difference between C-MIMO with t = 0.5, r = 0.84and D-MIMO with t = r = 0.5 is 0.26dB and the performance of D-MIMO with t = r = 0.8is slightly better than that of C-MIMO with t = 0.8, r = 0.95. Simulation results verify thatin realistic systems with restricted space at receiver, space correlation among antennas is animportant factor. The proposed bit combining scheme for a (M,N,L) D-MIMO system providesa comparable performance to a M ×NL C-MIMO system with complexity suitable for practicalapplications, especially when space correlation increases.


120 130 140 150 160 17010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

DMIMO d(1)=1km d(2)=1km

DMIMO d(1)=1km d(2)=1.2kmabsolute valueabsolute square value

MMSE−OSIC

SD−ML

Fig. 4.2 Absolute vs Absolute square for DMIMO(4,2,4) with 16QAM (f=1,g=1),where Eb is transmit bit power


120 130 140 150 160 17010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R



SD−ML

MMSE−OSIC



120 130 140 150 160 17010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R



SD−ML

MMSE−OSIC



120 130 140 150 160 17010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R



SD−ML

MMSE−OSIC



120 130 140 150 160 170 18010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R



MMSE−OSIC

SD−ML



120 130 140 150 160 170 18010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R



MMSE−OSIC

SD−ML



100 110 120 130 140 15010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO d=1kmDMIMOd(1)=1km,d(2)=1km

CMIMO d=1.2kmDMIMOd(1)=1km,d(2)=1.2kmSD−MLMMSE−OSIC

Fig. 4.8 DMIMO(4,2,4) and 4 × 4 CMIMO with QPSK (f=1,g=1), where Eb istransmit bit power


100 110 120 130 140 15010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO d=1kmDMIMOd(1)=1km,d(2)=1kmCMIMO d=1.2kmDMIMOd(1)=1km,d(2)=1.2kmSD−MLMMSE−OSIC

Fig. 4.9 DMIMO(4,2,4) and 4 × 4 CMIMO with QPSK (f=1,g=0), where Eb istransmit bit power


100 110 120 130 140 15010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO d=1kmDMIMOd(1)=1km,d(2)=1kmCMIMO d=1.2kmDMIMOd(1)=1km,d(2)=1.2kmSD−MLMMSE−OSIC

Fig. 4.10 DMIMO(4,2,4) and 4× 4 CMIMO with QPSK (f=0,g=1), where Eb istransmit bit power


120 130 140 150 160 17010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO d=1kmDMIMO d(1)=1km d(2)=1kmCMIMO d=1.2kmDMIMO d(1)=1km d(2)=1.2kmSD−MLMMSE−OSIC

Fig. 4.11 DMIMO(4,2,4) and 4 × 4 CMIMO with 16QAM (f=1,g=1), where Eb

is transmit bit power


120 130 140 150 160 17010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R





120 130 140 150 160 17010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R





120 130 140 150 160 17010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO d=1km


CMIMO d=1.2km

DMIMO d(1)=1km d(2)=1.2km

SD−MLMMSE−OSIC




120 130 140 150 160 17010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO d=1km

DMIMO d(1)=1km d(2)=1kmCMIMO d=1.2km

DMIMO d(1)=1km d(2)=1.2kmSD−MLMMSE−OSIC




120 130 140 150 160 17010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO d=1km

DMIMO d(1)=1km d(2)=1kmCMIMO d=1.2km

DMIMO d(1)=1km d(2)=1.2kmSD−MLMMSE−OSIC




125 135 145 155 165 17510

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R





125 135 145 155 165 17510

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R





125 135 145 155 165 17510

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R





120 130 140 150 16010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO d=1kmDMIMO d(1)=1km d(2)=1kmSD−MLSD−ML γ2=1

SD−ML γ2=10

Fig. 4.20 Effect of channel estimation error: DMIMO(4,2,4) d(1) = d(2) = 1kmand 4×4 CMIMO d=1km with SD-ML for 16QAM, where Eb is transmit bit power


125 135 145 155 16510

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO d=1.2kmDMIMO d(1)=1km d(2)=1.2kmSD−MLSD−ML γ2=1

SD−ML γ2=10

Fig. 4.21 Effect of channel estimation error: DMIMO(4,2,4) d(1) = 1km, d(2) =1.2km and 4× 4 CMIMO d=1.2km with SD-ML for 16QAM, where Eb is transmitbit power


125 135 145 155 16510

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO d=1kmDMIMO d(1)=1km d(2)=1kmSD−MLSD−ML γ2=1

SD−ML γ2=10

Fig. 4.22 Effect of channel estimation error: DMIMO(4,2,4) d(1) = d(2) = 1kmand 4×4 CMIMO d=1km with SD-ML for 64QAM, where Eb is transmit bit power


130 140 150 160 17010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO d=1.2kmDMIMO d(1)=1km d(2)=1.2kmSD−MLSD−ML γ2=1

SD−ML γ2=10

Fig. 4.23 Effect of channel estimation error: DMIMO(4,2,4) d(1) = 1km, d(2) =1.2km and 4× 4 CMIMO d=1.2km with SD-ML for 64QAM, where Eb is transmitbit power


120 130 140 150 160 170 18010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO d=1kmDMIMO d(1)=d(2)=1kmMMSE−OSICMMSE−OSIC γ2=1

MMSE−OSIC γ2=10

Fig. 4.24 Effect of channel estimation error: DMIMO(4,2,4) d(1) = d(2) = 1kmand 4 × 4 CMIMO d=1km with MMSE-OSIC for 16QAM, where Eb is transmitbit power


120 130 140 150 160 170 18010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO d=1.2kmDMIMO d(1)=1km d(2)=1.2kmMMSE−OSICMMSE−OSIC γ2=1

MMSE−OSIC γ2=10

Fig. 4.25 Effect of channel estimation error: DMIMO(4,2,4) d(1) = 1km, d(2) =1.2km and 4 × 4 CMIMO d=1.2km with MMSE-OSIC for 16QAM, where Eb istransmit bit power


130 140 150 160 170 180 19010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO d=1kmDMIMO d(1)=1km d(2)=1kmMMSE−OSICMMSE−OSIC γ2=1

MMSE−OSIC γ2=10

Fig. 4.26 Effect of channel estimation error: DMIMO(4,2,4) d(1) = d(2) = 1kmand 4 × 4 CMIMO d=1km with MMSE-OSIC for 64QAM, where Eb is transmitbit power


130 140 150 160 170 180 19010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO d=1.2kmDMIMO d(1)=1km d(2)=1.2kmMMSE−OSICMMSE−OSIC γ2=1

MMSE−OSIC γ2=10

Fig. 4.27 Effect of channel estimation error: DMIMO(4,2,4) d(1) = 1km, d(2) =1.2km and 4 × 4 CMIMO d=1.2km with MMSE-OSIC for 64QAM, where Eb istransmit bit power


120 125 130 135 140 145 15010

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO t=r=0DMIMO t=r=0CMIMO t=0.6,r=0DMIMO t=0.6,r=0CMIMO t=r=0.6DMIMO t=r=0.6

Fig. 4.28 Effect of channel spatial correlation: DMIMO(4,2,4) d(1) = d(2) = 1kmand 4×4 CMIMO d=1km with SD-ML for 16QAM, where Eb is transmit bit power


125 130 135 140 145 150 15510

−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

CMIMO t=r=0DMIMO t=r=0CMIMO t=0.6,r=0DMIMO t=0.6,r=0CMIMO t=r=0.6DMIMO t=r=0.6

Fig. 4.29 Effect of channel spatial correlation: DMIMO(4,2,4) d(1) = d(2) = 1kmand 4×4 CMIMO d=1km with SD-ML for 64QAM, where Eb is transmit bit power


120 125 130 135 140 145 15010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

C t=r=0D t=r=0C t=r=0.3D t=r=0.3C t=r=0.5D t=r=0.5C t=r=0.8D t=r=0.8

Fig. 4.30 Same space correlation for DMIMO(4,2,4) d(1) = d(2) = 1km (’D’ inlegend) and 4× 8 CMIMO d=1km (’C’ in legend) with SD-ML for 16QAM, whereEb is transmit bit power


120 125 130 135 140 145 15010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No(dB)

BE

R

C t=r=0D t=r=0C t=0.3, r=0.74D t=r=0.3C t=0.5, r=0.84D t=r=0.5C t=0.8, r=0.95D t=r=0.8

Fig. 4.31 Same space constraint at receiver of DMIMO(4,2,4) d(1) = d(2) = 1km(’D’ in legend) and 4× 8 CMIMO d=1km (’C’ in legend) with SD-ML for 16QAM,where Eb is transmit bit power

53

Chapter 5: D-MIMO Performance Analysis

In the previous chapter, we analyzed the BER performance of D-MIMO by using Monte-Carlosimulations. In this chapter, we present theoretical BER performance analysis for D-MIMOsystem. We consider a D-MIMO system with two base stations where bk ∈ {0, 1} is the kthtransmitted bit, k = 1, . . . ,MQ0. According to the bit level combining scheme implemented inthe Fusion Center (3.6)(3.7), the error probability for the kth bit of D-MIMO is:

Pr(ˆbk 6= bk) = Pr[(−1)b

(1)k w

(1)k + (−1)b

(2)k w

(2)k > 0|bk = 1] · Pr(bk = 1)

+ Pr[(−1)b(1)k w

(1)k + (−1)b

(2)k w

(2)k < 0|bk = 0] · Pr(bk = 0)

= A · Pr(bk = 1) +B · Pr(bk = 0) (5.1)

where w(n)k ≥ 0 is the kth element of w(n) (3.27), and

A = Pr[(−1)b(1)k w

(1)k + (−1)b

(2)k w

(2)k > 0|bk = 1] = A00 + A01 + A10 + A11. (5.2)

We have

A00 = Pr[w(1)k + w

(2)k > 0|bk = 1, b

(1)k = 0, b

(2)k = 0] · Pr(b(1)k = 0, b

(2)k = 0|bk = 1)

= Pr(b(1)k = 0|bk = 1)Pr(b

(2)k = 0|bk = 1) = p

(1)k · p(2)

k (5.3)

where the second equality is because b(1)k and b

(2)k are independent, w

(1)k , w

(2)k are positive with

probability 1, and p(n)k is the probability that b

(n)k is in error. Similarly, we have:

A01 = Pr[w(1)k − w

(2)k > 0|bk = 1, b

(1)k = 0, b

(2)k = 1] · Pr(b(1)k = 0, b

(2)k = 1|bk = 1)

= Pr[w(1)k − w

(2)k > 0|bk = 1, b

(1)k = 0, b

(2)k = 1] · p(1)

k · (1 − p(2)k ) (5.4)

A10 = Pr[w(1)k − w

(2)k < 0|bk = 1, b

(1)k = 1, b

(2)k = 0] · Pr(b(1)k = 1, b

(2)k = 0|bk = 1)

= Pr[w(1)k − w

(2)k < 0|bk = 1, b

(1)k = 1, b

(2)k = 0] · (1 − p

(1)k ) · p(2)

k (5.5)

A11 = Pr[−w(1)k − w

(2)k > 0|bk = 1, b

(1)k = 1, b

(2)k = 1] · Pr(b(1)k = 1, b

(2)k = 1|bk = 1) = 0 (5.6)

since w(1)k , w

(2)k cannot be negative. Similarly, we have

B = Pr[(−1)b(1)k w

(1)k + (−1)b

(2)k w

(2)k < 0|bk = 0] = B00 +B01 +B10 +B11 (5.7)

B00 = Pr[w(1)k + w

(2)k < 0|bk = 0, b

(1)k = 0, b

(2)k = 0] · Pr(b(1)k = 0, b

(2)k = 0|bk = 0) = 0 (5.8)

B01 = Pr[w(1)k − w

(2)k < 0|bk = 0, b

(1)k = 0, b

(2)k = 1] · Pr(b(1)k = 0, b

(2)k = 1|bk = 0)

= Pr[w(1)k − w

(2)k < 0|bk = 0, b

(1)k = 0, b

(2)k = 1] · (1 − p

(1)k ) · p(2)

k (5.9)

B10 = Pr[w(1)k − w

(2)k > 0|bk = 0, b

(1)k = 1, b

(2)k = 0] · Pr(b(1)k = 1, b

(2)k = 0|bk = 0)

= Pr[w(1)k − w

(2)k > 0|bk = 0, b

(1)k = 1, b

(2)k = 0] · p(1)

k · (1 − p(2)k ) (5.10)

B11 = Pr[−w(1)k − w

(2)k < 0|bk = 0, b

(1)k = 1, b

(2)k = 1] · Pr(b(1)k = 1, b

(2)k = 1|bk = 0)

= p(1)k · p(2)

k (5.11)

5 D-MIMO Performance Analysis 54

Since w(n)k , the bit reliability information, is related to the type of MIMO detection employed

at the Ports, we analyze the BER performance of D-MIMO based on the type of MIMO detectionemployed. In this thesis, we analyze the BER performance of D-MIMO with SD-ML detection.

5.1 Performance Analysis for C-MIMO with SD-ML Detection

To analyze BER performance of D-MIMO, we need to first find out p(n)k , the probability of

bit error for C-MIMO over a composite Rayleigh-lognormal environment. According to [9], thepairwise error probability (PEP) that the receiver mistakes the transmitted signal vector si foranother signal vector sj in an L×M MIMO system with SD-ML detection scheme is:

P (si → sj |H, si) = Q

(√

‖Hdij‖2

2σ2n

)

(5.12)

where dij = si − sj . Based on channel model (2.16), ‖Hdij‖2 = |hsh(d)|2 · ‖Hssfdij‖2. LetW = |hsh(d)|2 and V = ‖Hssfdij‖2, due to the independence of W and V , the average PEP is:

PEPavg = EW

[

EV

[

Q

(√

W · V2σ2

n

)]]

(5.13)

Let a = Hssfdij, since Hssf = R1/2R HwR

1/2T with Hw an L ×M matrix with i.i.d. CN (0, 1)

entries. Then a is complex normal distributed with mean vector E[a] = 0. Define R1/2T dij =

u = [u1, . . . , uM ]T . We have ‖R1/2T dij‖2 =

∑Mi=1 |ui|2. Then the covariance matrix:

Ra = E[aaH ] = E[

R1/2R HwR

1/2T dij · dH

ij (R1/2T )HHH

w (R1/2R )H

]

= R1/2R E

[HwuuHHH

w

](R

1/2R )H = R

1/2R E

[M∑

i=1

(hw)iui ·M∑

j=1

(hw)Hj u

∗j

]

(R1/2R )H

= R1/2R · I

M∑

i=1

|ui|2 · (R1/2R )H = ‖R1/2

T dij‖2 · RR (5.14)

where we used E[(hw)i · (hw)H

j

]=

{I i = j0 i 6= j

.

Since V = ‖a‖2 = aHQa is a Hermitian quadratic form in zero mean complex Gaussianrandom variables, where Q = I, we have the characteristic function of random variable V [35]:

ϕV (z) =1

det(I− z · Ra)=

1∏L

k=1(1 − zζk)(5.15)

where ζk are the eigenvalues of Ra. According to the receive correlation matrix model in Chapter2, the Hermitian matrix RR is positive definite, making the Hermitian matrix Ra is also positivedefinite, and hence, ζk are positive.

With the representation of the Q function obtained by Craig [36], we have:

Q

(√

W · V2σ2

n

)

=1

π

∫ π2

0

exp

[

− 1

2 sin2(φ)· W · V

2σ2n

]

dφ, (5.16)


and the first expectation in (5.13) becomes

EV

[

Q

(√

W · V2σ2

n

)]

=1

π

∫ π2

0

EV

[

exp

(

− W · V4 sin2(φ) · σ2

n

)]

dφ

=1

π

∫ π2

0

ϕV

(

− W

4 sin2(φ) · σ2n

)

dφ =1

π

∫ π2

0

1∏L

k=1

(

1 + W ·ζk

4σ2n sin2(φ)

)dφ. (5.17)

where the last equality is obtained using (5.15). To solve the integration in (5.17), we need to

perform partial fraction expension for rational function1

∏Lk=1

(1 + ck

x

) , where ck =Wζk4σ2

n

and

x = sin2(φ). The complete derivation of partial fraction expension is provided in Appendix A.

