System Architectures for Space-Time Communications - CiteSeer

University of California

Los Angeles

System Architectures for Space-Time

Communications over Frequency Selective

Fading Channels

A dissertation submitted in partial satisfaction

of the requirements for the degree

Doctor of Philosophy in Electrical Engineering

by

Jean-Francois Frigon

2004

c© Copyright by


2004

The dissertation of Jean-Francois Frigon is approved.

Gregory J. Pottie

Ali H. Sayed

James S. Gibson

Babak Daneshrad, Committee Chair

University of California, Los Angeles

2004

ii

To Quyen,

to my father and my mother,

and to my sister,

for their continuous encouragement and support ...

iii

Table of Contents

List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Scope and Objectives of the Dissertation . . . . . . . . . . . . . . 3

1.2 Dissertation Outline . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 Capacity of MIMO Frequency Selective Fading Channels . . . 7

2.1 Channel Capacity Model . . . . . . . . . . . . . . . . . . . . . . . 7

2.2 Capacity without Channel State Knowledge . . . . . . . . . . . . 9

2.3 Capacity with Channel State Knowledge . . . . . . . . . . . . . . 11

2.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.4.1 Basic Model . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.4.2 Colored Noise Model . . . . . . . . . . . . . . . . . . . . . 28

2.4.3 Channel Correlation and Ricean Distribution Model . . . . 37

2.4.4 Channel Knowledge . . . . . . . . . . . . . . . . . . . . . . 46

3 Smart Antenna Array Space-Time System . . . . . . . . . . . . . 49

3.1 Testbed Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.2 Test Environment . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.3 Field Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.1 Time Processing Performance . . . . . . . . . . . . . . . . 61

3.3.2 Spatial Processing Performance . . . . . . . . . . . . . . . 64

iv


3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4 MIMO Equalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

4.1 MIMO Equalization Model . . . . . . . . . . . . . . . . . . . . . . 73

4.2 MMSE Solution for the MIMO DFE Receiver with Cancellation . 77


4.3.1 Frequency Selective Fading Channels . . . . . . . . . . . . 81

4.3.2 Correlated and Ricean Fading Channels . . . . . . . . . . 92

4.3.3 Colored Noise . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.4 Flat Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . 100

4.4.1 Zero-Forcing Channel Inversion . . . . . . . . . . . . . . . 100

4.4.2 VBLAST . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

4.4.3 MMSE VBLAST . . . . . . . . . . . . . . . . . . . . . . . 103

4.4.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . 103

5 Multi-Carrier MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.1 MIMO OFDM Model . . . . . . . . . . . . . . . . . . . . . . . . . 108

5.2 MIMO OFDM MMSE Solution . . . . . . . . . . . . . . . . . . . 114


5.3.1 Comparison with MIMO Equalization . . . . . . . . . . . . 117

5.3.2 Interpolation Effects . . . . . . . . . . . . . . . . . . . . . 119

6 Spread Spectrum MIMO . . . . . . . . . . . . . . . . . . . . . . . . 129

v

6.1 MIMO CDMA Model . . . . . . . . . . . . . . . . . . . . . . . . . 130

6.2 MIMO Generalized RAKE Receiver MMSE Solution . . . . . . . 136

6.3 Combiner Spread Spectrum MIMO Receivers . . . . . . . . . . . . 139


6.4.1 Comparison with MIMO Equalization . . . . . . . . . . . . 145

6.4.2 Performance of Combiner Architectures . . . . . . . . . . . 150

7 Adaptive MIMO Algorithms . . . . . . . . . . . . . . . . . . . . . 152

7.1 Stochastic Gradient Algorithms . . . . . . . . . . . . . . . . . . . 152

7.1.1 LMS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 152

7.2 Recursive Least-Squares Adaptive Filters . . . . . . . . . . . . . . 154

7.2.1 RLS Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 154

7.2.2 Inverse QR Algorithm . . . . . . . . . . . . . . . . . . . . 156

7.2.3 Elementary Circular Rotations . . . . . . . . . . . . . . . . 161


7.3.1 Adaptive MIMO Equalization . . . . . . . . . . . . . . . . 167

7.3.2 Adaptive MIMO Channel Estimation . . . . . . . . . . . . 171

7.3.3 Adaptive MIMO OFDM . . . . . . . . . . . . . . . . . . . 174

7.3.4 Adaptive MIMO CDMA . . . . . . . . . . . . . . . . . . . 176

8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

vi

List of Figures

1.1 95% Outage Capacity at 20 dB SNR for Narrowband Channels. . 2

1.2 Block Diagram of Space-Time Communication System. . . . . . . 3

2.1 MIMO Channel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Capacity of the MIMO Frequency Selective Fading Channel. . . . 18

2.3 CDF of MIMO Channel Capacity. . . . . . . . . . . . . . . . . . . 19

2.4 Comparison of Techniques for Capacity Calculation. . . . . . . . . 20

2.5 Effect of Nb and Nf on the Capacity. . . . . . . . . . . . . . . . . 21

2.6 Effect of τ rms on the Capacity. . . . . . . . . . . . . . . . . . . . . 22

2.7 Capacity with Receive Diversity. . . . . . . . . . . . . . . . . . . . 23

2.8 Effect of the Number of Receive Antennas on the Capacity for

M = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.9 Effect of the Number of Transmit Antennas on the Capacity for

N = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.10 Comparison of the Effect of the Number of Transmit and Receive

Antennas on the MIMO Channel Capacity. . . . . . . . . . . . . . 26

2.11 Effect of the SNR on the Capacity for M = N = 4. . . . . . . . . 28

2.12 MIMO Channel Model for Colored Noise. . . . . . . . . . . . . . . 29

2.13 MIMO Channel Capacity in the Presence of Colored Noise. . . . . 31

2.14 MIMO Channel Capacity in the Presence of Colored Noise with

Mint = 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

vii

2.15 Effect of the Number of Receive Antennas on the Capacity in the

Presence of Colored Noise for M = 4. . . . . . . . . . . . . . . . . 33

2.16 Effect of the Number of Transmit Antennas on the Capacity in the

Presence of Colored Noise for N = 4. . . . . . . . . . . . . . . . . 34

2.17 Effect of γint on the Capacity for M = N = 4. . . . . . . . . . . . 35

2.18 Effect of Mint on the Capacity for M = N = 4. . . . . . . . . . . . 36

2.19 Effect of Nc,int on the Capacity for M = N = 4. . . . . . . . . . . 36

2.20 Autocorrelation of the Signal Envelope. . . . . . . . . . . . . . . . 38

2.21 MIMO Channel Capacity in the Presence of Correlation. . . . . . 41

2.22 Effect of Correlation Factor on the Capacity for M = N = 4. . . . 43

2.23 Effect of τ rms on the Relative Degradation of the Capacity in the

Presence of Correlation for M = N = 4. . . . . . . . . . . . . . . 44

2.24 MIMO Channel Capacity for a Ricean Fading Channel. . . . . . . 45

2.25 Effect of K on the Capacity for a Ricean Fading Channel for M =

N = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.26 Effect of τ rms on the Relative Degradation of the Capacity for a

Ricean Fading Channel for M = N = 4. . . . . . . . . . . . . . . 47

2.27 CDF of Capacity Improvement with Channel Knowledge at the

Transmitter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.1 Block Diagram of the Receiver. . . . . . . . . . . . . . . . . . . . 50

3.2 Block Diagram of the DiverQAM Space-Time Baseband Receiver. 52

3.3 Partial Plan of the 5th Floor of the UCLA Engineering IV Building. 54

viii

3.4 Plan of the Cubicle Area on the 5th Floor of the UCLA Engineering

IV Building. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 30 Mbps Slicer-SNR with 4 FFF DFE Taps for Small Range Trans-

mission in Cubicle Area. . . . . . . . . . . . . . . . . . . . . . . . 59

3.6 Comparison of T- and T/2-Spaced Equalizer with 4 FFF Taps. . . 62

3.7 Slicer-SNR CDF for Different Equalizer Sizes with a Single Antenna. 63

3.8 Slicer-SNR CDF for Different Equalizer Sizes with a Four Elements

Smart Antenna Array. . . . . . . . . . . . . . . . . . . . . . . . . 64

3.9 Slicer-SNR CDF when Using a 4 FFF Taps DFE. . . . . . . . . . 65

3.10 Slicer-SNR CDF Using Smart Antenna Array Only. . . . . . . . . 66

3.11 Simulated Slicer-SNR for Different Array Sizes and a 4-Taps FFF

DFE. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

3.12 Simulated Slicer-SNR for Different DFE Sizes. . . . . . . . . . . . 70

4.1 MIMO Channel Model. . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2 Architecture of the MIMO DFE Receiver with Cancellation. . . . 76

4.3 Slicer-SNR of MIMO DFE Receiver with Cancellation. . . . . . . 82

4.4 BER Performance of MIMO DFE Receiver with Cancellation. . . 84

4.5 CDF with Receive Diversity for M = 4. . . . . . . . . . . . . . . . 87

4.6 CDF for Different τ rms for M = N = 4. . . . . . . . . . . . . . . . 88

4.7 CDF for Various MIMO DFE Algorithms with M = N = 4. . . . 89

4.8 CDF for Various MIMO DFE Algorithms with M = N = 8. . . . 90

4.9 Simulated Capacity for Uncoded MIMO DFE with Cancellation

System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

ix

4.10 CDF for Correlated Channels for M = N = 4. . . . . . . . . . . . 93

4.11 CDF for Ricean Channel for M = N = 4. . . . . . . . . . . . . . . 94

4.12 CDF in the Presence of Colored Noise for M = N = 4. . . . . . . 96

4.13 Effect of γint on the Slicer-SNR CDF for M = N = 4. . . . . . . . 97

4.14 Effect of Mint on the Slicer-SNR CDF for M = N = 4. . . . . . . 98

4.15 BER Performance in the Presence of Colored Noise for M = N = 4. 99

4.16 VBLAST Receiver. . . . . . . . . . . . . . . . . . . . . . . . . . . 102

4.17 CDF for Flat Fading Channel for M = N = 4. . . . . . . . . . . . 104

4.18 CDF for Flat Fading Channel in the Presence of Colored Noise for

M = N = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.19 BER Performance for MIMO Flat Fading Channel for M = N = 4. 106