5.1.1 PEP in case of spatial correlation at receiver

In case of spatial correlation at receiver, RR is a Hermitian Toeplitz matrix. By using functioneig(·) in matlab to get eigenvalues of a 4 × 4 receive correlation matrix with different fadingcorrelation coefficient r = 0.1, . . . , 0.9, we find that the eigenvalues of each test are distinct. Fur-thermore, [37] mentioned that for a matrix with Hermitian Toeplitz structure, the eigenvalues areseldomly exactly equal. This leads to assume Hermitian Toeplitz matrix Ra has distinct eigenval-

ues ζk, k = 1, . . . , L, then the intergrand of (5.17)1

∏Lk=1

(

1 + W ·ζk

4σ2n sin2(φ)

) has L simple poles, with

partial fraction expansion (A.1):1

∏Lk=1

(

1 + W ·ζk

4σ2n sin2(φ)

) = 1+L∑

i=1

(−ci)L

∏Lk=1,k 6=i(ck − ci)(sin

2(φ) + ci),

where ck =Wζk4σ2

n

. The PEP averaged over small scale fading (5.17) will be:

EV

[

Q

(√

W · V2σ2

n

)]

=1

π

(

π

2+

L∑

i=1

(−ci)L

∏Lk=1,k 6=i(ck − ci)

∫ π2

0

1

sin2(φ) + cidφ

)

(5.18)

where

∫ π2

0

1

sin2(φ) + cidφ =

∫ π2

0

112

+ ci − 12cos(2φ)

dφ =1

2

∫ π

0

112

+ ci − 12cos t

dt =π

2√

c2i + ci(5.19)

where the first equality is because of sin2(φ) =1

2(1−cos(2φ)) and the third equality is according

to [38] (eq.3.661.4). With (5.19) and substitution of ck =Wζk4σ2

n

, (5.18) will be:

EV

[

Q

(√

W · V2σ2

n

)]

=1

2

(

1 +

L∑

i=1

(−ci)L


· 1√

c2i + ci

)

=1

2

(

1 +L∑

i=1

(−ζi)L

∏Lk=1,k 6=i(ζk − ζi)

·√

W

ζ2i ·W + 4σ2

nζi

)

(5.20)

According to (2.13) in Chapter 2, we have:

W = |hsh(d)|2 =κ · φdτ

(5.21)


where 10 log10 φ ∼ N (0, σ2φdB

). Taking the logarithm of (5.21) we have:

10 log10 W = 10 log10(κ

dτ) + 10 log10 φ = µ+ 10 log10 φ (5.22)

where µ = 10 log10(κ

dτ) is a constant and we have: 10 log10W ∼ N (µ, σ2

φdB) ⇒ lnW ∼

N (λµ, (λσφdB)2), where λ =

ln 10

10. So random variable W is log-normally distributed with

pdf:fW (w) =

1√2π(λσφdB

)wexp

[

−(lnw − λµ)2

2(λσφdB)2

]

w > 0. (5.23)

Then the average PEP including large-scale fading (shadowing) is:

PEPavg = EW

[

EV

[

Q

(√

W · V2σ2

n

)]]

=

∫ ∞

0

EV

[

Q

(√

w · V2σ2

n

)]

· fW (w)dw

=1

2+

1

2

L∑

i=1

(−ζi)L


∫ ∞

0

√w

ζ2i · w + 4σ2

nζifW (w)dw, (5.24)

where∫ ∞

0

√w

ζ2i · w + 4σ2

nζifW (w)dw =

1√2π(λσφdB

)

∫ ∞

0

√

1

ζ2i w

2 + 4σ2nζiw

exp

[

−(lnw − λµ)2

2(λσφdB)2

]

dw

=1√2π

∫ ∞

−∞

√

exp(λσφdBt+ λµ)

ζ2i · exp(λσφdB

t+ λµ) + 4σ2nζi

· exp

(

−t2

2

)

dt (5.25)

in the second equality we use the substitution t =lnw − λµ

λσφdB

.

Using Gauss-Hermite integration

∫ ∞

−∞

e−x2

f(x)dx =n∑

p=1

wpf(xp) where n is the order of

Hermite polynomials, xp are the zeros of Hermite polynomials and wp are the weights [39], then

PEPavg =1

2+

1

2√π

L∑

i=1

(−ζi)L


n∑

p=1

wp

√

exp (λσφdB·√

2xp + λµ)

ζ2i exp (λσφdB

·√

2xp + λµ) + 4σ2nζi

(5.26)

5.1.2 PEP in case of no spatial correlation at receiver

In case of no spatial correlation at receiver, covariance matrix Ra has an eigenvalue ζ =

‖R1/2T dij‖2 with multiplicity L, the integrand in (5.17) will be

1(

1 + W ·ζ4σ2

n sin2(φ)

)Lwhich is a func-

tion with a pole at −W · ζ4σ2

n

of order L. Let c =W · ζ4σ2

n

, its partial fraction expansion (A.3) is:

1(

1 + W ·ζ4σ2

n sin2(φ)

)L= 1 +

L∑

i=1

(Li

)(−c)i

(sin2(φ) + c)i. The PEP averaged over small scale fading (5.17) is:

EV

[

Q

(√

W · V2σ2

n

)]

=1

π

(

π

2+

L∑

i=1

(L

i

)

(−c)i

∫ π2

0

1

(sin2(φ) + c)idφ

)

(5.27)


Similarly, according to [38] (eq.3.661.4), we have:∫ π

2

0

1

(sin2(φ) + c)idφ =

∫ π2

0

1

(12

+ c− 12cos(2φ))i

dφ =1

2

∫ π

0

1

(12

+ c− 12cos t)i

dt

=π

2· 1

(2c)i−1√c2 + c

i−1∑

k=0

(2(i− 1) − 2k − 1)!!(2k − 1)!!

(i− 1 − k)!k!·(

c

1 + c

)k

(5.28)

where (2n− 1)!! =∏n

i=1(2i− 1), (−1)!! = 1. Let j = i− 1 and substitude (5.28) in (5.27), then:

EV

[

Q

(√

W · V2σ2

n

)]

=1

2

[

1 +

L−1∑

j=0

(L

j + 1

)(−c)j+1

(2c)j√c2 + c

j∑

k=0

(2j − 2k − 1)!!(2k − 1)!!

(j − k)!k!·(

c

1 + c

)k]

=1

2

[

1 +

L−1∑

j=0

j∑

k=0

(L

j + 1

)

(−1)j+1

(1

2

)j(2j − 2k − 1)!!(2k − 1)!!

(j − k)!k!·(

Wζ

4σ2n +Wζ

)k+ 12

]

(5.29)

And the average PEP including large scale fading is:

PEPavg = EW

[

EV

[

Q

(√

W · V2σ2

n

)]]

=1

2+

1

2

L−1∑

j=0

j∑

k=0

(L

j + 1

)

(−1)j+1

(1

2

)j

·

(2j − 2k − 1)!!(2k − 1)!!

(j − k)!k!·∫ ∞

0

(wζ

4σ2n + wζ

)k+ 12

fW (w)dw (5.30)

where∫ ∞

0

(wζ

4σ2n + wζ

)k+ 12

fW (w)dw =1√

2π(λσφdB)

∫ ∞

0

(wζ

4σ2n + wζ

)k+ 12

· 1

wexp

[

−(lnw − λµ)2

2(λσφdB)2

]

dw

=1√2π

∫ ∞

−∞

(ζ · exp(λσφdB

t+ λµ)

4σ2n + ζ · exp(λσφdB

t+ λµ)

)k+ 12

· exp

(

−t2

2

)

dt (5.31)

Using Gauss-Hermite integration and with (5.31) we have

PEPavg =1

2+

1

2√π

L−1∑

j=0

j∑

k=0

(L

j + 1

)

(−1)j+1

(1

2

)j(2j − 2k − 1)!!(2k − 1)!!

(j − k)!k!·

n∑

p=1

wp

(

ζ · exp(λσφdB

√2xp + λµ)

4σ2n + ζ · exp(λσφdB

√2xp + λµ)

)k+ 12

. (5.32)

We have σ2n =

σ2s

γ=

1

M · γ . Numerical results show that n = 7 is enough to provide accurate

evaluation for PEPavg .

5.1.3 BER performance for C-MIMO over space uncorrelated channels

Consider the co-located MIMO system r = Hs + n, where s ∈ {s1, s2, . . . , sQM}. H is L ×Mcomposite fading channel matrix (2.16) with M ≤ L following Kronecker model,

H = hsh(d) · HSSF =√WR

1/2R HwR

1/2T (5.33)


where the small scale fading matrix HSSF = R1/2R HwR

1/2T with receive correlation matrix RR

and transmit correlation matrix RT deterministic and Hermitian [27], Hw is an L×M randommatrix with i.i.d complex Gaussian components. The shadowing coefficient

√W = hsh(d) is a

random variable with lognormal distribution.The conditional union bound on BER for a given transmit symbol vector si is [40]:

Pb,ub|W,Hw,si=

1

M · log2Q

QM∑

j=1,j 6=i

i(si, sj)P (si → sj|W,Hw, si) (5.34)

where i(si, sj) denotes the number of information bit errors corresponding to the pair of symbols(si, sj), P (si → sj |W,Hw, si) is the conditioned pairwise error probability (5.12) that receivermistakes transmit signal vector si for another signal vector sj for a given W , Hw and si.

Averaging (5.34) over Hw and W , and also over all possibilities of transmit symbol vector si,result in the average union bound on BER

Pb,ub =1

QM ·M · log2Q

QM∑

i=1

QM∑

j=1,j 6=i

i(si, sj)P (si → sj |si) (5.35)

where P (si → sj|si) is the average PEP over a composite Rayleigh-lognormal channel H (5.13).

BER evaluation technique for uncorrelated channels

For uncorrelated channels, the average PEP, obtained by using (5.32), is a function of ‖dij‖2

where dij = si − sj. Let d1 < d2 <, . . . , < dMmax be the distance spectrum of ‖dij‖. The doublesum in (5.35) can be simplified by using a transfer function approach [40],

Pb,ub =1

QM ·M · log2Q

Mmax∑

m=1

a(m)P2(dm) (5.36)

where Mmax indicates the largest Euclidean distance between si and sj, P2(dm) denotes av-erage PEP for uncorrlated channels, obtained by using (5.32) with ζ = d2

m, and a(m) =QM∑

i=1

QM∑

j=1,j 6=i

[i(si, sj)|(‖dij‖2 = d2m)] indicates the total number of information bit errors associ-

ated with a pair of symbol vectors satisfying ‖dij‖2 = d2m, that can be obtained by using the

transfer function [40],

a(m) =

[1

m!

dm

dDm

([d

dIT (D, I)

]

I=1

)]

D=0

. (5.37)

In [40], transfer functions for square QAM constellations with Gray code mapping are pre-sented. We extend the transfer function to a truncated square QAM constellation with quasi-Gray code mapping such as 32QAM in Appendix B. Table 5.1 shows the transfer function T (D, I)for QPSK, 16QAM, 32QAM and 64QAM respectively, with the assumption that same modula-tion schemes are used in all transmit antennas.

In Fig.(5.1), we see that the largest squared Euclidean distance between two signal pointsequals 2(

√Q− 1)2|dmin|2 for square QAM, and 34|dmin|2 for truncated 32QAM (solid line) with

dmin indicating the distance between two nearest neighbor symbols in the constellation. Letsk

i and skj be the kth component of si and sj respectively, k = 1, . . . ,M . Hence the largest

squared Euclidean distance between si and sj is the case that for all values of k, satisfying


T (D, I)

QPSK (2 + 2ID)2M

16QAM (4 + 6ID + 4I2D4 + 2ID9)2M

32QAM 4M · {[3 + 5ID + 4I2D4 + (I + 2I3)D9 + 2I2D16 + ID25] · [2 + 3ID + 2I2D4 + ID9]+2 + (4I2 + 3I)D + 4(I + I3)D2 + 2(I + I2 + I3)D4 + (12I2 + 2I4)D5 + (4I3 + 2I)D8

+(I + 2I2 + 2I4)D9 + 12I3D10 + (4I2 + 6I4)D13 + 4I3D16 + (6I2 + 4I4)D17 + 2(I3

+I5)D18 + (6I3 + 2I5)D20 + (2I + 6I4)D25 + 3I2D26 + 2I3D29 + 2I3D32 + I2D34}M

64QAM [8 + 14ID + 12I2D4 + (6I + 4I3)D9 + 8I2D16 + (2I + 4I3)D25 + 4I2D36 + 2ID49]2M

Table 5.1 Transfer function for QPSK, 16QAM, 32QAM and 64QAM

32 QAM 16 QAM

Fig. 5.1 Largest Euclidean distance in 32 QAM and square QAM constellation

|ski − sk

j |2 =

{2(√Q− 1)2|dmin|2 square QAM

34|dmin|2 32QAM. Hence d2

Mmax= Mmax|dmin|2 with

Mmax =

{2M(

√Q− 1)2 square QAM constellation

34M 32 QAM(5.38)

The distance spectrum dm in 32QAM and a square QAM constellation can be expressed asd2

m = m|dmin|2, allowing a(m) = 0 for certain values of m.The union upper bound which is averaged over the composite fading channel H could be very

loose, especially at low SNR, since there is no dominant error event in PEP and many termscontribute significantly to the summations in (5.36). In [41], a much tighter upper bound isobtained by limiting the conditional union upper bound before averaging over H. However, dueto the minimization, the order of integration and summation cannot be changed and (ML+1)-foldintegration has to be carried out numerically, becomeing compuationally expensive. Furthermore,to obtain full union bound by computing all values of a(m) and P2(dm) for large number oftransmit antennas and symbol constellation size may be complex. Hence, we consider BERevaluation by using truncated union bound,

Pb =1

QM ·M · log2Q

K∑

m=1

a(m)P2(dm) K ≤Mmax (5.39)

K indicates the number of terms needed to evaluate the BER. Numerical results in the following


section show the values of K required for tight BER approximation at different ranges of BER.

Numerical Results

In this section, we present BER numerical evaluation and simulation results for uncorrelatedC-MIMO. In all schemes, BER is plotted versus Eb/No, where Eb is averaged receive bit power.

Computation of the average receive Eb/No

According to the composite fading channel model in Chapter 2 and considering the independenceof small-scale fading and large-scale fading, the averaged receive signal power on the j th receiveantenna is given by:

σ2s j = E

[M∑

i=1

M∑

k=1

hjih∗jksis

∗k

]

=M∑

i=1

M∑

k=1

E[hjih

∗jk

]E[sis

∗k]

=

M∑

i=1

E[|hji|2] ·PT

M=PT

M· E[h2

sh(d)] ·M∑

i=1

E[

|hji|2]

(5.40)

where hji is the small-scale fading coefficient from the ith transmit antenna to the j th receiveantenna, the second equality is because the independence of H from s, the third equality uses

E[sis∗k] =

{PT

Mi = k

0 i 6= k. According to (2.13), we have: E[h2

sh(d)] =κ

dτ·E[φ], indicating that the

large-scale fading gain is a function of d, the distance between transmitter and receiver. Sincelnφ ∼ N (0, (λσφdB

)2), with λ = ln 1010

, we have

E[φ] =

∫ ∞

0

xfφ(x)dx =

∫ ∞

0

1√2πλσφdB

exp

[

− (ln x)2

2(λσφdB)2

]

dx

=e

12(λσφdB

)2

√2πλσφdB

∫ ∞

−∞

exp

(

− 1

2(λσφdB)2

[t− (λσφdB)2]2)

dt

=e

12(λσφdB

)2

√2πλσφdB

∫ ∞

−∞

e− t′2

2(λσφdB)2 dt′ = e

12(λσφdB

)2 (5.41)

where we substitute ln x = t in the third equality, t′ = t − (λσφdB)2 in the fourth equality and

∫ ∞

−∞

e− t′2

2(λσφdB)2 dt′ =

√2πλσφdB

[38] (eq. 3.323.2). Assume d = 1000m and with parameters:

fc = 1.9GHz, d0 = 100m, τ = 4 and σφdB= 4, we have

E[h2sh(d)] =

κ

dτ· e 1

2(λσφdB

)2 = 1.5788 × 10−12 · 1.5283 = 2.4129 × 10−12. (5.42)

According to (2.9), the small-scale fading matrix of C-MIMO is represented by a Kronecker

model, Hssf = R1/2R HwR

1/2T . Let ei be a unit vector with 1 in the ith position and 0 elsewhere.

Define R1/2T ei = g, we have ‖g‖2 = eH

i (R1/2T )HR

1/2T ei = (RT )ii = 1:

E[|hji|2] = E[eHj R

1/2R HwR

1/2T ei · eH

i (R1/2T )H(Hw)H(R

1/2R )Hej ]

= eHj R

1/2R · E[Hwg · gH(Hw)H ] · (R1/2

R )Hej = eHj R

1/2R ·E[

M∑

k=1

(hw)kgk ·M∑

l=1

(hw)Hl g

∗l ] · (R1/2

R )Hej

=M∑

k=1

|gk|2 · eHj R

1/2R (R

1/2R )Hej = ‖g‖2 · (RR)jj = 1 (5.43)


where we used E[(hw)k(hw)Hl ] =

{I k = l0 k 6= l

. Hence∑M

i=1E[

|h2ji

]

= M , and the averaged

receive symbol power on each receive antenna (5.40) will be:

σ2s j = σ2

s =PT

M·M ·E[h2

sh(d)] = PT · κdτ

· e 12(λσφdB

)2 (5.44)

andEb

No=

σ2s

Q0No.

Fig. (5.2) ∼ Fig. (5.7) present BER evaluation results for uncorrelated 2 × 2 C-MIMO withQPSK, 2× 2 C-MIMO with 16QAM, 4× 4 C-MIMO with QPSK, 4× 4 C-MIMO with 16QAM,2 × 4 C-MIMO with QPSK and 2 × 4 C-MIMO with 16QAM. For each system, the BER wasevaluated with different values of K (5.39). We see that BER results with K = 1 provides a lowerbound, becoming tighter as the number of receive antennas L increases, and becoming looser asthe size of constellation Q increases. BER evaluation results tend to converge as K increases,especially at range of low BER.

Fig. (5.8) ∼ Fig. (5.13) illustrate clearer the tightness of BER evaluations with differentvalues of K. In all figures, the solid lines denote the values of Eb/No required to achieve BER=10−2, BER= 10−4 and BER= 10−6 based on simulation results. Marker-”*”, marker-”+” andmarker-”x” stand for the required values of Eb/No based on (5.39) for different values of K toachieve BER= 10−2, BER= 10−4 and BER= 10−6 respectively. Considering the uncorrelated2 × 2 C-MIMO with 16QAM (Fig. 5.9), we see that BER evaluation with K = 2 provides tightapproximation at range of high BER, BER evaluations with K ≥ 5 provide tight approximationsat range of moderate-to-low BER. However, since the complexity of BER evaluation increasesas K increases, it seems that K = 5 is optimum to provide tight approximations at range ofmoderate-to-low BER. Furthermore, K = 3 provides good approximation for whole range ofBER. Similar analysis (Table. 5.2) can be extended to other C-MIMO systems presented in Fig.(5.8) ∼ Fig. (5.13).

high BER moderate BER low BER full BER range2 × 2 QPSK 2 2 2 22 × 2 16QAM 2 5 5 34 × 4 QPSK 2 3 4 34 × 4 16QAM 2 4 7 32 × 4 QPSK 1 2 2 22 × 4 16QAM 1 2 2 2

Table 5.2 Values of K required for tight BER approximation of uncorrelatedC-MIMO

Conclusion

For uncorrelated C-MIMO, the analysis of union bound for BER can be simplified by using atransfer function approach. We propose a BER evaluation technique based on truncated unionbound. Numerical results indicate that at different ranges of BER, different values of K haveto be used to provide tight BER approximation. The value of K for low BER is larger than forhigh BER. However, one can identify a common value of K that provides satisfactory accuracyfor an extended BER range for a several configurations.