5.1 Architecture of MIMO OFDM Transmitter. . . . . . . . . . . . . 109

5.2 MIMO OFDM Channel Model. . . . . . . . . . . . . . . . . . . . 110

5.3 Architecture of MIMO OFDM Receiver. . . . . . . . . . . . . . . 111

5.4 Comparison of CDF for MIMO OFDM and MIMO DFE Receivers. 117

5.5 CDF for Interpolated MIMO OFDM Receiver for τ rms = 0.5Ts. . . 121

5.6 CDF for Interpolated MIMO OFDM Receiver for τ rms = Ts. . . . 121

5.7 CDF for Interpolated SISO OFDM Receiver for τ rms = 0.5Ts. . . . 122

5.8 Effect of Ordering Algorithm. . . . . . . . . . . . . . . . . . . . . 124

5.9 Performance of Different MIMO OFDM Interpolation Algorithms. 126

5.10 CDF for Different Cancellation Algorithms for MIMO OFDM Re-

ceivers Using Interpolation. . . . . . . . . . . . . . . . . . . . . . 128

x

6.1 Architecture of MIMO CDMA Transmitter. . . . . . . . . . . . . 131

6.2 MIMO Generalized RAKE Receiver with Cancellation. . . . . . . 133

6.3 Combining MIMO Generalized RAKE Receiver with Cancellation. 140

6.4 CDF for MIMO Generalized RAKE Receiver with Cancellation. . 146

6.5 Slicer-SNR CDF for Outdoor Channel. . . . . . . . . . . . . . . . 148

6.6 Slicer-SNR CDF for Ls = 32. . . . . . . . . . . . . . . . . . . . . 149

6.7 Slicer-SNR CDF for the Combiner MIMO RAKE Receivers with

Cancellation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150

7.1 MSE Learning Curves for the MIMO DFE Receiver with Cancel-

lation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167

7.2 SNR Learning Curves for the MIMO DFE Receiver with Cancel-

lation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

7.3 Comparison of RLS and Inverse QR Algorithms. . . . . . . . . . . 170

7.4 Learning Curves in the Presence of Colored Noise. . . . . . . . . . 171

7.5 Architecture of the MIMO Channel Estimation Receiver. . . . . . 172

7.6 Performance of Channel Estimation Algorithms. . . . . . . . . . . 174

7.7 Learning Curves for the MIMO OFDM Receiver with Cancellation. 176

7.8 Learning Curves for the MIMO Generalized RAKE Receiver with

Cancellation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

7.9 Learning Curves with Spreading Sequence Length Ls = 32. . . . . 180

xi

List of Tables

2.1 Frequency Selective Fading Channel Parameters . . . . . . . . . . 17

3.1 Environment Description . . . . . . . . . . . . . . . . . . . . . . . 56

3.2 Receiver Configurations . . . . . . . . . . . . . . . . . . . . . . . 57

3.3 4-QAM 5% Outage SNR . . . . . . . . . . . . . . . . . . . . . . . 60

3.4 Comparison of 5% Outage SNR for a 4 FFF Taps DFE . . . . . . 61

5.1 5% Outage Slicer-SNR Decrease for Interpolated OFDM . . . . . 123

7.1 Convergence Properties for CDMA RLS . . . . . . . . . . . . . . 179

7.2 Convergence Properties for CDMA LMS . . . . . . . . . . . . . . 179

xii

List of Abbreviations

ADSL Asymmetric Digital Subscriber Line

BER Bit Error Rate

BERT Bit Error Rate Tester

CCI Co-Channel Interference

CDF Cumulative Distribution Function

CDMA Code Division Multiple Access

CMA Constant Modulus Algorithm

DFE Decision Feedback Equalization

DFT Discrete Fourier Transform

DMT Discrete Multitone

DSL Digital Subscriber Line

DSSS Direct Sequence Spread Spectrum

FBF Feedback Filter

FFF Feedforward Filter

FIR Finite Impulse Response

FFT Fast Fourier Transform

IDFT Inverse DFT

xiii

IID Independently Identically Distributed

ISI Inter-Symbol Interference

LMS Least-Mean Square

MAC Multiple Access Control

MIMO Multiple Input Multiple Output

MMSE Minimum Mean Squared Error

MRC Maximum Ratio Combining

MSE Mean Squared Error

OFDM Orthogonal Frequency Division Multiplexing

OVSF Orthogonal Variable Spreading Factor

QAM Quadrature Amplitude Modulation

RLC Radio Link Control

RLS Recursive Least Square

rms root-mean square

SER Symbol Error Rate

SIMO Single Input Multiple Output

SISO Single Input Single Output

SNR Signal to Noise Ratio

SVD Singular Value Decomposition

xiv

UMTS Universal Mobile Telecommunications System

VBLAST Vertical Bell Labs Layered Space-Time

VGA Variable Gain Amplifier

WCDMA Wideband CDMA

xv

Acknowledgments

I am thankful to my advisor Dr. Babak Daneshrad for providing several years

of constant academic and technical guidance. His multi-disciplinary approach

and global vision of research problems was instrumental in defining my research

directions. I would also like to thank him for his numerous editorial comments on

the publications we wrote together and helping me expressing clearly my ideas on

paper. Finally, I am grateful for the financial support he provided to me during

my studies at UCLA.