10 15 20 25 30 35 4010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R

simulation BERBER−truncated2,3,4

1

Fig. 5.2 BER evaluation for uncorrelated 2×2 C-MIMO with SD-ML for QPSK.Numbers denote the values of K in (5.39), Eb is averaged receive bit power

15 20 25 30 35 40 4510

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R

simulation BERBER−truncated3,. . . ,8

2

1

Fig. 5.3 BER evaluation for uncorrelated 2×2 C-MIMO with SD-ML for 16QAM.Numbers denote the values of K in (5.39), Eb is averaged receive bit power


5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R

simulation BERBER−truncated4,. . . ,8

1

23


10 15 20 25 30 3510

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R

simulation BERBER−truncated

4

3

2

1

5,. . .,10



5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R

simulation BERBER−truncated2,. . .,4

1


5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R


1

2

3,. . .,8



1 2 3 410

15

20

25

30

35

40

K

Eb/N

o(d

B)

10−2

10−4

10−6

BER=10−6

BER=10−4

BER=10−2

Fig. 5.8 BER evaluation for uncorrelated 2 × 2 C-MIMO with QPSK, Eb isaveraged receive bit power

1 2 3 4 5 6 7 815

20

25

30

35

40

45

K

Eb/N

o(d

B)

10−2

10−4

10−6

BER=10−2

BER=10−4

BER=10−6

Fig. 5.9 BER evaluation for uncorrelated 2 × 2 C-MIMO with 16QAM, Eb isaveraged receive bit power


1 2 3 4 5 6 7 810

15

20

25

30

K

Eb/N

o(d

B)

10−2

10−4

10−6

BER=10−2

BER=10−4

BER=10−6


1 2 3 4 5 6 7 8 9 1010

15

20

25

30

35

K

Eb/N

o(d

B)

10−2

10−4

10−6

BER=10−2

BER=10−4

BER=10−6



1 2 3 45

10

15

20

25

K

Eb/N

o(d

B)

10−2

10−4

10−6

BER=10−2

BER=10−6

BER=10−4


1 2 3 4 5 6 7 810

15

20

25

30

K

Eb/N

o(d

B)

10−2

10−4

10−6

BER=10−6

BER=10−4

BER=10−2



5.1.4 BER performance for C-MIMO over space correlated channels

In this section, we consider the BER performance of C-MIMO over space correlated channels.Three types of spatial correlation are considered: spatial correlation at receive side only, spatialcorrelation at transmit side only and spatial correlation at both transmit and receive sides.

Spatial correlation at receiver only

When there’s spatial correlation only at receiver, the average PEP obtained by using (5.26) is afunction of ζi, the ith eigenvalue of ‖dij‖2 ·RR, and hence it depends on ‖dij‖2 with dij = si−sj .The truncated union bound is similar to the case of no space correlation,

PRb =

1

QM ·M · log2Q

K∑

m=1

a(m)PR2 (dm) K ≤Mmax (5.45)

where PR2 (dm) denotes the average PEP in case of spatial correlation at receiver only, obtained by

using (5.26) with ζi be the ith eigenvalue of d2m ·RR, dm denotes the mth component of distance

spectrum of ‖dij‖ with d1 < d2 <, . . . , < dMmax , a(m) is the total number of information biterrors of all possible pairs (si, sj) satisfying ‖dij‖2 = d2

m.

Spatial correlation at transmitter only

When there’s spatial correlation only at transmitter, the average PEP obtained by using (5.32)

is a function of ‖R1/2T dij‖2. Let dT

1 < dT2 · · · < dT

MTmax

be the Euclidean distance spectrum of

‖R1/2T dij‖. The union bound will be:

P Tb,ub =

1

QM ·M · log2Q

MTmax∑

m=1

aT (m)P T2 (dT

m) (5.46)

where aT (m) is the total number of information bit errors of all possible pairs of (si, sj) satisfying

‖R1/2T dij‖2 = (dT

m)2, P T2 (dT

m) denotes the average PEP when spatial correlation at transmitteronly, obtained by using (5.32) with ζ = (dT

m)2. Due to RT , aT (m) can not be obtained by usingtransfer function. Hence it is complex to obtain aT (m) and dT

m for system with large number oftransmit antennas and large constellation size.

Consider dij = [s1i − s1

j , s2i − s2

j , . . . , sMi − sM

j ]T = [η1, η2, . . . , ηM ]T =∑M

k=1 ηkek, where smi

and smj denote the mth component of symbol vector si and sj respectively, em denotes the unit

vector with 1 in the mth component and 0 elsewhere. Hence, we have,

‖R1/2T dij‖2 = dH

ijRTdij =M∑

k=1

η∗keHk · RT ·

M∑

k=1

ηkek

=

M∑

k=1

|ηk|2(RT )kk +

M∑

k=1

M∑

l=1,l 6=k

η∗kηl(RT )kl = ‖dij‖2 +

M∑

k=1

M∑

l=1,l 6=k

η∗kηl(RT )kl (5.47)

where (RT )ij denotes the ij th element of RT and (RT )kk = 1. We see that when there is only

one non-zero element in dij , then ‖R1/2T dij‖2 = ‖dij‖2, otherwise

∑Mk=1

∑Ml=1,l 6=k η

∗kηl(RT )kl 6= 0.

Furthermore, we see that there could be multiple values of ηm, m = 1, . . . ,M correspondingto a specific ‖dij‖, hence, there are multiple values of ‖R1/2

T dij‖2 corresponding to a specificvalue of ‖dij‖2. This can be verified in Fig.(5.14) (5.15), where marker-’o’ denotes the values


of ‖R1/2T dij‖2, dash line indicates the values when ‖R1/2

T dij‖2 = ‖dij‖2 and x-axis denotes the

distance spectrum of ‖dij‖2. For square QAM, d2m = m|dmin|2 and |dmin|2 =

6σ2s

Q− 1=

6

M(Q− 1)[29]. In the following, two techniques are proposed so that BER evaluation can be performed byusing a transfer function approach despite the presence of RT .

0 2 4 6 80

2

4

6

8

10

12

d2

m

‖R1/2

Td

ij‖2

Fig. 5.14 ‖R1/2T dij‖2 and dm for C-MIMO with M = 2, Q = 16, t = 0.5

BER evaluation by using minimum value of ‖R1/2T dij‖2

Let b = ‖R1/2T dij‖−1a, where a = R

1/2R HwR

1/2T dij with covariance matrix E[aaH ] given by

(5.14). Hence, b is complex normal distributed CN (0, Cov(b)), and

Cov(b) = E[bbH ] = ‖R1/2T dij‖−2 · E[aaH ] = RR (5.48)

The conditional PEP in (5.12) can be written as

P (si → sj |W,Hw, si) = Q

(√

W · ‖a‖2

2σ2n

)

= Q

√

W · ‖R1/2T dij‖2 · ‖b‖2

2σ2n

(5.49)

Since b is complex normal distributed with zero mean vector and covariance matrix RR, we seethat the distribution of ‖b‖2 does not depend on ‖R1/2

T dij‖2. Hence, the conditional PEP is a

monotonic non-increasing function of ‖R1/2T dij‖2. The Q-function is monotonic non-increasing,

then Q(√Aδ1) > Q(

√Aδ2) when δ1 < δ2, taking the expectation of Q-function over A, we have

EA[Q(√

Aδ1) −Q(√

Aδ2)] > 0 ⇒ EA[Q(√

Aδ1)] > EA[Q(√

Aδ2)] (5.50)

Hence the average PEP is also a monotonic non-increasing function of ‖R1/2T dij‖2. Define dm =

min‖dij‖=dm

‖R1/2T dij‖, where the minimum is over all pairs of symbol vectors (si, sj) with ‖dij‖ = dm.


0 1 2 3 40

2

4

6

8

d2

m

‖R1/2

Td

ij‖2

Fig. 5.15 ‖R1/2T dij‖2 and dm for C-MIMO with M = 4, Q = 4, t = 0.5

The average PEP with ‖dij‖ = dm satisfies PEPavg(‖R1/2T dij‖) < PEPavg(dm). This indicates

that PEPavg(dm) will dominant BER performance over all possible (si, sj) with ‖dij‖ = dm. Anapproximation to the BER is given by

P Tb,min =

1

QM ·M · log2Q

K∑

m=1

a(m)P T2 (dm) K ≤Mmax (5.51)

where a(m) indicates total number of bit errors of all possible pairs of (si, sj) satisfying ‖dij‖2 =d2

m obtained by using the transfer function, P T2 (dm) is the average PEP obtained by using (5.32)

with ζ = d2m, and K indicates the number of terms used in BER evaluation.

BER evaluation by using average value of ‖R1/2T dij‖2

Let A = a1, . . . , aQ be the set of signal constellation points. Consider a pair (s′

i, s′

j) with

d′

ij = s′

i − s′

j , where its’ kth componet equals η′

k = (ski )

′ − (skj )

′

= am − an = µ, 1 ≤ m,n ≤ Q.

Then there exists another pair (s′′

i , s′′

j ) with d′′

ij = s′′

i − s′′

j , where the kth component of d′′

ij has

η′′

k = (ski )

′′ − (skj )

′′

= an − am = −µ and the other components of d′′

ij are the same as those

of d′

ij , indicating that ‖d′

ij‖2 = ‖d′′

ij‖2. Hence, all possible pairs (si, sj) can be divided into

groups, each group contains two pairs of vectors with the same property as (s′

i, s′

j) and (s′′

i , s′′

j ).

Assume there are total N possible vectors d(1)ij , . . . ,d

(N)ij with the same Euclidean norm square

‖d(n)ij ‖2 = |η(n)

1 |2 + |η(n)2 |2 + · · ·+ |η(n)

M |2 = d2m, n = 1, . . . , N . Using (5.47) the arithmatic mean of

‖R1/2T dij‖2 with a specific d2

m is,

E‖dij‖2=d2m[‖R1/2

T dij‖2] =1

N

N∑

n=1

‖R1/2T d

(n)ij ‖ =

1

N

N∑

n=1

‖d(n)ij ‖2+

1

N

M∑

k=1

M∑

l=1,l 6=k

N∑

n=1

[(η(n)k )∗(η

(n)l )](RT )kl

(5.52)


For each specific value of k and l, all these N possible dnij can be divided into N/2 groups

with each group contains a pair of vectors (d(p1)ij ,d

(p2)ij ) only differ in the kth component with

η(p1)k = −η(p2)

k , 1 ≤ p1, p2 ≤ N . Hence the conditioned summation(∑N

n=1[(η(n)k )∗(η

(n)l )]

) ∣∣∣k,l

in

(5.52) can be considered as a summation of all these N/2 groups, each group has the property,

(η(p1)k )∗(η

(p1)l ) + (η

(p2)k )∗(η

(p2)l ) = (η

(p1)k )∗(η

(p1)l ) + (−η(p1)

k )∗(η(p1)l ) = 0 (5.53)

Hence(∑N

n=1[(η(n)k )∗(η

(n)l )]

) ∣∣∣k,l

= 0, and the arithmetic mean in (5.52) is,

E‖dij‖2=dm[‖R1/2

T dij‖2] = d2m + 0 = d2

m (5.54)

Then, another method for BER evaluation is based on using the average value of ‖R1/2T dij‖2:

P Tb,avg =

1

QM ·M · log2Q

K∑

m=1

a(m)P T2 (dm) K ≤ Mmax (5.55)

where a(m) indicates total number of bit errors of all possible pairs of (si, sj) satisfying ‖dij‖2 =d2

m obtained by using transfer function, P T2 (dm) is the average PEP obtained by using (5.32)

with ζ = d2m, where d2

m is the mth element of the distance spectrum of ‖dij‖2, and K indicatesthe number of terms needed to approximate BER.

Furthermore, we notice that in square QAM when ‖dij‖2 = |dmin|2, indicating that for

each possible dij there is only one non-zero element so that ‖R1/2T dij‖2 = ‖dij‖2, hence BER

evaluations with K = 1 by using ’BER-min’ are same as those by using ’BER-avg’.

Spatial correlation at both transmitter and receiver

When there is spatial correlation at both transmitter and receiver, the average PEP obainedby using (5.26) is a function of ζi, the ith eigenvalue of ‖R1/2

T dij‖2 · RR. This average PEP

is a function of ‖R1/2T dij‖2. Hence, similar to the case of spatial correlation at transmitter, we

propose two techniques to evaluate BER performance.

BER evaluation by using minimum value of ‖R1/2T dij‖2

P TRb,min =

1

QM ·M · log2Q

K∑

m=1

a(m)P TR2 (dm) K ≤Mmax (5.56)

with dm = min‖dij‖=dm

‖R1/2T dij‖, a(m) indicates total number of bit errors of all possible pairs

(si, sj) satisfying ‖dij‖2 = d2m that can be obtained by using the transfer function, d2

m is themth element of the distance spectrum of ‖dij‖2, P TR

2 (dm) denotes the average PEP in case ofspatial correlation at both transmitter and receiver, obtained by using (5.26) with ζi be the itheigenvalue of d2

m ·RR, and K indicates the number of terms needed to evaluate the BER.

BER evaluation by using average value of ‖R1/2T dij‖2

P TRb,avg =

1

QM ·M · log2Q

K∑

m=1

a(m)P TR2 (dm) K ≤Mmax (5.57)

with a(m) indicates total number of bit errors of all possible pairs (si, sj) satisfying ‖dij‖2 = d2m


that can be obtained by using the transfer function, P TR2 (dm) denotes the average PEP when

space correlation at both transmitter and receiver, obtained by using (5.26) with ζi be the itheigenvalue of d2

m ·RR, d2m is the mth element of the distance spectrum of ‖dij‖2, and K indicates

the number of terms needed to evaluate the BER.

Numerical Results

In this section, we present BER numerical and simulation results for C-MIMO over space corre-lated channel. We consider both low and moderate correlation cases with correlation coefficientequals 0.3 and 0.5 correspondingly. Three types of space correlation are considered: space corre-lation at receiver, space correlation at transmitter and space correlation at both transmitter andreceiver. For cases with space correlation at transmitter and space correlation at both transmit-ter and reciever, we present numerical results for two types of BER evaluation techniques: BERevaluation by using minimum value of ‖R1/2

T dij‖2 noted as ’BER-min’, and BER evaluation by

using average value of ‖R1/2T dij‖2 noted as ’BER-avg’. In all schemes, BER is plotted versus

Eb/No, where Eb denotes averaged receive bit power.Fig.(5.16) ∼ Fig.(5.20) present BER numerical and simulation results for a correlated 2 × 2

C-MIMO system with QPSK and correlation coefficients equal 0.3. We see that in all figures,BER evaluation results with K = 1 provides lower bounds, which have a Eb/No gap of 1.2dBto simulation result at BER= 10−5, BER evaluation with K = 3 has almost the same value asK = 4, which is the full union bound. Fig.(5.16) presents the results for space correlation only atreceiver. We see that BER evaluation with K = 2 provides a tight approximation when BER≤10−3 with Eb/No gap less than 0.2dB at BER= 10−5. BER evaluation withK = 3 provides a tightapproximation at range of low BER with Eb/No gap less than 0.2dB at BER= 10−5. Fig.(5.17)and (5.18) present results for space correlation at transmitter using the ’BER-min’ and ’BER-avg’ technique respectivly. We see that using the ’BER-min’ technique, the results with ’K=2’provides an upper bound with an Eb/No gap around 0.6dB at BER= 10−5, BER evaluation withK = 3 and K = 4 provide looser upper bounds with an Eb/No gap around 1dB at BER= 10−5.For ’BER-avg’ technique, BER evaluation with K = 2 provides a tight approximation whenBER≤ 10−3 and Eb/No gap is less than 0.1dB at BER= 10−5, BER evaluations with K = 3 andK = 4 provide upper bound with Eb/No gap less than 0.4dB at BER= 10−5. Fig.(5.19) andFig.(5.20) present results for space correlation at both transmitter and receiver using the ’BER-min’ and ’BER-avg’ technique respectivly. The results are similar to those of space correlationat transmitter.

Fig.(5.21) ∼ Fig.(5.25) present BER numerical and simulation results for a correlated 2×2 C-MIMO system with 16QAM and correlation coefficients equal 0.3. We see that in all figures, BERevaluations withK = 1 provide lower bounds, which have a Eb/No gap of 3dB to simulation resultat BER= 10−5. BER evaluations with K = 6, 7, 8 are almost the same. Fig.(5.21) presents theresults for space correlation at receiver. We see that BER evaluation with K = 2 provides a lowerbound when BER≤ 10−2 and it becomes looser as Eb/No increases. The Eb/No gap is around1.1dB at BER= 10−5. BER evaluation with K = 3, . . . , 8 provide a tighter approximations.When BER decreases, the approximation becomes tighter as K increases. The minimum Eb/No

gap among them is less than 0.2dB at BER= 10−5. Fig.(5.22) and (5.23) present results forspace correlation at transmitter using the ’BER-min’ and ’BER-avg’ technique respectivly. Wesee that using the ’BER-min’ technique, the results with K = 2 provide a lower bound when


BER≤ 10−3 with Eb/No gap around 0.4dB at BER= 10−5. Results with K = 3, . . . , 8 provideupper bounds and they become looser as K increases, with minimum Eb/No gap around 0.3dBat BER= 10−5. Using the ’BER-avg’ technique, evaluation with K = 2 provides a lower boundwhen BER≤ 10−2 and it become looser as Eb/No increases, and Eb/No gap around 1.4dB atBER= 10−5. Results with K = 3, . . . , 8 provide a tighter BER approximations. When BERdecreases, the approximation becomes tighter as K increases, with minimum Eb/No gap lessthan 0.2dB at BER= 10−5. Fig.(5.24) and Fig.(5.25) present the results for space correlation atboth transmitter and receiver using the ’BER-min’ and ’BER-avg’ technique respectivly. Theresults are similar to those of space correlation at transmitter.