I would like to thank my other committee members, Dr. Gregory Pottie,

Dr. Ali Sayed, and Dr. James Gibson, for their insightful suggestions and their

precious time. I also had the opportunity, during my years at UCLA, to learn

from many other professors through courses and discussions. My special thanks

go to Professors Rick Wesel, Kung Yao, Izhak Rubin, and Ken Yang.

I also appreciated the technical interaction with my fellow students at UCLA.

I am specially thankful to Jeffrey Putnam for the great ASIC he provided me to

perform the field measurements. I also want to thank Erik Berg, Edward Roth,

and Kyung-Ho Cha for the help with the testbed. I also say thank you for the

enjoyable technical discussions and the great time we had together to my colleges

and friends from the Wireless Integrated System Research Group, including the

old-timers Eugene Grayver, Ahmed El-Tawil, Hanli Zou, and Danijela Cabric,

and the new-comers Stephan Lang, Jingming Wang, Christian Oberli, Raghu

Rao, Alireza Mehrnia, Steve Hsu, and Jatin Bathia. I also want to thank other

friends from UCLA like Alireza Tarighat and Kambiz Shoarinejad .

I would not have been able to accomplish this work without the support and

encouragement from my family. I would like to take this opportunity to thank

xvi

my parents to educating me and giving me the mean to pursue my goals. I also

thank my sister for her help and encouragement. Last, but not least, I would

like to thank my beloved wife Quyen. Your love, support, understanding and

encouragement were what made the completion of this dissertation possible.

Finally, I acknowledge DARPA and the National Sciences and Engineering

Research Council of Canada for the financial support of this work.

xvii

Vita

June 28, 1973 Born, Montreal, Canada

1994-1995 Research Assistant, Fractal Analysis Research Group, Ecole

Polytechnique de Montreal, Montreal, Canada

1995 Intern Engineer, SPAR Aerospace, Montreal, Canada

1995-1996 Research Assistant, Communications Research Group, Ecole

Polytechnique de Montreal, Montreal, Canada

1996 B.Eng., Electrical Engineering, Ecole Polytechnique de

Montreal, Montreal, Canada

1996 System Engineer, Nortel, Montreal, Canada

1997-1998 Teaching Assistant, University of British Columbia, Vancouver,

Canada

1998 M.A.Sc., Electrical Engineering, University of British

Columbia, Vancouver, Canada

1998-2000 Graduate Student Researcher, Electrical Engineering Depart-

ment, University of California, Los Angeles, CA

2001-2003 Director of Wireless Communications Systems, Innovics Wire-

less Inc., Los Angeles, CA

2004 Ph.D., Electrical Engineering, University of California, Los An-

geles

xviii

Publications

J.F. Frigon, and V.C.M. Leung, “A Pseudo Bayesian Aloha Algorithm with Mixed

Priorities for Wireless ATM,” in Proceedings of Ninth IEEE International Sym-

posium on Personal, Indoor, and Mobile Radio Communications (PIMRC’98),

Boston, MA, vol. 1, pp. 45-49, September 1998.

J.F. Frigon, H.C.B. Chan, and V.C.M. Leung, “Data and Voice Integration in DR-

TDMA for Wireless ATM Networks,” in Proceedings of 1999 IEEE International

Conference on Communications (ICC’99), Vancouver, Canada, vol. 3, pp. 1696-

1700, June 1999.

J.F. Frigon, H.C.B. Chan, and V.C.M. Leung, “A Variable Bit Rate Resource

Allocation Algorithm for Wireless ATM,” in Proceedings of Globecom Telecom-

munications Conference (GLOBECOM’99), Rio de Janeiro, Brazil, vol. 5, pp.

2673-2677, December 1999.

J.F. Frigon, and B. Daneshrad, “Field Measurements of High Speed QAM Wireles

Transmission Using Equalization and Real-Time Beamforming,” in Proceedings

of Globecom Telecommunications Conference (GLOBECOM’99), Rio de Janeiro,

Brazil, vol. 4, pp. 2102-2106, December 1999.

J.F. Frigon, B. Daneshrad, J. Putnam, E. Berg, R. Kim, W. Sun, and H. Samueli,

“Field Trial Results for High Speed Wireless Indoor Data Communications,”

IEEE Journal on Selected Areas in Communications, vol. 18, no. 3, pp. 297-309,

March 2000.

xix

J.F. Frigon, and V.C.M. Leung, “A Pseudo Bayesian Aloha Algorithm with Mixed

Priorities,” ACM/Baltzer Wireless Networks, vol. 7, pp. 55-63, January 2001.

J.F. Frigon, V.C.M. Leung, and H.C.B. Chan, “Dynamic Reservation TDMA

Protocol for Wireless ATM Networks,” IEEE Journal on Selected Areas in Com-

munications, vol. 19, no. 2, pp. 370-383, February 2001.

J.F. Frigon, and B. Daneshrad, “A Multiple Input-Multiple Output (MIMO)

Receiver for Wideband Space-Time Communications,” in Proceedings of 12th

IEEE International Symposium on Personal, Indoor, and Mobile Radio Commu-

nications (PIMRC’01), San Diego, CA, vol. 1, pp. 164-168, September 2001.

J.F. Frigon, and B. Daneshrad, “Field Measurements of an Indoor High-Speed

QAM Wireless System Using Decision Feedback Equalization and Smart Antenna

Array,” IEEE Transactions on Wireless Communications, vol. 1, no. 1, pp. 134-

144, January 2002.

J.F. Frigon, and B. Daneshrad, “A Multiple Input-Multiple Output (MIMO)

Adaptive Decision Feedback Equalizer (DFE) with Cancellation for Wideband

Space-Time Communications,” International Journal of Wireless Information

Networks, vol.9, no. 1, pp.13-23, January 2002.

A. ElTawil, E. Grayver, H. Zou, J.F. Frigon, G. Poberezhskiy, and B. Daneshrad,

“Dual Antenna UMTS Mobile Station Transceiver ASIC for 2 Mb/s Data Rate,”

in Proceedings of the IEEE International Solid-State Conference, San Francisco,

CA, vol. 46, pp. 146-147, February 2003.

xx

J..F. Frigon, A. ElTawil, E. Grayver, Y. Li, and B. Daneshrad, “Space-Time

Wireless Communication Systems for Indoor and Outdoor Environments, ” Sub-

mitted to IEEE Communications Magazine, October 2003.

xxi

Abstract of the Dissertation

System Architectures for Space-Time

Communications over Frequency Selective

Fading Channels

by


Doctor of Philosophy in Electrical Engineering

University of California, Los Angeles, 2004

Professor Babak Daneshrad, Chair

Space-time MIMO communication is a powerful technique that provides a sig-

nificant improvement in spectral efficiency. The application of MIMO communi-

cation systems to frequency selective fading channels is studied in this disserta-

tion. The research is oriented toward communication systems with a complexity

suitable for implementation. These systems exploit the multipaths and the mul-

tiple transmit/receive antennas to deliver high channel throughput to the user.

The research presented in this dissertation covers the theoretical aspect, opti-

mal implementation, and practical adaptive algorithm for wideband MIMO. It

thus provides a solid basis for the designer of MIMO communication systems for

frequency selective fading environments.

The theoretical capacity of wideband MIMO systems under colored noise is

derived and used to demonstrate the multi-fold increase in capacity offered by

these systems. The effect on the MIMO wideband capacity of frequency di-

versity, transmit/receive antennas configurations, multi-users and multi-devices

environments, channel correlation, line-of-sight propagation, and channel knowl-

xxii

edge at the transmitter is characterized. Field measurements of indoor high speed

wireless communications realized using a testbed featuring real-time equalization

and smart antenna array technology are also presented. The field measurements

demonstrate the improvement provided by a smart antenna array in realistic

frequency selective fading channel conditions.

Different system architectures for wideband MIMO are studied. Equaliza-

tion, multi-carrier and spread spectrum forms of MIMO receivers are considered

in this dissertation. Novel receivers are introduced such as an optimal finite

length MIMO DFE receiver with cancellation, a generalized MIMO RAKE re-

ceiver with cancellation, and low complexity combiner MIMO RAKE receivers.

The performance of the wideband MIMO receivers is studied and compared for

various system configurations and channel environments. Several simulation re-

sults show that their behavior is conformed to the theoretical wideband MIMO

channel capacity.

Novel LMS and RLS adaptive algorithms are proposed for the MIMO DFE

receiver with cancellation. A new MIMO inverse QR RLS algorithm with bet-

ter stability properties and lower complexity than the RLS algorithm is also

introduced. Extension of the adaptive algorithms to MIMO OFDM and the gen-

eralized MIMO RAKE receiver are finally derived.

xxiii

CHAPTER 1

Introduction

Communication system engineers are currently facing the challenge of provid-

ing robust high data rate links in a limited radio spectrum. This calls for the

development of sophisticated signal-processing techniques providing spectral ef-

ficient communication links. Furthermore, in order to provide an alternative

to traditional high speed wired networks such as Digital Subscriber Line (DSL)

services and Ethernet networks, wideband signaling techniques should be used.