Fig.(5.26) ∼ Fig.(5.30) present BER numerical and simulation results for a correlated 4 × 4C-MIMO with QPSK and correlation coefficients equal 0.3. We see that in all figures, BERevaluation results with K = 1 provide lower bounds, which have a Eb/No gap of 1.4dB tosimulation result at BER= 10−5. BER evaluation with K = 4, . . . , 8 are almost same whenBER≤ 10−3. Fig.(5.26) presents the results for space correlation at receiver. We see thatevaluation with K = 2 provide a lower bound when BER≤ 10−3 and it becomes looser asEb/No increases, with Eb/No gap around 0.35dB at BER= 10−5. Evaluation with K = 3, . . . , 8provide a tight BER approximation at range of low BER, with Eb/No gap less than 0.1dB atBER= 10−6. Fig.(5.27) and (5.28) present results for space correlation at transmitter using the’BER-min’ and ’BER-avg’ technique respectivly. For the ’BER-min’ technique, BER evaluationwith K = 2, . . . , 8 provide upper bounds and they become looser as K increases, the minimumEb/No gap is around 0.3dB at BER= 10−5. Using the ’BER-avg’ technique, evaluation withK = 2 provides a lower bound when BER≤ 10−3 and it becomes looser as Eb/No increases, withEb/No gap around 0.6dB at BER= 10−5. Evaluations with K = 3, . . . , 8 provide upper boundsat range of high-to-moderate BER and it become looser as K increases. On the other hand, theyprovide lower bounds at range of moderate-to-low BER and becoming tighter as K increases,with Eb/No gap less than 0.3dB at BER= 10−5. Fig,(5.29) and Fig,(5.30) present results forspace correlation at both transmitter and receiver using the ’BER-min’ and ’BER-avg’ techniquerespectivly. The results are similar to those of space correlation at transmitter.

Fig.(5.31) ∼ Fig.(5.35) present BER numerical and simulation results for a correalted 2 × 4C-MIMO with QPSK and correlation coefficients equal 0.3. We see that in all figures, BERevaluation with K = 1 provides a lower bound, with a Eb/No gap of 0.36dB to simulation resultat BER= 10−5. BER evaluations with K = 3, 4 are almost the same. Fig.(5.31) presents resultsfor space correlation at receiver. We see that evaluations with K = 2, 3, 4 provide upper boundsand they converge to simulation results as Eb/No increases. The Eb/No gap is less than 0.2dB atBER= 10−5. Fig.(5.32) and (5.33) present results for space correlation at transmitter using the’BER-min’ and ’BER-avg’ technique respectivly. For the ’BER-min’ technique, evaluations withK = 2, 3, 4 provide upper bounds that converge as Eb/No increases. The Eb/No gap is around0.67dB at BER= 10−5. For the ’BER-avg’ technique, evaluations with K = 2, 3, 4 provide tightapproximations for BER≤ 10−4 with Eb/No gap less than 0.1dB at BER= 10−5. Fig.(5.34) andFig.(5.35) present results for space correlation at both transmitter and receiver using the ’BER-min’ and ’BER-avg’ technique respectivly. The results are similar to those of space correlationat transmitter.

Fig.(5.36) ∼ Fig.(5.40) present BER numerical and simulation results for a correlated 2 × 4C-MIMO with 16QAM and correlation coefficients equal 0.3. We see that in all figures, BERevaluation results with K = 1 provides a lower bound, with a Eb/No gap around 0.7dB to


simulation result at BER= 10−5. Evaluations withK = 3, . . . , 8 are almost the same when BER≤10−4. Fig.(5.36) presents the results for space correlation at receiver, showing that evaluationswith K = 2 provides a tight approximation when BER≤ 10−3 with Eb/No gap less than 0.1dBat BER= 10−5. BER evaluations with K = 3, . . . , 8 converge to the results for K = 2 whenBER≥ 10−5. Fig.(5.37) and (5.38) present results for space correlation at transmitter using the’BER-min’ and ’BER-avg’ technique respectivly. For the ’BER-min’ technique, evaluations withK = 2, . . . , 8 provide upper bounds converging when BER≤ 10−4. The minimum Eb/No gapamong these bounds is around 0.7dB at BER= 10−5. For the ’BER-avg’ technique, evaluationswith K = 2 provides a tight approximation when BER≤ 10−3 with Eb/No gap less than 0.1dBat BER= 10−5. Evaluations with K = 3, . . . , 8 converge when BER≤ 10−3 and they provide agood BER approximation at range of low BER with Eb/No gap less than 0.1dB at BER= 10−5.Fig.(5.39) and Fig.(5.40) present results for space correlation at both transmitter and receiverusing the ’BER-min’ and ’BER-avg’ technique respectivly. The results are similar to those ofspace correlation at transmitter.

Fig.(5.41) ∼ Fig.(5.45) present BER numerical and simulation results for a correlated 2 × 2C-MIMO with 16QAM and correlation coefficients equal 0.5. We see that in all figures, BERevaluation results with K = 1 provides a lower bound, with a Eb/No gap around 3.5dB tosimulation result at BER= 10−5. Evaluations with K = 6, . . . , 18 are almost the same whenBER≤ 10−1. Fig.(5.41) presents the results for space correlation at receiver, showing that eval-uations with K = 3, . . . , 18 provide a tight approximation when BER≤ 10−3. The maximumEb/No gap among these is less than 0.5dB at BER= 10−5. Fig.(5.42) and (5.43) present resultsfor space correlation at transmitter using the ’BER-min’ and ’BER-avg’ technique respectivly.For the ’BER-min’ technique, evaluations with K = 2, . . . , 18 provide upper bounds and thesebounds become looser as K increases. Evaluations with K = 2 provides tight upper boundwhen BER≤ 10−4 with Eb/No gap less than 0.05dB at BER= 10−5, however it can not providegood approximation at range of high BER with Eb/No gap around 1dB at BER= 10−2. Forthe ’BER-avg’ technique, evaluation with K = 6, . . . , 18 converge when BER≤ 10−1 and theyprovide a good BER approximation when BER≤ 10−3 with the maximum Eb/No gap less than0.05dB at BER= 10−5. Fig.(5.44) and Fig.(5.45) present results for space correlation at bothtransmitter and receiver using the ’BER-min’ and ’BER-avg’ technique respectivly. The resultsare similar to those of space correlation at transmitter.

Fig.(5.46) ∼ Fig.(5.50) present BER numerical and simulation results for a correlated 4 × 4C-MIMO with QPSK and correlation coefficients equal 0.5. We see that in all figures, BERevaluation results with K = 1 provides a lower bound, with a Eb/No gap around 1.8dB tosimulation result at BER= 10−5. BER evaluations with K = 4, . . . , 8 are almost the samewhen BER≤ 10−4. Fig.(5.46) presents the results for space correlation at receiver, showing thatevaluations with K = 3 provides a good approximation when BER≤ 10−4 with Eb/No gap lessthan 0.1dB. K = 4, . . . , 8 provide a tight approximation at range of high BER. Fig.(5.47) and(5.48) present results for space correlation at transmitter using the ’BER-min’ and ’BER-avg’technique respectivly. For the ’BER-min’ technique, evaluations with K = 2, . . . , 8 provide upperbounds and these bounds become looser as K increases, the minimum Eb/No gap among thesebounds is aroung 1dB at BER= 10−5. For the ’BER-avg’ technique, evaluation with K = 4, . . . , 8provide upper bounds at range of high-to-moderate BER and they become looser as K increases,on the other hand, they provide lower bounds at range of moderate-to-low BER and they becometighter as K increases, the minimum Eb/No gap is around 0.4dB at BER= 10−5. Fig.(5.49) and


Fig.(5.50) present results for space correlation at both transmitter and receiver using the ’BER-min’ and ’BER-avg’ technique respectivly. The results are similar to those of space correlationat transmitter.

As previously shown, BER evaluation results tend to converge as K increases, making itdifficult to see the differences for different values of K. Hence, we summarize the BER evaluationresults with different values of K in a more clear format presented in Fig.(5.51) ∼ Fig.(5.65). Foreach MIMO system, a group of three figures is presented, corresponding to space correlation atreceiver, space correlation at transmitter and space correlation at both transmitter and receiver.In all figures, the three solid lines denote the values of Eb/No required to achieve BER= 10−2,BER= 10−4 and BER= 10−6 based on simulation results. In figures corresponding to spacecorrelation at reciever (Fig.(5.51),(5.54),(5.57),(5.60),(5.63)), markers - ’*’,’+’ and ’x’ denotethe values of Eb/No using (5.45) for different values of K to achieve BER= 10−2, BER= 10−4

and BER= 10−6 respectively. In figueres corresponding to space correlation at transmitter(Fig.(5.52),(5.55),(5.58),(5.61),(5.64)), two sets of markers are used. Markers - ’o’,’♦’ and ’�’denote the values of Eb/No using the ’BER-min’ technique (5.51) for different values of K toachieve BER= 10−2, BER= 10−4 and BER= 10−6 respectively, and markers - ’*’,’+’ and ’x’denote the values of Eb/No using the ’BER-avg’ technique (5.55) for different values of K toachieve BER= 10−2, BER= 10−4 and BER= 10−6 respectively. In figures corresponding to spacecorrelation at both transmitter and reciever (Fig.(5.53),(5.56),(5.59),(5.62),(5.65)), we also usetwo sets of markers. Markers - ’o’,’♦’ and ’�’ denote the values of Eb/No using the ’BER-min’ technique (5.56) for different values of K to achieve BER= 10−2, BER= 10−4 and BER=10−6 respectively, and markers - ’*’,’+’ and ’x’ denote the values of Eb/No using the ’BER-avg’technique (5.57) for different values of K to achieve BER= 10−2, BER= 10−4 and BER= 10−6

respectively.We first investigate the BER evaluation with different values of K for MIMO systems with

light space correlation. Consider a group of three figures for a 2 × 2 C-MIMO with 16QAMand correlation coefficients equals 0.3 (Fig.5.54, Fig.5.55 and Fig.5.56 ). We see that for casewith spatial correlation at receiver only (Fig.5.54), BER evaluations with K = 2, K = 5 andK = 6 provide tight BER approximation at range of high BER, moderate BER and low BERrespectively. Furthermore, K = 3 provides reasonable approximation for whole range of BER.Fig.(5.55) presents the results for space correlation at transmitter only, we see that by using ’BER-avg’ technique, evaluations with K = 2, K = 5 and K = 6 provide tight BER approximationat range of high BER, moderate BER and low BER, evaluation with K = 3 provides goodapproximation for whole range of BER. By using ’BER-min’ technique, evaluations with K = 2provide better approximation over other values ofK for whole range of BER. It is seen that resultsfor case with space correlation at both transmitter and receiver (Fig.5.56) are similar to the casewith space correlation at transmitter only. Hence, by applying similar anlysis to other MIMOsystems, we summarize the values of K required to provide tight BER approximation at differentrange of BER in Table. 5.3, 5.4 and 5.5, which corresponding to space correlation at receiveronly, space correlation at transmit only by using ’BER-avg’ technique and space correlationat transmit only by using ’BER-min’ technique respectively. Furthermore, we see that in allschemes, the approximations using the ’BER-avg’ technique are tighter than using the ’BER-min’ technique. The probable reason behind this is the value of dm in the ’BER-min’ technique(5.51) (5.56) is smaller than dm in the ’BER-avg’ technique (5.55) (5.57), hence, according to theproperty of Q-function, the average PEP in the ’BER-min’ technique is larger than that in the


’BER-avg’ technique, making the BER evaluations using the ’BER-min’ technique larger thanthose using the ’BER-avg’ technique. This shows that it’s difficult to obtain good approximationby adjusting K in the ’BER-min’ technique.

high BER moderate BER low BER full BER range2 × 2 QPSK 2 2 2 22 × 2 16QAM 2 5 6 34 × 4 QPSK 2 3 4 32 × 4 QPSK 1 2 2 22 × 4 16QAM 1 2 2 2

Table 5.3 Values of K required for tight BER approximation of C-MIMO whenspace correlation at receiver only, r = 0.3


Table 5.4 Values of K required for tight BER approximation of C-MIMO whenspace correlation at transmitter only with ’BER-avg’ technique, t = 0.3


Table 5.5 Values of K required for tight BER approximation of C-MIMO whenspace correlation at transmitter only with ’BER-min’ technique, t = 0.3

We next investigate the BER evaluation with different values of K for MIMO systems withmoderate space correlation. Consider a group of three figures for a 2× 2 C-MIMO with 16QAMand correlation coefficients equals 0.5 (Fig. 5.66, Fig. 5.67 and Fig. 5.68). We see that whenthere is space correlation at receiver only (Fig. 5.66), BER evaluation with K = 2, K = 5 andK = 6 provide tight BER approximation at range of high BER, moderate BER and low BERrespectively. Furthermore, K = 3 provides reasonable approximation for the whole range of BER.For case of space correlation at transmitter only (Fig. 5.67), we see that by using ’BER-avg’technique, BER evaluation with K = 3, K = 10 and K = 14 provide tight BER approximation atrange of high BER, moderate BER and low BER respectively. Evaluation with K = 6 providesgood approximation for the whole range of BER. By using ’BER-min’ technique, evaluationwith K = 2 provide better approximation over other values of K for the whole range of BER,however, the approximations are looser than those by using the ’BER-avg’ technique. Similarresults can be obtained for case with space correlation at both transmitter and receiver (Fig.


5.68). Similarly, we summarize the values of K required to provide tight BER approximation atdifferent range of BER for other MIMO systems in Table. 5.6, 5.7 and 5.8, which correspondingrespectively to space correlation at receiver only, space correlation at transmitter only by using’BER-avg’ technique and space correlation at transmitter only by using ’BER-min’ technique.Furthermore, we see that when space correlation increases, BER evaluation using the ’BER-avg’technique need larger K to provide good approximation at range of moderate-to-low BER. Thisis because when using the ’BER-avg’ technique, the average PEP doesn’t depend on the levelof space correlation at transmitter ’t’, making the BER evaluation not affected by ’t’. However,according to the simulation result, we see that for a given Eb/No, BER increases as ’t’ increases.Hence, in order to provide good approximation, we need to consider more terms (larger K) inthe ’BER-avg’ technique as space correlation at transmitter increases.

high BER moderate BER low BER full BER range2 × 2 16QAM 2 5 6 34 × 4 QPSK 2 3 4 3

Table 5.6 Values of K required for tight BER approximation of C-MIMO whenspace correlation at receiver only, r = 0.5


Table 5.7 Values of K required for tight BER approximation of C-MIMO whenspace correlation at transmitter only with ’BER-avg’ technique, t = 0.5


Table 5.8 Values of K required for tight BER approximation of C-MIMO whenspace correlation at transmitter only with ’BER-min’ technique, t = 0.5

Conclusion

In this section, we present BER evaluation techniques for C-MIMO with space correlation atreceiver only, space correlation at transmitter only and space correlation at both transmitterand receiver. For cases with space correlation at transmitter only and space correlation at bothtransmitter and receiver, two techniques for BER evaluation, the ’BER-min’ technique and the’BER-avg’ technique, have been proposed. Numerical results show that the BER evaluationtechnique for space correlation at receiver only provides good approximation at different range ofBER by considering different values of K. When there is space correlation at transmitter only orat both transmitter and receiver, the ’BER-avg’ technique provides tighter approximation thanthe ’BER-min’ technique. When space correlation increases, the ’BER-avg’ technique needslarger K to provide good approximation. Furthermore, BER evaluation with K = 1 providelower bound of BER performance, and this lower bound tends to be tighter as L increase andlooser as Q increases.


10 15 20 25 30 35 4010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R


1

Fig. 5.16 BER evaluation for 2× 2 C-MIMO with QPSK and r = 0.3. Numbersdenote the values of K in (5.45), Eb is averaged receive bit power.

10 15 20 25 30 35 4010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R

simulation BERBER−truncated−min

3,4

1

2

Fig. 5.17 BER evaluation using ’BER-min’ for 2 × 2 C-MIMO with QPSK andt = 0.3. Numbers denote the values of K in (5.51), Eb is averaged receive bitpower.


10 15 20 25 30 35 4010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R

simulation BERBER−truncated−avg2,3,4

1

Fig. 5.18 BER evaluation using ’BER-avg’ for 2 × 2 C-MIMO with QPSK andt = 0.3. Numbers denote the values of K in (5.55), Eb is averaged receive bitpower.

10 15 20 25 30 35 4010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R


3,4

2

1

Fig. 5.19 BER evaluation using ’BER-min’ for 2 × 2 C-MIMO with QPSK andt = r = 0.3. Numbers denote the values of K in (5.56), Eb is averaged receive bitpower.


10 15 20 25 30 35 4010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R

simulation BERBER−truncated−avg

1

2,3,4

Fig. 5.20 BER evaluation using ’BER-avg’ for 2 × 2 C-MIMO with QPSK andt = r = 0.3. Numbers denote the values of K in (5.57), Eb is averaged receive bitpower.

10 20 30 40 5010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R


3,. . .,8

1

2

Fig. 5.21 BER evaluation for 2×2 C-MIMO with 16QAM and r = 0.3. Numbersdenote the values of K in (5.45), Eb is averaged receive bit power.


10 20 30 4010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R


3,. . .,8

2

1

Fig. 5.22 BER evaluation using ’BER-min’ for 2× 2 C-MIMO with 16QAM andt = 0.3. Numbers denote the values of K in (5.51), Eb is averaged receive bitpower.

10 20 30 4010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R

simulation BERBER−truncated−avg3,. . .,8

1

2

Fig. 5.23 BER evaluation using ’BER-avg’ for 2× 2 C-MIMO with 16QAM andt = 0.3. Numbers denote the values of K in (5.55), Eb is averaged receive bitpower.


10 20 30 4010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R


1

2

3,. . .,8

Fig. 5.24 BER evaluation using ’BER-min’ for 2× 2 C-MIMO with 16QAM andt = r = 0.3. Numbers denote the values of K in (5.56), Eb is averaged receive bitpower.

10 20 30 4010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R

simulation BERBER−truncated−avg3,. . .,8

2

1

Fig. 5.25 BER evaluation using ’BER-avg’ for 2× 2 C-MIMO with 16QAM andt = r = 0.3. Numbers denote the values of K in (5.57), Eb is averaged receive bitpower.


5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R


4,. . .,8

1

23


5 10 15 20 25 30 3510

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R


4,. . . ,8

1

2

3



5 10 15 20 25 30 3510

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R


1

2

3

4,. . . ,8


5 10 15 20 25 30 3510

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R


1

2

3

4,. . .,8



5 10 15 20 25 30 3510

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R


1

2

3

4,. . .,8


5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R


1

Fig. 5.31 BER evaluation for 2× 4 C-MIMO with QPSK and r = 0.3. Numbersdenote the values of K in (5.45), Eb is averaged receive bit power


5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R


1

2,3,4


5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R


1

2,3,4



5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R


2,3,4

1


5 10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R

simulation BERBER−truncated−avg2,3,4

1



5 10 15 20 25 30 3510

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R


1

2

3,. . .,8

Fig. 5.36 BER evaluation for 2×4 C-MIMO with 16QAM and r = 0.3. Numbersdenote the values of K in (5.45), Eb is averaged receive bit power

5 10 15 20 25 30 3510

−8

10−7

10−6

10−5

10−4

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10−2

10−1

100

Eb/No (dB)

BE

R


3,. . .,8

1

2



5 10 15 20 25 30 3510

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10−7

10−6

10−5

10−4

10−3

10−2

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100

Eb/No (dB)

BE

R


1

2

3,. . .,8


5 10 15 20 25 30 3510

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10−7

10−6

10−5

10−4

10−3

10−2

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100

Eb/No (dB)

BE

R


3,. . .,8

1

2



5 10 15 20 25 30 3510

−8

10−7

10−6

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10−3

10−2

10−1

100

Eb/No (dB)

BE

R


3,. . .,8

1

2


10 20 30 4010

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Eb/No (dB)


1

2

3

4,. . .,18

Fig. 5.41 BER evaluation for 2×2 C-MIMO with 16QAM and r = 0.5. Numbersdenote the values of K in (5.45), Eb is averaged receive bit power.


10 20 30 4010

−7

10−6

10−5

10−4

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10−1

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101

Eb/No (dB)

BE

R


1

2

3

4,. . .,18


10 20 30 4010

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101

Eb/No (dB)

BE

R


1

2

3

4,. . .,18



10 20 30 4010

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101

Eb/No (dB)

BE

R


3

4,. . .,18

2

1


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101

Eb/No (dB)

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R


1

2

3

4,. . .,18



5 10 15 20 25 30 3510

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Eb/No (dB)

BE

R

simulation BERBER−truncated4,. . .,8

1

2

3


10 15 20 25 30 3510

−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R

simulation BERBER−truncated−min4,. . . ,8

1

2

3



10 15 20 25 3010

−7

10−6

10−5

10−4

10−3

10−2

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Eb/No (dB)

BE

R


1

2

3

4,. . . ,8


5 10 15 20 25 30 3510

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Eb/No (dB)

BE

R


4,. . .,8

3

2

1



5 10 15 20 25 30 3510

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Eb/No (dB)

BE

R


1

2

3

4,. . .,8


1 2 3 410

15

20

25

30

35

40

K

Eb/N

o(d

B)

10−2

10−4

10−6

BER=10−2

BER=10−4

BER=10−6

Fig. 5.51 BER evaluation for 2 × 2 C-MIMO with QPSK and r = 0.3. Eb isaveraged receive bit power.


1 2 3 410

15

20

25

30

35

40

K

Eb/N

o(d

B)

Avg 10−2; Avg 10−4; Avg 10−6;

Min 10−2; Min 10−4; Min 10−6;

BER=10−2

BER=10−4

BER=10−6

Fig. 5.52 BER evaluation for 2 × 2 C-MIMO with QPSK and t = 0.3. ’Avg’denotes BER evaluation with ’BER-avg’ scheme, ’Min’ denotes BER evaluationwith ’BER-min’ scheme. Eb is averaged receive bit power.

1 2 3 410

15

20

25

30

35

40

K

Eb/N

o(d

B)

Avg 10−2; Avg 10−4; Avg 10−6;

Min 10−2; Min 10−4; Min 10−6;

BER=10−2

BER=10−4

BER=10−6

Fig. 5.53 BER evaluation for 2× 2 C-MIMO with QPSK and t = r = 0.3. ’Avg’denotes BER evaluation with ’BER-avg’ scheme, ’Min’ denotes BER evaluationwith ’BER-min’ scheme. Eb is averaged receive bit power.


1 2 3 4 5 6 7 815

20

25

30

35

40

45

K

Eb/N

o(d

B)

10−2

10−4

10−6

BER=10−2

BER=10−4

BER=10−6

Fig. 5.54 BER evaluation for 2 × 2 C-MIMO with 16QAM and r = 0.3. Eb isaveraged receive bit power.

1 2 3 4 5 6 7 815

20

25

30

35

40

45

K

Eb/N

o(d

B)

Avg 10−2; Avg 10−4; Avg 10−6;

Min 10−2; Min 10−4; Min 10−6;

BER=10−2

BER=10−4

BER=10−6

Fig. 5.55 BER evaluation for 2 × 2 C-MIMO with 16QAM and t = 0.3. ’Avg’denotes BER evaluation with ’BER-avg’ scheme, ’Min’ denotes BER evaluationwith ’BER-min’ scheme. Eb is averaged receive bit power.


1 2 3 4 5 6 7 815

20

25

30

35

40

45

K

Eb/N

o(d

B)

Avg 10−2; Avg 10−4; Avg 10−6;

Min 10−2; Min 10−4; Min 10−6;

BER=10−2

BER=10−4

BER=10−6

Fig. 5.56 BER evaluation for 2×2 C-MIMO with 16QAM and t = r = 0.3. ’Avg’denotes BER evaluation with ’BER-avg’ scheme, ’Min’ denotes BER evaluationwith ’BER-min’ scheme. Eb is averaged receive bit power.

1 2 3 4 5 6 7 810

15

20

25

30

K

Eb/N

o(d

B)

10−2

10−4

10−6

BER=10−2

BER=10−4

BER=10−6



1 2 3 4 5 6 7 810

15

20

25

30

K

Eb/N

o(d

B)

Avg 10−2; Avg 10−4; Avg 10−6;

Min 10−2; Min 10−4; Min 10−6;

BER=10−6

BER=10−4

BER=10−2


1 2 3 4 5 6 7 810

15

20

25

30

K

Eb/N

o(d

B)

Avg 10−2; Avg 10−4; Avg 10−6;

Min 10−2; Min 10−4; Min 10−6;

BER=10−6

BER=10−4

BER=10−2



1 2 3 4

10

15

20

25

K

Eb/N

o(d

B)

10−2

10−4

10−6

BER=10−2

BER=10−4

BER=10−6


1 2 3 45

10

15

20

25

30

K

Eb/N

o(d

B)

Avg 10−2; Avg 10−4; Avg 10−6;

Min 10−2; Min 10−4; Min 10−6;

BER=10−2

BER=10−4

BER=10−6



1 2 3 45

10

15

20

25

30

K

Eb/N

o(d

B)

Avg 10−2; Avg 10−4; Avg 10−6;

Min 10−2; Min 10−4; Min 10−6;

BER=10−2

BER=10−4

BER=10−6


1 2 3 4 5 6 7 810

15

20

25

30

K

Eb/N

o(d

B)

10−2

10−4

10−6

BER=10−2

BER=10−4

BER=10−6



1 2 3 4 5 6 7 810

15

20

25

30

K

Eb/N

o(d

B)

Avg 10−2; Avg 10−4; Avg 10−6;

Min 10−2; Min 10−4; Min 10−6;

BER=10−2

BER=10−4

BER=10−6


1 2 3 4 5 6 7 810

15

20

25

30

K

Eb/N

o(d

B)

Avg 10−2; Avg 10−4; Avg 10−6;

Min 10−2; Min 10−4; Min 10−6;

BER=10−2

BER=10−4

BER=10−6



1 2 3 4 5 6 7 8 9 1015

20

25

30

35

40

45

K

Eb/N

o(d

B)

10−2

10−4

10−6

BER=10−4

BER=10−6

BER=10−2


1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1815

20

25

30

35

40

45

K

Eb/N

o(d

B)

Avg 10−2; Avg 10−4; Avg 10−6;

Min 10−2; Min 10−4; Min 10−6;

BER=10−4

BER=10−6

BER=10−2



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 1815

20

25

30

35

40

45

K

Eb/N

o(d

B)

Avg 10−2; Avg 10−4; Avg 10−6;

Min 10−2; Min 10−4; Min 10−6;

BER=10−2

BER=10−6

BER=10−4


1 2 3 4 5 6 7 810

15

20

25

30

K

Eb/N

o(d

B)

10−2

10−4

10−6

BER=10−6

BER=10−4

BER=10−2



0 2 4 6 810

15

20

25

30

35

K

Eb/N

o(d

B)

Avg 10−2; Avg 10−4; Avg 10−6;

Min 10−2; Min 10−4; Min 10−6;

BER=10−4

BER=10−6

BER=10−2


1 2 3 4 5 6 7 810

15

20

25

30

35

K

Eb/N

o(d

B)

Avg 10−2; Avg 10−4; Avg 10−6;

Min 10−2; Min 10−4; Min 10−6;

BER=10−2

BER=10−4

BER=10−6



5.2 Performance Analysis for D-MIMO with SD-ML Detection

We see from the simulation results in Chapter 4 that the best performance of D-MIMO withSD-ML detection is achieved by using reliability parameter setting of f = 1 and g = 0. Hencewe use this parameter setting for analytical performance evaluation of D-MIMO. According to(3.26) and the channel model (2.16), the reliability information at Portn of D-MIMO is:

w(n)k,ML = ‖h(n)

i ‖2 = |hsh,n(d(n))|2‖h(n)

ssf,i‖2. (5.58)

where h(n)ssf,i is the ith column vector of H

(n)ssf . Let X(n) = |hsh,n(d

(n))|2 and Y(n)i = ‖h(n)

ssf,i‖2.Then A01 from (5.4) can be written as:

A01 = Pr[X(1)Y(1)i −X(2)Y

(2)i > 0|bk = 1, b

(1)k 6= bk, b

(2)k = bk] · p(1)

k · (1 − p(2)k ) (5.59)

Due to the presence of the product of random variables X(n) and Y(n)i , it is difficult to compute

the conditional probability of (5.59). Consider the expectation of Z written as:

E[Z] = E[Z|Z > D] · Pr[Z > D] + E[Z|Z < D] · Pr[Z < D]. (5.60)

We see that the receive SNR for symbol si on Portn is:σ2

s‖h(n)i ‖2

σ2n

= γX(n)Y(n)i , where γ =

σ2s

σ2n

.

Large scale fading, represented by X(n), affects all receiver antennas equally. Hence when γX(n)

for a symbol is higher, it’s more likely that the detection of the bit associated with that symbolwill be correct. Consider a threshold C that has the property:

b(n)k = bk : γX(n) > C with probability α and

γX(n) < C with probability 1 − α

b(n)k 6= bk : γX(n) < C with probability α and

γX(n) > C with probability 1 − α

(5.61)

where 1/2 ≤ α ≤ 1 and C > 0.

Then by using (5.60), we have the expectation of X(n) when b(n)k = bk, denoted as X

(n)noerr:

X(n)noerr = E[X(n)|b(n)

k = bk]

= E[X(n)|X(n) > C/γ, b(n)k = bk] · Pr[X(n) > C/γ|b(n)

k = bk] +

E[X(n)|X(n) < C/γ, b(n)k = bk] · Pr[X(n) < C/γ|b(n)

k = bk]. (5.62)

Now using the approximation:

E[X(n)|X(n) > C/γ, b(n)k = bk] ≈ E[X(n)|X(n) > C/γ]

E[X(n)|X(n) < C/γ, b(n)k = bk] ≈ E[X(n)|X(n) < C/γ] (5.63)

and since from (5.61), we have

Pr[X(n) > C/γ|b(n)k = bk] = α, Pr[X(n) < C/γ|b(n)

k = bk] = 1 − α (5.64)

then:X(n)

noerr = X(n)>C · α+X(n)

<C · (1 − α) (5.65)

where X(n)>C = E[X(n)|X(n) > C/γ] and X(n)

<C = E[X(n)|X(n) < C/γ]. Similarly, when


b(n)k 6= bk, we have the expectation of X(n), denoted as X

(n)err :

X(n)err = E[X(n)|b(n)

k 6= bk] = X(n)>C · (1 − α) +X(n)

<C · α (5.66)

To simplify the analysis, we approximate (5.59) by replacing X(n) with a suitable deterministic

variable. We substitute X(n)noerr for X(n) when b

(n)k = bk (5.65), and X

(n)err for X(n) when b

(n)k 6= bk

(5.66). Hence, the condition of b(n)k 6= bk and b

(n)k = bk in (5.59) can be absorbed in X

(n)err and

X(n)noerr correspondingly, and A01 from (5.59) can be approximated as:

A01 ≈ Pr[

X(1)err · Y (1)

i −X(2)noerr · Y (2)

i > 0|bk = 1]

· p(1)k · (1 − p

(2)k ) (5.67)

Similar approximations are used for A10, B01 and B10 from (5.5), (5.9) and (5.10)

A10 ≈ Pr[

X(1)noerr · Y (1)

i −X(2)err · Y (2)

i < 0|bk = 1]

· (1 − p(1)k ) · p(2)

k (5.68)

B01 ≈ Pr[

X(1)noerr · Y (1)

i −X(2)err · Y (2)

i < 0|bk = 0]

· (1 − p(1)k ) · p(2)

k (5.69)

B10 ≈ Pr[

X(1)err · Y (1)


i > 0|bk = 0]

· p(1)k · (1 − p

(2)k ) (5.70)

Hence, the probability of bit error for D-MIMO (5.1) is approximated as:

Pr(ˆbk 6= bk)

≈ p(1)k · p(2)

k +(

Pr[X(1)err · Y (1)


i > 0, bk = 1] + Pr[X(1)err · Y (1)


i

> 0, bk = 0])

· p(1)k · (1 − p

(2)k ) +

(

Pr[X(1)noerr · Y (1)

i −X(2)err · Y (2)

i < 0, bk = 1]+

Pr[X(1)noerr · Y (1)

i −X(2)err · Y (2)

i < 0, bk = 0])

· (1 − p(1)k ) · p(2)

k

≈ p(1)k · p(2)

k + Pr[X(1)err · Y (1)


i > 0] · p(1)k · (1 − p

(2)k )

+Pr[X(1)noerr · Y (1)

i −X(2)err · Y (2)

i < 0] · (1 − p(1)k ) · p(2)

k . (5.71)

where we use the property Pr(bk = 0) + Pr(bk = 1) = 1. Tightening of this approximation, andits accuracy will be addressed in next sections.

5.2.1 Computation of X(n)<C and X(n)

>C

In order to evaluate (5.71), we need to calculate X(n)>C and X(n)

<C first. In this section, wepresent the computation of these two conditional expectations.

With the definition of X(n)>C in previous sub-section, we have:

X(n)>C = E[X(n)|X(n) > C/γ] =

∫ ∞

−∞

x(n)fX(n)|X(n)>C/γ(x(n))dx(n). (5.72)

The conditional CDF is:

FX(n)|X(n)>C/γ(x(n)) = Pr[X(n) < x(n)|X(n) > C/γ] =

Pr[X(n) < x(n), X(n) > C/γ]

Pr[X(n) > C/γ]

=

{F

X(n)(x(n))−F

X(n)(C/γ)

1−FX(n)(C/γ)

x(n) > C/γ

0 x(n) < C/γ. (5.73)


We can obtain the conditional pdf byd

dx(n)FX(n)|X(n)>C/γ(x

(n)):

fX(n)|X(n)>C/γ(x(n)) =

{1

1−FX(n)(C/γ)

fX(n)(x(n)) x(n) > C/γ

0 x(n) < C/γ. (5.74)

So X(n)>C in (5.72) is:

X(n)>C =

1

1 − FX(n)(C/γ)

∫ ∞

C/γ

x(n)fX(n)(x(n))dx(n). (5.75)

Similarly, we have X(n)<C as:

X(n)<C = E[X(n)|X(n) < C/γ] =

∫ ∞

−∞

x(n)fX(n)|X(n)<C/γ(x(n))dx(n). (5.76)

The conditional CDF is:

FX(n)|X(n)<C/γ(x(n)) = Pr[X(n) < x(n)|X(n) < C/γ] =

Pr[X(n) < x(n), X(n) < C/γ]

Pr[X(n) < C/γ]

=

{F

X(n)(x(n))

FX(n)(C/γ)

x(n) < C/γ

1 x(n) > C/γ. (5.77)

The conditional pdf will be:

fX(n)|X(n)<C/γ(x(n)) =

{1

FX(n)(C/γ)

fX(n)(x(n)) x(n) < C/γ

0 x(n) > C/γ. (5.78)

Since the random number X(n) = |hsh,n(d(n))|2 ≥ 0, we have:

X(n)<C =

1

FX(n)(C/γ)

∫ C/γ

0

x(n)fX(n)(x(n))dx(n). (5.79)

According to (2.13) in Chapter 2, we have: X(n) = |hsh,n(d(n))|2 =

κ · φn

[d(n)]τ, where 10 log10 φn ∼

N (0, σ2φdB

). Similar to the derivation of (5.21) ∼ (5.23) in Section 5.1.1, we have the probability

density function of X(n) as:

fX(n)(x(n)) =1√

2π(λσφdB)x(n)

e−

(ln x(n)−λµn)2

2(λσφdB)2 x(n) > 0 (5.80)

where µn = 10 log10

(κ

[d(n)]τ

)

is a constant corresponding to Portn and λ =ln 10

10.