Traditional solutions for the physical layer realization include systems based on:

Quadrature Amplitude Modulation (QAM) with Decision Feedback Equalization

(DFE), Direct Sequence Spread Spectrum (DSSS) Code Division Multiple Ac-

cess (CDMA) and Orthogonal Frequency Division Multiplexing (OFDM). These

techniques use temporal signal processing to mitigate the Inter-Symbol Inter-

ference (ISI) introduced by the wideband frequency selective fading channel.

However, they offer a limited spectral-efficiency. One way to improve these sys-

tems is to combine temporal processing with spatial processing that exploits the

spatial dimension. Such space-time processing operates with multiple transmit

antennas and/or multiple receive antennas and improve the link capacity and

quality by reducing the Co-Channel Interference (CCI), mitigating the ISI and

taking advantage of the rich channel diversity provided by the extra wireless

links [PP97, PL98].

Foschini has shown that by exploiting the fading properties of the wireless

1

1 2 3 4 5 6 7 8 9 100

5

10

15

20

25

30

35

40

45

50

Number of Receive Antennas

Cap

acity

(bp

s/H

z)Traditional system (number of receive/ transmit antennas=1)

Number of transmit antennas=Number of receiver antennas

Smart antenna array (number of transmit antenna fixed to 1)

Figure 1.1: 95% Outage Capacity at 20 dB SNR for Narrowband Channels.

environment, a communication structure with multiple transmit/receive anten-

nas can deliver enormous capacities [Fos96, FG98]. Figure 1.1 illustrates the

potential of space-time communication systems in narrowband channels com-

pared to a traditional Single Input Single Output (SISO) architecture (number of

transmit and receive antenna equal to one) and a Single Input Multiple Output

(SIMO) smart antenna array system (one transmit antenna and multiple receive

antennas). The capacity of the Multiple Input Multiple Output (MIMO) commu-

nication system where the number of transmit antennas is equal to the number of

receive antennas increases linearly with the number of receive antennas whereas

the capacity of the smart antenna array system increases logarithmically. The

theoretical capacity of the MIMO channel shows that the spectral efficiency of

the wireless communication link can be increased by several order of magnitude

by using multiple transmit/receive antennas. The theoretical MIMO capacity is

2

Space-TimeChannel Encoder

Space-TimeModulator

MIMO FrequencySelective Fading

Channel

Space-TimeDetector

Space-TimeChannel Decoder

Figure 1.2: Block Diagram of Space-Time Communication System.

obtained by combining space-time coding and signal processing techniques, and

several architectures have been proposed to implement space-time systems for the

narrowband flat fading wireless environment [FGV99, NTS98, TSC98]. It should

also be noted that the spectral efficiency per transmit antenna is approximately

5 bps/Hz and therefore requires multi-level signaling techniques. This should be

taken into account when designing the system architecture.

1.1 Scope and Objectives of the Dissertation

In order to deliver spectral efficient high data rates in wireless networks, the

concept of MIMO space-time communication needs to be extended to the wide-

band frequency selective fading channel and is the subject of this dissertation.

Figure 1.2 shows the general block diagram of a space-time communication sys-

tem. The research presented in this dissertation is focused on the blocks inside

the dashed box. The capacity of the MIMO frequency selective fading channel

is studied to understand the channel potential and behavior, and architectures

for the realization of wideband space-time modulator and demodulator is inves-

tigated. The research is specifically oriented toward communication and receiver

architectures with a complexity suitable for implementation. These systems are

designed to efficiently exploit the multipath and multiple transmit/receive anten-

nas to provide the best estimation of the transmitted data. A correctly designed

3

wideband MIMO modem allows, with a reasonable implementation complexity,

the overall communication systems to approach the theoretical capacity and be-

havior. Space-time channel coding exploit the space and time dimensions to

efficiently protect the data and is also an important component of the space-time

communication system. However, it is not in the scope of this dissertation.

Different space-time structures have been studied for the MIMO dispersive

channel. Approaches using multi-variable equalization techniques [YR94, TAS95,

AS99, MHC99, CC99], OFDM [RC98, ATN98], and DSSS-CDMA [HVF99] have

been reported. The objective of the research presented in this dissertation is to

investigate more efficient wideband MIMO techniques with low complexity, to

compare and validate the performance of these algorithms, and to provide a solid

basis for the designer of MIMO communication systems for frequency selective

fading environments. The specific objectives of the dissertation are:

• Derive the channel capacity of the MIMO frequency selective fading channel

in the presence of colored noise;

• Study the effect of the MIMO system configuration, the parameters of the

MIMO frequency selective fading channel, and the noise properties on the

theoretical wideband MIMO channel capacity;

• Provide field measurements of a wideband smart antenna array communica-

tion system to characterize the performance of a multiple antennas system

under realistic channel conditions and validate the space-time channel mod-

els;

• Find the optimal solution for an efficient MIMO equalization based system

with low complexity and suitable to adaptive techniques;

4

• Study the performance of the MIMO equalizer for various system configu-

rations and compare with theoretical capacity;

• Derive a MIMO system based on OFDM techniques and study its perfor-

mance and the impact of frequency interpolation;

• Find the optimal solution for an efficient MIMO system using the general-

ized RAKE receiver technique;

• Derive low complexity sub-optimal CDMA MIMO receivers;

• Compare the performance of MIMO receivers using equalization, OFDM

and CDMA techniques;

• Find stable and efficient Least-Mean Square (LMS) and Recursive Least

Square (RLS) adaptive algorithms for the equalization, OFDM and CDMA

MIMO receivers.

The research presented in this dissertation therefore covers the theoretical as-

pects, optimal implementation, and practical adaptive algorithms of wideband

MIMO communication systems.

1.2 Dissertation Outline

The dissertation is organized as follows. In Chapter 2, the capacity of MIMO

systems for frequency selective fading channels in the presence of colored noise

is derived and analyzed. Field measurements and simulation results of a smart

antenna array testbed are presented in Chapter 3 and illustrate, through a real

system, the feasibility and potential of wideband communications space-time sys-

tems. In Chapter 4, a novel MIMO DFE receiver with cancellation is described

5

and studied. MIMO communication systems based on OFDM and CDMA tech-

niques are introduced in Chapter 5 and Chapter 6, respectively. Adaptive al-

gorithms for the MIMO receivers for frequency selective fading channels are de-

scribed in Chapter 7. Finally, the dissertation is concluded in Chapter 8.

6

CHAPTER 2

Capacity of MIMO Frequency Selective Fading

Channels

In this chapter the theoretical capacity of a communications system with multiple

transmit and receive antenna over a frequency selective fading channel in the

presence of colored noise is derived. The results presented in this chapter show

the significant improvement in capacity that MIMO systems can yield compare

to traditional single input single output systems. The large capacity promised by

information theory is one of the major motivation for the research in the area of

wideband communications space-time systems.

2.1 Channel Capacity Model

Figure 2.1 illustrates a general space-time communication system over a frequency

selective fading channel. The system consists of M transmit antennas and N

receive antennas. Assume that for every channel use i, the MIMO channel input is

given by xi = [x1,i · · ·xM,i], and the channel output is given by yi = [y1,i · · · yN,i].

The data is transmitted over a frequency selective fading channel Ht that can be

7

x1,i

xM,i

z1,i

zN,i

y1,i

yN,i

Ht

Figure 2.1: MIMO Channel.

expressed as:

Ht =

h1,1,0 · · · h1,N,0

.... . .

...

hM,1,0 · · · hM,N,0

.... . .

...

h1,1,Nc · · · h1,N,Nc

.... . .

...

hM,1,Nc · · · cM,N,Nc

,

where Nc + 1 is the number of taps of the multipath channel, and hm,n,nc is the

nc + 1th tap of the channel joining the mth transmit antenna to the nth receive

antenna. Furthermore, the output of the frequency selective fading channel is

corrupted by additive noise given by zi = [z1,i · · · zN,i]. The noise is assumed to

be jointly Gaussian and independent of the transmit data x.

Now, let assume that this system is use Nb consecutive time and is initialized

8

with a zero state (i.e., the channel delay lines are initialized with zeros). Let

define a 1×NbM transmit data vector x = [xNb · · ·x1], a 1×NbN receive vector

y = [yNb · · ·y1], and a 1×NbN noise vector z = [zNb · · · z1]. The noise is assumed

to be colored with a correlation matrix Rz = E[z∗z]. The receive vector is then

given by:

y = xH + z. (2.1)

The channel matrix H consists of the first NbM rows of the matrix [Ht0 · · ·

HtNb−1], where

Hti =

0iM×N

Ht

0(Nb−i−1)M×N

.In the following sections, the capacity of this system is derived for the cases where

the transmitter does not have or has information about the channel state.

2.2 Capacity without Channel State Knowledge

In this section, the capacity of the MIMO wideband channel is derived for the

case where the transmitter has no a priori knowledge about the channel state.

From information theory, it is known that the capacity of this channel is given

by [CT91]:

C =1

Nb

(h(y)− h(y|x)

)bits/use. (2.2)

The conditional entropy of the received data y given the transmit data x is:

h(y|x) = h(xH + z|x)

= h(z)

=1

2log2

((2πe)NNb |Rz|

).

(2.3)

9

The channel capacity is maximized when the entropy of the received data y

is maximized. For a given Ry, h(y) is maximized if y is jointly Gaussian, which

occurs if the transmitted data x is also jointly Gaussian. In this case, the entropy

of the received data y is given by:

h(y) =1

2log2

((2πe)NNb|Ry|

)=

1

2log2

((2πe)NNb|H∗RxH + Rz|

).

(2.4)

Since the transmitter has no knowledge about the channel state, the best it

can do is to transmit a data vector x which is jointly Gaussian, and Independently

Identically Distributed (IID) in both the space and time dimensions. Assuming

that the total transmit power per channel used is P , Rx is given by:

Rx =P

MIMNb. (2.5)

The capacity of the MIMO wideband channel when the transmitter has no a

priori knowledge about the channel state is then given by:

C =1

2Nblog2

(|H∗RxH + Rz|

|Rz|

)bits/use

=1

2Nblog2

(∣∣ PM

H∗H + Rz

∣∣|Rz|

)bits/use.