With (5.75) and (5.80), we have:

X(n)>C =

1

1 − FX(n)(C/γ)

∫ ∞

C/γ

1√2π(λσφdB

)e− (ln x(n)

−λµn)2

2(λσφdB)2 dx(n)

=1

1 − FX(n)(C/γ)

∫ ∞

ln(C/γ)−λµnλσφdB

1√2πe−

s2

2+(λσφdB

s+λµn)ds

=e[

12(λσφdB

)2+λµn]

1 − FX(n)(C/γ)

∫ ∞


1√2πe−

(s−λσφdB)2

2 ds

=e[

12(λσφdB

)2+λµn]

1 − FX(n)(C/γ)

∫ ∞


−λσφdB

1√2πe−

t2

2 dt

=e[

12(λσφdB

)2+λµn]

1 − FX(n)(C/γ)Q

(ln(C/γ) − λµn − (λσφdB

)2

λσφdB

)

(5.81)

where in the second equality we use the substitution s =lnx(n) − λµn

λσφdB

and in the fourth equality

we use the substitution t = s− λσφdB. Q(·) is Q-function with Q(x) =

1√2π

∫ ∞

x

e−u2

2 du.

Similarly, with (5.79) and (5.80), we have:

X(n)<C =

1

FX(n)(C/γ)

∫ C/γ

0

1√2π(λσφdB

)e−

(ln x(n)−λµn)2

2(λσφdB)2 dx(n)

=1

FX(n)(C/γ)

∫ ln(C/γ)−λµnλσφdB

−∞

1√2πe(−

s2

2+λσφdB

s+λµn)ds

=e[

12(λσφdB

)2+λµn]

FX(n)(C/γ)


−∞

1√2πe−

(s−λσφdB)2

2 ds

=e[

12(λσφdB

)2+λµn]

FX(n)(C/γ)


−λσφdB

−∞

1√2πe−

t2

2 dt

=e[

12(λσφdB

)2+λµn]

FX(n)(C/γ)

[

1 −Q

(ln(C/γ) − λµn − (λσφdB

)2

λσφdB

)]

(5.82)

where FX(n)(C/γ) in (5.81) and (5.82) is:

FX(n)(C/γ) =

∫ C/γ

0

1√2π(λσφdB

)x(n)e−

(ln x(n)−λµn)2

2(λσφdB)2 dx(n)

=


−∞

1√2πe−

s2

2 ds = 1 −Q

(ln(C/γ) − λµn

λσφdB

)

. (5.83)

5.2.2 Computation of the characteristic function of Y(n)i

Let U1 = X(1)err · Y (1)

i − X(2)noerr · Y (2)

i , U2 = X(1)noerr · Y (1)

i − X(2)err · Y (2)

i . According to (5.71), toobtain the bit error probability of D-MIMO, we need to compute the probability Pr[U1 > 0]


and Pr[U2 < 0], which can be obtained by finding the characteristic functions of U1 and U2

respectively, where:

ϕU1(jω) = E[ejωU1 ] = E[

ejωX(1)err ·Y

(1)i · e−jωX

(2)noerr·Y

(2)i

]

= ϕY

(1)i

(jωX(1)err ) · ϕY

(2)i

(−jωX(2)noerr) (5.84)

where the last equality is because of the independence of Y(1)i and Y

(2)i . Similarly,

ϕU2(jω) = E[ejωU2 ] = E[

ejωX(1)noerr·Y

(1)i · e−jωX

(2)err ·Y

(2)i

]

= ϕY

(1)i

(jωX(1)noerr) · ϕY

(2)i

(−jωX(2)err ) (5.85)

Hence to find ϕU1(jω) and ϕU2(jω), we need to firstly find the characteristic function of Y(n)i ,

where Y(n)i = ‖h(n)

ssf,i‖2.

According to small-scale fading channel matrix of D-MIMO (2.9), H(n)SSF = (R

(n)R )1/2H(n)

w (RT )1/2,where H(n)

w is an L ×M matrix with i.i.d. random complex entries with CN (0, 1). With given

correlation matrixes RnR and RT , h

(n)ssf,i, the ith column vector of H

(n)SSF , is complex normal dis-

tributed with mean vector E[h(n)ssf,i] = 0. Let ei be a unit vector with 1 in the ith position and 0

elsewhere, and define R1/2T ei = u = [u1, . . . , uM ]T . We have ‖u‖2 = eH

i (R1/2T )HR

1/2T ei = (RT )ii =

1. Then the covariance matrix:

Bni = E[h

(n)ssf,i(h

(n)ssf,i)

H ] = E[

(R(n)R )1/2H(n)

w R1/2T ei · eH

i (R1/2T )H(H(n)

w )H [(R(n)R )1/2]H

]

= (R(n)R )1/2E[H(n)

w uuH(H(n)w )H ][(R

(n)R )1/2]H = (R

(n)R )1/2E

[M∑

i=1

(h(n)w )iui ·

M∑

j=1

(h(n)w )H

j u∗j

]

[(R(n)R )1/2]H

= (R(n)R )1/2 · I · ‖u‖2 · [(R(n)

R )1/2]H = R(n)R (5.86)

where we used E[

(h(n)w )i · (h(n)

w )Hj

]

=

{I i = j0 i 6= j

.

Since Y(n)i = ‖h(n)

ssf,i‖2 is a Hermitian quadratic form in complex Gaussian random variables

with E[h(n)ssf,i] = 0, similar to the previous derivation in (5.15), we have the characteristic function

of quadratic form [35]:ϕ

Y(n)i

(jω) =L∏

k=1

(1 − jωλ(n)k )−1 (5.87)

where λ(n)k are the eigenvalues of Bn

i . According to (5.86), it is seen that Bni is a Hermitian and

positive definite matrix, so λ(n)k are positive. In case of space correlation at receiver, Bn

i is a

Hermitian Toeplitz matrix. As previously discussed in Section 5.1.1, we assume λ(n)k are distinct.

In case of no space correlation at receiver, λ(n)k = 1, k = 1, . . . , L.

5.2.3 BER Evaluation for D-MIMO

In this section, we first evaluate Pr[U1 > 0] and Pr[U2 < 0] based on the charateristic function

of Y(n)i , then we provide BER analysis for D-MIMO using (5.71).Using (5.84) and (5.87), we have the characteristic function of random variable U1:

ϕU1(jω) =L∏

k=1

[

(1 − jωX(1)errλ

(1)k )(1 + jωX(2)

noerrλ(2)k )]−1

(5.88)


The pdf fU1(u1) is given by the Fourier transform of ϕU1(jω). Let z = jω, the probability:

Pr[U1 > 0] =

∫ ∞

0

fU1(u1)du1 =

∫ ∞

0

1

2πj

∫ j∞

−j∞

ϕU1(z)e−zu1dzdu1

=1

2πj

∫ j∞+ǫ

−j∞+ǫ

ϕU1(z)

[∫ ∞

0

e−zu1du1

]

dz =1

2πj

∫ j∞+ǫ

−j∞+ǫ

ϕU1(z)

zdz (5.89)

where ǫ is a small positive number has been inserted in order to move the path of integrationaway from the singularity at z = 0 and allow for the interchange in the order of intergration [42].

Similarly, according to (5.85) and (5.87), the characteristic function of random variable U2 isgiven by:

ϕU2(jω) =

L∏

k=1

[

(1 − jωX(1)noerrλ

(1)k )(1 + jωX(2)

errλ(2)k )]−1

(5.90)

and

Pr[U2 < 0] =

∫ 0

−∞

fU2(u2)du2 =

∫ 0

−∞

1

2πj

∫ j∞

−j∞

ϕU2(z)e−zu2dzdu2

=1

2πj

∫ j∞−ǫ

−j∞−ǫ

ϕU2(z)

[∫ 0

−∞

e−zu2du2

]

dz = − 1

2πj

∫ j∞−ǫ

−j∞−ǫ

ϕU2(z)

zdz (5.91)

The integrals

∫ j∞+ǫ

−j∞+ǫ

ϕU1(z)

zdz and

∫ j∞−ǫ

−j∞−ǫ

ϕU2(z)

zdz can be evaluated by using coutour inte-

gration. Since

{ϕUk

(z)

z

}

k=1,2

are analytic on those closed coutours except for a finite number of

singular points, these coutour integrations can be evaluated by using the residue theorem [43].Detail process of evaluation of these two intergral by using residue theorem are represented inAppendix C.

When there is spatial correlation at receiver, using (C.3) and (C.6) in Appendix C, we havethe bit error probability for D-MIMO with SD-ML detection (5.71):

Pr(ˆbk 6= bk) ≈ p

(1)k · p(2)

k +

1 −

L∑

l=1

1∏L

k=1(1 +X

(1)err λ

(1)k

X(2)noerrλ

(2)l

)∏L

k′=1,k′ 6=l(1 − λ(2)

k′

λ(2)l

)

· p(1)

k · (1 − p(2)k )

+

L∑

l=1

1∏L

k=1(1 +X

(1)noerrλ

(1)k

X(2)err λ

(2)l

)∏L

k′=1,k′ 6=l(1 − λ(2)

k′

λ(2)l

)

· (1 − p

(1)k ) · p(2)

k (5.92)

When there is no spatial correlation at receiver, using (C.4) and (C.7) in Appendix C, thebit error probability for D-MIMO with SD-ML detection (5.71) will be:


(1)k · p(2)

k +

1 +1

(L− 1)!(X(2)noerr)L

limz→− 1

X(2)noerr

dL−1

dzL−1

(

1

z · (1 −X(1)err z)L

)

· p(1)k · (1 − p

(2)k )

− 1

(L− 1)!(X(2)err )L

limz→− 1

X(2)err

dL−1

dzL−1

[

1

z(1 −X(1)noerrz)L

]

· (1 − p(1)k ) · p(2)

k (5.93)


where X(n)noerr and X

(n)err can be obtained from (5.65) and (5.66), p

(n)k is the BER evaluation of

C-MIMO which has been analized in Section 5.1. Let f(z) =dL−1

dzL−1

(

1

z · (1 −X(1)err z)L

)

, we see

from (5.98) that f(z) is continuous at − 1

X(2)noerr

, hence limz→− 1

X(2)noerr

f(z) = f

(

− 1

X(2)noerr

)

. Similar

computation applies to limz→− 1

X(2)err

dL−1

dzL−1

[

1

z(1 −X(1)noerrz)L

]

.

Computation ofdL−1

dzL−1

[z−a(1 − cz)−b

]

In this section, we first computedL−1

dzL−1

[z−a(1 − cz)−b

]for L = 1, 2, 3, 4, then derive the general

form of the (L-1 )th derivative of z−a(1 − cz)−b by using mathematical induction.When L = 1, we have:

dL−1

dzL−1

[z−a(1 − cz)−b

]= z−a(1 − cz)−b. (5.94)

When L = 2, we have:

d

dz

[z−a(1 − cz)−b

]= (−a)z−(a+1)(1 − cz)−b + (−b)(−c)z(−a)(1 − cz)−(b+1). (5.95)


d2

dz2

[z−a(1 − cz)−b

]=

d

dz

[(−a)z−(a+1)(1 − cz)−b + (−b)(−c)z(−a)(1 − cz)−(b+1)

]

= (−a)[−(a + 1)]z−(a+2)(1 − cz)−b + 2(−a)(−b)(−c)z−(a+1)(1 − cz)−(b+1)

+(−b)[−(b + 1)](−c)2z−a(1 − cz)−(b+2). (5.96)


d3

dz3

[z−a(1 − cz)−b

]

=d

dz

[(−a)[−(a + 1)]z−(a+2)(1 − cz)−b + 2(−a)(−b)(−c)z−(a+1)(1 − cz)−(b+1)

+(−b)[−(b + 1)](−c)2z−a(1 − cz)−(b+2)]

= (−a)[−(a + 1)][−(a+ 2)]z−(a+3)(1 − cz)−b + 3(−a)[−(a + 1)](−b)(−c)z−(a+2)(1 − cz)−(b+1)

+3(−a)(−b)[−(b + 1)](−c)2z−(a+1)(1 − cz)−(b+2) + (−b)[−(b + 1)][−(b+ 2)](−c)3z−a

·(1 − cz)−(b+3). (5.97)

From (5.94) ∼ (5.97), we derive the general form for the (L-1 )th derivative of z−a(1− cz)−b as:

dL−1

dzL−1

[z−a(1 − cz)−b

]

=

L−1∑

k=0

d(L−1)k (−1)L−1 (a+ L− 2 − k)!

(a− 1)!· (b+ k − 1)!

(b− 1)!(−c)kz−(a+L−1−k)(1 − cz)−(b+k) (5.98)

where d(L−1)k is the coefficient of kth term in (L-1 )th derivative of z−a(1−cz)−b, with the property,


d(L−1)k =

{1 k = 0, L− 1

d(L−2)k−1 + d

(L−2)k k = 1, . . . , L− 2

(5.99)

Proof : Let L = M , we have (M-1 )th derivative of z−a(1 − cz)−b from (5.98),

dM−1

dzM−1

[z−a(1 − cz)−b

]

=

M−1∑

k=0

d(M−1)k (−1)M−1 (a+M − 2 − k)!

(a− 1)!· (b+ k − 1)!

(b− 1)!(−c)kz−(a+M−1−k)(1 − cz)−(b+k) (5.100)

we can get the M th derivative of z−a(1 − cz)−b from (5.100),

dM

dzM

[z−a(1 − cz)−b

]

=d

dz

[M−1∑

k=0

d(M−1)k (−1)M−1 (a+M − 2 − k)!

(a− 1)!· (b+ k − 1)!

(b− 1)!(−c)kz−(a+M−1−k)(1 − cz)−(b+k)

]

=M−1∑

k=0

d(M−1)k (−1)M−1 (a+M − 2 − k)!

(a− 1)!· (b+ k − 1)!

(b− 1)!(−c)k ·

[−(a +M − 1 − k)z−(a+M−1−k)−1(1 − cz)−(b+k) + (−(b+ k))(−c)z−(a+M−1−k)(1 − cz)−(b+k)−1

]

=M−1∑

k=0

d(M−1)k (−1)M (a +M − 1 − k)!

(a− 1)!· (b+ k − 1)!

(b− 1)!(−c)kz−(a+M−1−k)−1(1 − cz)−(b+k)

+

M−1∑

k=0

d(M−1)k (−1)M (a+M − 2 − k)!

(a− 1)!· (b+ k)!

(b− 1)!(−c)k+1z−(a+M−1−k)(1 − cz)−(b+k)−1

=M−1∑

k=0

d(M−1)k (−1)M (a +M − 1 − k)!

(a− 1)!· (b+ k − 1)!

(b− 1)!(−c)kz−(a+M−k)(1 − cz)−(b+k)

+

M∑

k′=1

d(M−1)k′−1 (−1)M (a+M − 1 − k′)!

(a− 1)!· (b+ k′ − 1)!

(b− 1)!(−c)k′

z−(a+M−k′)(1 − cz)−(b+k′)

=

k = 0︷︸︸︷

(−1)M (a+M − 1)!

(a− 1)!z−(a+M)(1 − cz)−b +

k = M︷︸︸︷

(−1)M (b+M − 1)!

(b− 1)!(−c)Mz−a(1 − cz)−(b+M) +

M−1∑

k=1

[d(M−1)k + d

(M−1)k−1 ](−1)M (a+M − 1 − k)!

(a− 1)!· (b+ k − 1)!

(b− 1)!(−c)kz−(a+M−k)(1 − cz)−(b+k)(5.101)

where we use k′ = k + 1 in the fourth equality and d(M−1)0 = d

(M−1)M−1 = 1 in the last equality.


Let L = M + 1, we can get the M th derivative of z−a(1 − cz)−b from general form (5.98),

dM

dzM

[z−a(1 − cz)−b

]

=M∑

k=0

d(M)k (−1)M (a+M − 1 − k)!

(a− 1)!· (b+ k − 1)!

(b− 1)!(−c)kz−(a+M−k)(1 − cz)−(b+k)

=

k = 0︷︸︸︷

(−1)M (a+M − 1)!

(a− 1)!z−(a+M)(1 − cz)−b +

k = M︷︸︸︷

(−1)M (b+M − 1)!

(b− 1)!(−c)Mz−a(1 − cz)−(b+M)

+M−1∑

k=1

d(M)k (−1)M (a+M − 1 − k)!

(a− 1)!· (b+ k − 1)!

(b− 1)!(−c)kz−(a+M−k)(1 − cz)−(b+k) (5.102)

where we use d(M)k =

{1 k = 0,M

d(M−1)k + d

(M−1)k−1 k = 1, . . . ,M − 1

. With (5.101) and (5.102), we see

that the general form (5.98) holds for all L. Substitute this general form into (5.93) with a = 1,

b = L, c = X(1)err or c = X

(1)noerr, we obtain,


(1)k · p(2)

k +

[

1 +1


·

limz→− 1

X(2)noerr

(L−1∑

k=0

d(L−1)k (−1)L−1(L− 1 − k)!

(L+ k − 1)!

(L− 1)!(−X(1)

err )kz−(L−k)(1 −X(1)

err z)−(L+k)

)

·p(1)k · (1 − p

(2)k ) − 1

(L− 1)!(X(2)err )L

·

limz→− 1

X(2)err

(L−1∑

k=0

d(L−1)k (−1)L−1(L− 1 − k)!

(L+ k − 1)!

(L− 1)!(−X(1)

noerr)kz−(L−k)(1 −X(1)

noerrz)−(L+k)

)

·(1 − p(1)k ) · p(2)

k with d(L−1)k =

{1 k = 0, L− 1

d(L−2)k−1 + d

(L−2)k k = 1, . . . , L− 2

. (5.103)

5.2.4 Numerical results for Analytical BER evaluation for D-MIMO

In this section, we present BER numerical results for an uncorrelated D-MIMO system withSD-ML detection. BER evaluation technique for uncorrelated D-MIMO is implemented by using(5.103), where p

(n)k is the BER evaluation for uncorrelated C-MIMO using (5.39). According to

the BER evaluation results for uncorrelated C-MIMO in Section 5.1.3, we see that K = 3 providereasonable results for all the systems we considered, especially for the range of moderate-to-lowBER. As an example, we use K = 3 for C-MIMO and evaluate the BER of D-MIMO.

Tight BER results for D-MIMO can be obtained by choosing suitable values of C and α (5.61).We first find suitable C and α for a (4, 2, 4) D-MIMO system, and then apply these values to otherD-MIMO systems such as a (2, 2, 2) and a (2, 2, 4) D-MIMO systems. The process of finding theoptimal C and α can be done in two steps: firstly, we find the regions for parameters C and α byimplementing experiments of comparing numerical and simulation BER for D-MIMO at differentBER points: BER= 10−3, BER= 10−4, BER= 10−5 and BER= 10−6, secondly, based on theseregions of C and α, we find optimal C and α by minimizing the difference between numerical


and simulation BER over the region from BER= 10−3 to BER= 10−6. Table 5.9 shows the resultof the optimal setting of C and α for BER evaluation of an uncorrelated (4,2,4) D-MIMO withSD-ML detection for QPSK and 16QAM. We see that when constellation size increase, the value

QPSK 16QAMC 3.0 26α 0.886 0.85

Table 5.9 C and α for D-MIMO with different modulation schemes

of α tends to decrease slightly and value of C tends to increase.Fig. 5.72 presents the numerical and simulation BER results for an uncorrelated (4,2,4) D-

MIMO with SD-ML for QPSK and 16QAM, employing the parameter settings C and α fromTable 5.9. We see that for both QPSK and 16QAM, the BER evaluation technique provides tightapproximation at range of moderate-to-low BER with Eb/No gap less than 0.7dB at BER= 10−3

and almost no Eb/No gap when BER≤ 10−5. However, the approxiamtion is looser at range ofhight-to-moderate BER with Eb/No gap around 1.67dB at BER= 10−2, this could be related withthe BER evaluation of C-MIMO by using K = 3, which doesn’t provide good approximation atrange of high-to-moderate BER. Hence, one possible solution to obtain good BER evaluation forD-MIMO at whole range of BER is to apply BER evaluation of C-MIMO by employing differentvalues of K at different ranges of BER, such as K = 2 at high-to-moderate BER and K = 3at moderate-to-low BER. Numerical and simulation BER results for an uncorrelated (2,2,2) D-MIMO with QPSK and 16QAM are presented in Fig. 5.73. It is seen that the BER evaluationtechnique using the parameter settings C and α from Table 5.9 provide good approximation foran uncorrelated (2,2,2) D-MIMO system. At BER= 10−3, the Eb/No gap is around 0.6dB forQPSK and 0.46dB for 16QAM. The Eb/No gap is less than 0.25dB for both QPSK and 16QAMwhen BER≤ 10−5. Fig. 5.74 presents BER numerical and simulation results for uncorrelated(2,2,4) D-MIMO with QPSK and 16QAM. We see that the BER evaluation techinique usingthe parameter settings C and α from Table 5.9 doesn’t provide good approximation for anuncorrelated (2,2,4) D-MIMO system, and the approximation become looser as the constellationsize increases. At BER= 10−5, the Eb/No gap is around 0.8dB for QPSK and 2dB for 16QAM.This indicates that BER evaluation with parameter settings C and α for a (M,N,L) D-MIMOsystem with M = L can not provide good BER evaluation for a (M,N,L) D-MIMO with M < L,especially when the constellation size increases. A suggestion to improve the BER evaluation fora (M,N,L) D-MIMO with M < L is to find another possible optimal setting of C and α.

The importance of deriving theoretical BER analysis for D-MIMO is that it provides aneffiecient way to obtain system performance of D-MIMO rather than by using time consumingMonte-Carlo simulation, which is averaged over at least 100000 channel realizations and 500frame errors for each Eb/No in our scheme. Such time difference is more obvious at range oflow BER and the system with high constellation size. We found that for uncorrelated (4, 2, 4)D-MIMO, it takes 2.64 hrs at BER= 4.23×10−7 for QPSK and 9.4 hrs at BER= 1.82×10−7 for16QAM to accumulate 500 frame errors, using the computer model: HP Pavilion Elite HPE-235f(BK450AA), AMD Phenom II x6 1035T, 2.6GHz, 4000MHz FSB.

The analytical results confirm the significant performance improvement provided by D-MIMOwith bit level combining, demonstrated by computer simulations.


115 120 125 130 135 140 14510

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Eb/No (dB)

BE

R

simulation BERanalytical BER

16QAM

QPSK

Fig. 5.72 Numerical and simulation BER for uncorrelated (4,2,4) DMIMO d(1) =d(2) = 1km with SD-ML for QPSK and 16QAM. Parameter settings C and α areemployed from Table 5.9, Eb is transmit bit power


120 130 140 15010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R


QPSK

16QAM



115 120 125 130 135 14010

−7

10−6

10−5

10−4

10−3

10−2

10−1

Eb/No (dB)

BE

R


16QAM

QPSK


119

Chapter 6: Conclusions

In this thesis, we considered a model for a (M,N,L) D-MIMO scheme, and a bit level combin-ing technique aided by bit reliability information for an uplink cellular system over a compositeRayleigh-lognormal channel. Bit reliability information is derived based on a modification oflogarithmic likelihood ratio (LLR) and is adjusted to different MIMO detection schemes throughparameters f and g. We have shown by simulations that f = 0, g = 1 is more favorable toD-MIMO with MMSE-OSIC, and f = 1, g = 0 is more favorable to D-MIMO with SD-ML.Simulation results demonstrate that significant performance gains can be obtained by using thisbit level combining scheme for a (M,N,L) D-MIMO over a M ×L C-MIMO with perfect, as wellas imperfect, channel estimation, and also for channel with spatial correlation. Furthermore, em-ploying this bit level combining scheme for a (M,N,L) D-MIMO provides comparable performanceto a M×NL C-MIMO, especially when space correlation becomes significant. Therefore, this bitlevel combining scheme could be attractive for uplink D-MIMO, yielding significant performanceimprovement with complexity suitable for practical applications.

Another contribution of this thesis is the performance analysis of C-MIMO. We provided theo-retical analysis of averaged PEP for C-MIMO with SD-ML detection over a composite Rayleigh-lognormal channel, and proposed BER evaluation techniques by using truncated BER unionbounds where the cases with no space correlation, space correlation only at transmitter or re-ceiver and space correlation at both transmitter and reciever are investigated. When there isspace correlation at transmitter, we proposed two types of BER evaluation technique, ’BER-min’and ’BER-avg’. By comparing the numerical results with the simulation results, we have shownthat BER evaluations for C-MIMO with no spatial correlation and spatical correlation only atreciever provide tight approximation by using different value of truncations K at different rangeof BER. When there is space correlation at transmitter, the ’BER-avg’ technique provides betterapproximation than the ’BER-min’ technique and the number of terms K required to providegood approximation in the ’BER-avg’ technique increases as space correlation increases.

Furthermore, we provide the performance analysis and BER approximation technique for D-MIMO with SD-ML over a composite Rayleigh-lognormal channel, where both spatial correlationand no spatial correlation are investigated. By comparing the numerical results with the simula-tion results, we have shown that the BER approximation technique provides good approximationfor uncorrelated D-MIMO by choosing optimal values of C and α. Furthermore, we noticed thatdifferent values of C and α are required to provide good BER approximation for a D-MIMO sys-tem with different modulation schemes. The parameter settings of C and α which can providegood approxmiations for BER performance of a (M,N,L) D-MIMO system with M = L didn’tprovide good approximations for a D-MIMO system with M < L, especially when constellationsize increases. These numerical results confirmed that significant performance improvement canbe obtained for a D-MIMO system by employing the bit level combining scheme.

It would have been good to finish theoretical BER performance analysis for C-MIMO andD-MIMO over a composite Rayleigh-lognormal channel with MMSE-OSIC detection scheme also.Due to the time limited for a Master program, we couldn’t consider also this subject, and henceit is left for a future research project.

120

Appendix A: Partial fraction expansion in integration

According to [38] (pp.56), to integrate an arbitrary rational functionF (x)

f(x), we first need to

perform a partial fraction expansionF (x)

f(x)= E(x) +

ϕ(x)

f(x), with E(x) a polynomial and

ϕ(x)

f(x)a

proper rational function meaning the degree of numerator is less than the degree of denominator.

In this section, we provide partial fraction expansion for1

∏Lk=1(1 + ck

x).

A.1 ck are distinct

When c1 6= c2 6= . . . 6= cL, thenF (x)

f(x)=

1∏L

k=1(1 + ck

x)

=xL

∏Lk=1(x+ ck)

with E(x) = 1, F (x) = xL

and ϕ(x) = F (x) − E(x) · f(x) = xL −L∏

k=1

(x + ck). According to [38] (pp.56),ϕ(x)

f(x)can be

decomposed into:ϕ(x)

f(x)=

b1x+ c1

+b2

x+ c2+ . . . +

bLx+ cL

where bi =ϕ(−ci)f ′(−ci)

. Since f ′(x) =

L∏

k=1,k 6=1

(x+ ck) +

L∏

k=1,k 6=2

(x+ ck) + . . .+

L∏

k=1,k 6=L

(x+ ck), we have bi =(−ci)L


and

F (x)

f(x)=

1∏L

k=1(1 + ck

x)

= 1 +

L∑

i=1

(−ci)L


· 1

x+ ci(A.1)

A.2 ck are same

When c1 = c2 = . . . = cL = c, thenF (x)

f(x)=

1∏L

k=1(1 + cx)

=xL

(x+ c)Lwith E(x) = 1, F (x) = xL

and ϕ(x) = F (x) − E(x) · f(x) = xL − (x + c)L. According to [38] (pp.56),ϕ(x)

f(x)can be

decomposed into:ϕ(x)

f(x)=

bL(x+ c)L

+bL−1

(x+ c)L−1+ . . . +

b1x+ c

, where bL−i =ψ(i)(−c)

i!with

ψ(x) =ϕ(x) · (x+ c)L

f(x)= xL − (x+ c)L and ψ(i)(−c) =

di

dxiψ(x)

∣∣∣∣x=−c

. We have ψ(i)(x) by using

mathmetical induction:

ψ(i)(x) =L!

(L− i)!· [xL−i − (x+ c)L−i], (A.2)

Hence, bL−i =L!

i!(L− i)!· (−c)L−i =

(L

L− i

)

(−c)L−i and

F (x)

f(x)=

1∏L

k=1(1 + cx)

= 1 +L∑

k=1

(Lk

)(−c)k

(x+ c)k(A.3)

121

Appendix B: Derivation of Transfer Function

This section presents the derivation of transfer function for a square QAM constellation suchas QPSK, 16QAM and 64QAM with Gray code mapping and a truncated square constellationsuch as 32QAM with quai-Gray code mapping.

B.1 Transfer function for square constellation

The transfer function method keeps track of the number of bit errors through the exponent ofthe dummy variable I, hence I i indicates i different digits between two constellation points. Itkeeps track of the squared Euclidean distance through the exponent of the dummy variable D,hence Dn2

indicates the Euclidean distance between two constellation points is (n|dmin|)2 withdmin denotes the distance between two nearest neighbor symbols in constellation, and n = 0means the distance between a constellation point and itself (see Fig B.1).

Square QAM modulation with constellation size of Q can be viewed as two PAM modulationwith constellation size of

√Q over the in-phase and quadrature carriers. For 16QAM, signals

over in-phase and quadrature use Gray code mapping {00,01,11,10}, consider only I-component,

• There are 4 possible symbol pairs giving 0 bit errors with Dn2corresponding to n = 0




The transfer function for 16QAM by considering both I -component and Q-component [40]:

T (D, I)16QAM = (4 + 6ID + 4I2D4 + 2ID9)2. (B.1)

Similarly, the transfer function for QPSK with Gray code mapping {0,1} on each component is,

T (D, I)QPSK = (2 + 2ID)2. (B.2)

For 64QAM, we use Gray code mapping {000,001,011,010,110,111,101,100} as I -component andQ-component, the transfer function for 64QAM is:

T (D, I)64QAM = [8+14ID+12I2D4 +(6I+4I3)D9 +8I2D16 +(2I+4I3)D25 +4I2D36 +2ID49]2.(B.3)

We see that (B.3) is a bit different than [40], due to different Gray codes used.

B.2 Transfer function for 32QAM

For a truncated square constellation, such as 32QAM, due to its quasi-Gray mapping, the transferfunction can be computed in two parts (Fig. B.2). Part I consists of a rectangular QAM with

D0

D1

D4

D9

Fig. B.1 Structure of Dn2in one branch of constellation

B Derivation of Transfer Function 122

00000 00001

00010 00011

00100

00101

00110

00111

01000

01001

01010

01011

01100

01101

01110

01111

1000010001

1001010011

10100

10101

10110

10111

11000

11001

11010

11011

11100

11101

11110

11111

Part I Part IIFig. B.2 Computation of transfer function for 32QAM

I -component {001,011,010,110,111,101} and Q-component {00,01,11,10}. Part II consists of 8points on both top and bottom which are {00010,00011,10011,10010,00000,00001,10001,10000}.

We first consider the I -component of Part I for 32QAM,• There are 6 possible symbol pairs giving 0 bit errors with Dn2

corresponding to n = 0• There are 10 possible symbol pairs giving 1 bit errors with Dn2






corresponding to n = 5Next, we consider the Q-component of Part I for 32QAM,• There are 4 possible symbol pairs giving 0 bit errors with Dn2




corresponding to n = 3Hence, we have the transfer function of Part I for 32QAM by considering both I -component

and Q-component,

T (D, I)32QAM−PartI (B.4)

= (6 + 10ID + 8I2D4 + (4I3 + 2I)D9 + 4I2D16 + 2ID25) · (4 + 6ID + 4I2D4 + 2ID9)

To compute the transfer function for Part II of 32QAM, we first consider transfer functionof each component in Part II, then add them together. Due to construction of 32QAM from arectangular 32QAM (Fig.4.1), the symbol constellaltion points with quasi-Gray code mappinghave geometrical symmetry horizontally and vertically, the transfer functions associated withconstellation points {00010}, {00000}, {10010} and {10000} are same, similarly, the transferfunctions associated with constellation points {00011}, {10011}, {00001} and {10001} are same.Hence, the transfer function corresponding to Part II of 32QAM is:

T (D, I)32QAM−PartII = 4 · [T (D, I)00010 + T (D, I)00011] (B.5)

where multiplier 4 is due to geometrical symmetry.

B Derivation of Transfer Function 123

We first consider the transfer function associated with constellation point {00010} and eachconstellation point in Part I from left to right and up to down:

T (D, I)00010,PartI

= 2(ID2 + I2D5 + I3D10 + I2D17 + I2D + I3D4 + I4D9 + I3D16 + ID2 + I2D5 + I3D10

+I2D17 + I2D5 + I3D8 + I4D13 + I3D20 + I3D10 + I4D13 + I5D18 + I4D25 + I2D17

+I3D20 + I4D25 + I3D32)

= 2(I2D + 2ID2 + I3D4 + 3I2D5 + I3D8 + I4D9 + 3I3D10 + 2I4D13 + I3D16 + 3I2D17

+I5D18 + 2I3D20 + 2I4D25 + I3D32) (B.6)

where multiplier 2 is due to the pairs of symbols in Part II and symbols in Part I are ordered.We then consider the transfer function associated point {00010} and each point in Part II:

T (D, I)00010,PartII = 1 + ID + I2D4 + ID9 + ID25 + I2D26 + I3D29 + I2D34 (B.7)

Hence, the transfer function associated with constellation point {00010} of 32QAM is:

T (D, I)00010 = T (D, I)00010,PartI + T (D, I)00010,PartII (B.8)

Similarly, we have transfer function associated with constellation point {00011} and eachconstellation point in Part I:

T (D, I)00011,PartI

= 2(I2D5 + ID8 + I2D13 + I3D20 + I3D2 + I2D5 + I3D10 + I4D17 + I2D + ID4 + I2D9

+I3D16 + I3D2 + I2D5 + I3D10 + I4D17 + I4D5 + I3D8 + I4D13 + I5D20 + I3D10 +

I2D13 + I3D18 + I4D25)

= 2[I2D + 2I3D2 + ID4 + 3I2D5 + I4D5 + (I + I3)D8 + I2D9 + 3I3D10 + (2I2 + I4)D13

+I3D16 + 2I4D17 + I3D18 + (I3 + I5)D20 + I4D25]. (B.9)

And the transfer function associated with point {00011} and each point in Part II:

T (D, I)00011,PartII = ID + 1 + ID + I2D4 + I2D26 + ID25 + I2D26 + I3D29 (B.10)

Hence, the transfer function associated with constellation point {00011} of 32QAM is:

T (D, I)00011 = T (D, I)00011,PartI + T (D, I)00011,PartII (B.11)

Finally, we have the transfer funcation for 32QAM with quasi-Gray code mapping,

T (D, I)32QAM = T (D, I)32QAM−PartI + T (D, I)32QAM−PartII

= 4[(3 + 5ID + 4I2D4 + (2I3 + I)D9 + 2I2D16 + ID25) · (2 + 3ID + 2I2D4 + ID9)

+2 + (3I + 4I2)D + (4I + 4I3)D2 + (2I + 2I2 + 2I3)D4 + (12I2 + 2I4)D5 +

(4I3 + 2I)D8 + (I + 2I2 + 2I4)D9 + 12I3D10 + (4I2 + 6I4)D13 + 4I3D16 +

(6I2 + 4I4)D17 + (2I3 + 2I5)D18 + (6I3 + 2I5)D20 + (2I + 6I4)D25 + 3I2D26

+2I3D29 + 2I3D32 + I2D34] (B.12)

We see that derivation of transfer function for a square constellation with Gray code mappingbecomes more complex when the constellation size increases, and the derivation of transferfunction for a truncated square constellation such as 32QAM is even more complex due to itsquasi-Gray code mapping.

124

Appendix C: Integral Evaluation with Residue Theorem

C.1 Computation of

∫ j∞+ǫ

−j∞+ǫ

ϕU1(z)

zdz

Define a closed contour C1 that passes along the imaginary line from −jR+ ǫ to jR+ ǫ and thencounter-clockwise along the semicircle Γ1 of radius R which lies in the half-plane of ℜ(z) < 0and let R tend to infinity. (see Fig. C.1). We have:

∫

C1

ϕU1(z)

zdz =

∫ jR+ǫ

−jR+ǫ

ϕU1(z)

zdz +

∫

Γ1

ϕU1(z)

zdz (C.1)

withϕU1(z)

z=

1

z∏L

k=1(1 − zX(1)errλ

(1)k )(1 + zX

(2)noerrλ

(2)k )

.

Based on the estimation lemma [43], a contour integral is upper bounded by:

∣∣∣∣

∫

Γ1

ϕU1(z)

zdz

∣∣∣∣≦

Ml(Γ1), where l(Γ1) is the arc length of Γ1 with l(Γ1) =1

2(2πR) = πR and M is an upper bound

of

∣∣∣∣

ϕU1(z)

z

∣∣∣∣

along Γ1. Since

∣∣∣∣

ϕU1(z)

z

∣∣∣∣

=|ϕU1(jR)|

R≤ 1

cR2L, where c is a constant and we have:

∫

Γ1

ϕU1(z)

zdz ≤

∣∣∣∣

πR

cR2L

∣∣∣∣

R→∞−−−→ 0.

Let c(1)l = X

(1)errλ

(1)l , c

(2)l = X

(2)noerrλ

(2)l . Since X(n)

>C (5.81) and X(n)<C (5.82) are positive, we

see that X(n)noerr (5.65) and X

(n)err (5.66) are also positive. When there is spatial correlation, the

eigenvalues λ(n)l are positive and assumed to be distinct. Then both c

(1)l and c

(2)l are positive,

andϕU1(z)

zhas 2L+ 1 simple and real poles: 0,

1

c(1)l

, − 1

c(2)l

with l = 1, 2, . . . , L. Hence,ϕU1(z)

z

is analytic in contour C1 except for a number of isolated singularities, the integral of a complexfunction over a contour can be evaluated by using residue theorem [43], which equals 2πj timesthe sum of its residues at singularities inside contour. Then (C.1) becomes:

∫ j∞+ǫ

−j∞+ǫ

ϕU1(z)

zdz =

∫

C1

ϕU1(z)

zdz = 2πj

∑

pξp≤0

Res

(ϕU1(z)

z, z = ξp

)

(C.2)

where ξp are the singularities ofϕU1(z)

zinside contour C1. In [43], it is shown that at a simple pole

ξ, the residue ofϕU1(z)

zis given by: Res

(ϕU1(z)

z, ξ

)

= limz→ξ

(z − ξ) · ϕU1(z)

z. Hence, evaluation


ℑ(z)ℑ(z)

ℜ(z)ℜ(z)

ξ0 ξ1 ξ2L ξ′0 ξ′1 ξ′2Lǫǫ

Γ1 Γ2

Contour C1Contour C2

Fig. C.1 Contour C1 and C2 in evaluating integral

of the integral (C.2) involves the residues at the poles less than or equal to 0:

∫ j∞+ǫ

−j∞+ǫ

ϕU1(z)

zdz = 2πj

Res

(ϕU1(z)

z, 0

)

+∑

lc(2)l >0

Res

(

ϕU1(z)

z,− 1

c(2)l

)

= 2πj

1 −

L∑

l=1

1∏L

k=1(1 +X

(1)err λ

(1)k

X(2)noerrλ

(2)l

)∏L

k′=1,k′ 6=l(1 − λ(2)

k′

λ(2)l

)

(C.3)

When there is no spatial correlation, then λ(1)l = λ

(2)l = 1, l = 1, . . . , L and

ϕU1(z)

z=

1

z(1 − zX(1)err )L(1 + zX

(2)noerr)L

will have real poles: ξ0 = 0 of order 1, ξ1 =1

X(1)err

of order L and

ξ2 = − 1

X(2)noerr

of order L. From [43], we see that if ξ is a pole of order L, then the residue ofϕU1(z)

z

around z = ξ can be found by: Res

(ϕU1(z)

z, ξ

)

=1

(L− 1)!limz→ξ

dL−1

dzL−1

[

(z − ξ)L · ϕU1(z)

z

]

. The

integral (C.2) is evaluated by considering the residues at the poles less than or equal to 0,∫ j∞+ǫ

−j∞+ǫ

ϕU1(z)

zdz = 2πj

[

Res

(ϕU1(z)

z, 0

)

+ Res

(ϕU1(z)

z,− 1

X(2)noerr

)]

= 2πj

1 +1


limz→− 1

X(2)noerr

dL−1

dzL−1

(

1

z · (1 −X(1)err z)L

)

(C.4)


C.2 Computation of

∫ j∞−ǫ

−j∞−ǫ

ϕU2(z)

zdz

Similarly, define a closed contour C2 (see Fig. C.1), we have:

∫

C2

ϕU2(z)

zdz =

∫ jR−ǫ

−jR−ǫ

ϕU2(z)

zdz +

∫

Γ2

ϕU2(z)

zdz (C.5)

withϕU2(z)

z=

1

z∏L

k=1(1 − zX(1)noerrλ

(1)k )(1 + zX

(2)errλ

(2)k )

and

∫

Γ2

ϕU2(z)

zdz

R→∞−−−→ 0.

Let d(1)l = X

(1)noerrλ

(1)l , d

(2)l = X

(2)errλ

(2)l . For the case with spatial correlation, the eigenvalues λ

(n)l

are positive and are assumed to be distinct. Since X(n)noerr, X

(n)err are positive, we see that d

(1)l > 0,

d(2)l > 0, and

ϕU2(z)

zhave 2L+1 simple and real poles: 0,

1

d(1)l

, − 1

d(2)l

with l = 1, 2, . . . , L. Hence

ϕU2(z)

zis analytic in contour C2 except for a number of isolated singularities, and the integral

∫ j∞−ǫ

−j∞−ǫ

ϕU2(z)

zdz can be evaluated by using residue theorem:

∫ j∞−ǫ

−j∞−ǫ

ϕU2(z)

zdz =

∫

C2

ϕU2(z)

zdz = 2πj

∑

pξ′p<0

Res

(ϕU2(z)

z, z = ξ′p

)

= 2πj

∑

ld(2)l >0

Res

(

ϕU2(z)

z,− 1

d(2)l

)

= −2πj

L∑

l=1

1∏L

k=1(1 +X

(1)noerrλ

(1)k

X(2)err λ

(2)l

)∏L

k′=1,k′ 6=l(1 − λ(2)

k′

λ(2)l

)

(C.6)

where ξ′p are the singularities ofϕU2(z)

zinside coutour C2.

For the case with no spatial correlation, λ(1)l = λ

(2)l = 1, l = 1, . . . , L.

ϕU2(z)

zhas real poles:

ξ′0 = 0 of order 1, ξ′1 =1

X(1)noerr

of order L and ξ′2 = − 1

X(2)err

of order L. By considering the residue

at the poles less than 0, the integral

∫ j∞−ǫ

−j∞−ǫ

ϕU2(z)

zdz will be:

∫ j∞−ǫ

−j∞−ǫ

ϕU2(z)

zdz = 2πj · Res

(ϕU2(z)

z,− 1

X(2)err

)

= 2πj · 1

(L− 1)!(X(2)err )L

limz→− 1

X(2)err

dL−1

dzL−1

[

1

z(1 −X(1)noerrz)L

]

(C.7)

127

Appendix D: Computer Simulation Overview and Guide

This section presents an overview of the software that generate the simulation results in thethesis. The relavant C code softeware is contained in the attached CD and grouped into twodirectories. Directory MMSE-OSIC contains all files related to MIMO receiver with MMSE-OSIC detection. These files are listed in Table D.1. Directory SD-ML contains all files relatedto MIMO receiver with SD-ML detection, and are listed in Table D.2. All simulation programsload simulation parameters from the ”DMIMO−input.txt” file at the start of execution. TableD.3 lists all the simulation parameters along with their possible input values.

For the compilation and execution of the software, the GNU Scientific Library (GSL) soft-ware package version GSL-1.13 was used, which is available at www.gnu.org. The software wascompiled and ran on a Red Hat’s Fedora Linux environment with gcc version 4.4.5. To run thesimulation related to MMSE-OSIC detection, we need to access the derectory ”MMSE-OSIC”and compile the code using GCC command:

gcc MMSEOSIC−Main−V2.c −lgsl −lgslcblas −lm −o [executable name]Similarly, to run the simulation related to SD-ML detection, we need to access the derectory”SD-ML” and compile the code using GCC command:

gcc spheredecsimulator−Main−V2.c −lgsl −lgslcblas −lm −o [executable name]The SNR step is determined in main function of C program, MMSEOSIC−Main−V2.c or sphere-decsimulator−Main−V2.c, by an integer variable ’Step’ with default value equals 2.

After running the executable file with input parameter configurations in ”DMIMO−input.txt”, the results are saved to the file defined in ”DMIMO−input.txt” by parameter ”FILE-NAME−OUTPUT”. There are a total of 11 columns in the output file, each column correspondsto a vector of real numbers. Vectors name and descriptions corresponding to each elementof vectors are listed in Table D.4. The length of each vector is equal to the number of SNRpoints that were simulated. The elements of each vector are ordered in accordence to the SNRpoints. The data of the output file can be loaded to a Matlab workspace as a matrix by usingMatlab function ’load’, then at each SNR point, the BER performance can be computed by

BER=biterrors

iterations ·Nt ·Q0, where Nt is the number of transmit antennas and Q0 = log 2(Q). Then

BER can be plotted versus Eb/No by using Matlab function ’semilogy’ with Eb/No = SNR −mwhere m = 10 log 10(Q0).


File name DescriptionMMSEOSIC−Main−V2.c Main system C-file applied for MIMO with MMSE-OSIC

detection for square MQAM and 32QAMgrayencoder.h Perform modulation for square MQAM with

symbolconstellation.h Gray code mappingmodulator.h

grayencoder32qam.h Perform modulation for 32QAM withsymbolconstellation32qam.h quasi-Gray code mapping

modulator32qam.hSymbolConstellation−Bit−Set.h For each binary bit, group whole constellation points into

two subsets according to their corresponding bit valuedata−generator.h Generate transmit signal vector

DMIMO−channel−generator.h Generate composite fading channel matrixssf−channel−generator.h Generate small scale fading channel matrixsh−channel−generator.h Generate large scale fading coefficient

channel−estimate−generator.h Generate channel estimation error matrixnoise−generator.h Generate AWGN noise vector

MMSE−BLAST−lin.h Perform MMSE-OSIC detection for MIMOwith square MQAM

MMSE−BLAST−32QAM.h Perform MMSE-OSIC detection for MIMOwith 32QAM

MMSE32QAMresult.h Perform demodulation for symbol with 32QAMfirstdet.h Generate order pm and symbol estimate at the mth

iteration according to MMSE-OSIC algorithmmtx−inv.h Perform complex matrix inversion by using LU decomposition

Reliability−info−v2.h Generate bit reliability information matrix forD-MIMO with square MQAM

Reliability−info−32qam−v2.h Genereate bit reliability information matrix forD-MIMO with 32QAM

Harddetection−SymboltoBit.h Convert symbol level detection to bit level detectionFusionCenter−Detection.h Generate bit detection at the Fusion Center for D-MIMO

DMIMO−Biterr.h Count the number of bit errorDMIMO−input.txt File for input parameters

Table D.1 List of C software files for MMSE-OSIC detection scheme in enclosedCD


File name Descriptionspheredecsimulator−Main−V2.c Main system C-file applied for MIMO with SD-ML

detection for square MQAM and 32QAMgrayencoder.h Perform modulation for square MQAM with

symbolconstellation.h Gray code mappingmodulator.h

grayencoder32qam.h Perform modulation for 32QAM withsymbolconstellation32qam.h quasi-Gray code mapping

modulator32qam.hSymbolConstellation−Bit−Set.h For each binary bit, group whole constellation points into

two subsets according to their corresponding bit valuedata−generator.h Generate transmit signal vector

DMIMO−channel−generator.h Generate composite fading channel matrixssf−channel−generator.h Generate small scale fading channel matrixsh−channel−generator.h Generate large scale fading coefficient

channel−estimate−generator.h Generate channel estimation error matrixnoise−generator.h Generate AWGN noise vector

rings.h Generate concentric rings for square MQAMrings32qam.h Generate concentric rings for 32QAM

ZFBLASTordering.h Applying zero-forcing (ZF) ordering to H tomaximize the probability that the firstfeasible solution of SD-ML is optimal

ZFfirstdet.h Find index of minimum value of diag[(HHH)−1]pos−def−mtx−inv.h Perform complex matrix inversion by

using cholesky decompositionSD.h Perform SD-ML detection for square MQAM

SD32qam.h Perform SD-ML detection for 32QAMcandidates.h Find the candiates of each layer of MIMO with square

MQAM by searching the constellation points withineach layer’s Disk according to SD-ML detection algorithm

candidates32qam.h Find the candiates of each layer of MIMO with 32QAMby searching the constellation points within each

layer’s Disk according to SD-ML detection algorithmpMLoutput−normal−order.h Perform demodulation for symbol with 32QAM

Reliability−info−v2.h Generate bit reliability information matrix forD-MIMO with square MQAM

Reliability−info−32qam−v2.h Genereate bit reliability information matrix forD-MIMO with 32QAM

Harddetection−SymboltoBit.h Convert symbol level detection to bit level detectionFusionCenter−Detection.h Generate bit detection at the Fusion Center for D-MIMO

DMIMO−Biterr.h Count the number of bit errorDMIMO−input.txt File for input parameters

Table D.2 List of C software files for SD-ML detection scheme in enclosed CD


Parameter Description Data Input Range/ValuesType

FILENAME−OUTPUT Name for output file char AnyStart−SNR Start SNR int AnyEnd−SNR End SNR int Any

Number−of−BaseStations−N Number of Base Stations (BS) int > 0Number−of−Tx−antennas−Nt Number of transmit antennas int > 0Number−of−Rx−antennas−Nr Number of receive antennas int > 0

Modulation−size−Q Size of constellation int 4,16,32,64Exponent−a Parameter f for bit reliability double 0,1

information in (3.26)Exponent−b Parameter g for bit reliability double 0,1

information in (3.26)Gamma−square Level of channel estimation double [0,1]

error γ2

tx−cor Level of spatial correlation double [0,1]at the transmitter

rx−cor Level of spatial correlation double [0,1]at the receiver

Shadow−indication−1/Y−0/N Indication of shadowing fading int 0 (no shadowing fading)1 (shadowing fading)

Channel−estimate−indication Indication of channel int 0 (perfect channel estimation)

−1/Y−0/N estimation error 1 (imperfect channel estimation)Channel−correlation−1/Y−0/N Indication of spatial correlation int 0 (no spatial correlation)

1 (spatial correlation)Distance−vector−d Distance vector between double > 0

MS and BS

Table D.3 Simulation parameters of the input file

Column Vector Description of each elementnumber Name

1 SNR One SNR point2 iterations Total iterations, at least 1 million channel realizations and a

minimum of 500 frame errors are accumulated for each SNR point3 frameerrors Number of frame errors for each SNR point4 symbolerrors Number of symbol errors for each SNR point5 biterrors Number of bit errors for each SNR point6 timeused Time used for each SNR point7 a Values of parameter f for bit reliability infromation in (3.26)8 b Values of parameter g for bit reliability infromation in (3.26)9 gamma Level of channel estimation error10 tx−cor Level of spatial correlation at transmitter11 rx−cor Level of spatial correlation at receiver

Table D.4 Parameters of the output file

131

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