(2.6)

If it is also assumed that the noise z is IID in both the space and time dimensions,

and that the noise variance is given by σ2, Rz is

Rz = σ2INNb (2.7)

and the capacity is then given by:

C =1

2Nb

log2

∣∣∣∣ P

Mσ2H∗H + INNb

∣∣∣∣ bits/use. (2.8)

From this result, the well-known capacity formula of the MIMO flat fading

channel [Fos96] can easily be obtained by letting Nb = 1:

C =1

2log2

∣∣∣∣ P

Mσ2H∗H + IN

∣∣∣∣ bits/use. (2.9)

10

The previous approach used the time domain to compute the capacity of the

wideband channel considered as a whole. Another approach is to use a frequency

domain approach to divide the frequency selective fading channel into a collec-

tion of flat fading channels through a Fourier Transform. Assume that an Nf

Fourier Transform is employed. Note that a frequency transmission requires Nf

consecutive uses of the channel to transmit the information. Then, let define the

channel coefficient hpm,n, 1 ≤ m ≤M , 1 ≤ n ≤ N , 0 ≤ p < Nf , by:

hpm,n =Nc−1∑k=0

hm,n,ke−j 2πpk

Nf (2.10)

and the frequency channel matrix Hp, 0 ≤ p < Nf , by:

Hp =

hp1,1 · · · hp1,N

.... . .

...

hpM,1 · · · hpM,N

.Then, the capacity of the MIMO wideband channel when it is computed through

a frequency domain approach and the transmitter has no a priori knowledge about

the channel state is given by:

C =1

2Nf

Nf−1∑p=0

log2

(∣∣ PM

Hp∗Hp + Rzp∣∣

|Rzp|

)bits/use. (2.11)

Rzp is the auto-correlation of the noise for the pth sub-carrier. To simplify the

analysis, the noise is assumed to be independent in the frequency domain.

2.3 Capacity with Channel State Knowledge

In this section, the capacity of the MIMO wideband channel is derived for the

case where the transmitter has a priori knowledge about the channel state. That

is the transmitter knows the channel matrix H and the noise autocorrelation Rz.

11

From information theory, it is known that the capacity of this channel is given

by:

C = maxRx

1

Nb

(h(y)− h(y|x)

)bits/use. (2.12)

Again, the conditional entropy of the received data y given the transmit data x

is:

h(y|x) =1

2log2

((2πe)NNb|Rz|

), (2.13)

and the entropy of the received data y is given by:

h(y) =1

2log2

((2πe)NNb|H∗RxH + Rz|

). (2.14)

The capacity of the channel is therefore:

C = maxRx

1

2Nb

log2

(|H∗RxH + Rz|

|Rz|

)bits/use. (2.15)

Since the logarithmic function is monotically increasing and Rz is independent

of Rx, the capacity problem is to find Rx which maximizes |H∗RxH+Rz| under

the fixed transmit power constraint Tr(Rx) = PNb.

Let Rz = LL∗

(i.e., L is the Cholesky decomposition of Rz). The maximiza-

tion problem is then equivalent to:

maxRx

|L−1||H∗RxH + LL∗||L−∗| = max

Rx

|L−1H∗RxHL

−∗+ I|

= maxRx

|C∗RxC + I|(2.16)

where C = HL−∗

.

The Singular Value Decomposition (SVD) of C is given by C = USV∗, where

U and V are unitary (i.e., UU∗ = U∗U = I and VV∗ = V∗V = I),S has the

same dimensions as C, and the diagonal elements of SS∗ are denoted by λi,

1 ≤ i ≤MNb. The maximization problem then becomes:

maxRx

|VS∗U∗RxUSV∗ + I| (2.17)

12

which is equivalent to:

maxRx

|V∗||VS∗U∗RxUSV∗ + I||V| = maxRx

|S∗U∗RxUS + I|

= maxRx

|S∗RwS + I|(2.18)

where Rw = U∗RxU.

Now let consider the transmit power constraint Tr(Rx) = PNb. The following

property can be derived:

Tr(Rw) = Tr(U∗RxU)

= Tr(U∗URx)

= Tr(Rx)

= PNb.

(2.19)

Therefore, the initial problem is reduced to find Rw which maximizes |S∗RwS +

I| under the constraint Tr(Rw) = PNb. This is a classic water-filling prob-

lem [CT91, RC98]. |S∗RwS+I| is maximized when Rw is diagonal,∑MNb

i=1 Rwi,i =

PNb, and

Rwi,i =

(ν − 1

λi)+ if λi > 0

0 if λi = 0

(2.20)

where

(x)+ =

x if x > 0

0 if x ≤ 0.

(2.21)

That is, Rw is such that more power is allocated to channels with larger gain

(i.e., larger singular values).

Thus the capacity of the MIMO wideband channel when the transmitter has

a priori knowledge about the channel state is given by:

C =1

2Nb

log2

(|H∗RxH + Rz|

|Rz|

)bits/use, (2.22)

13

where Rx = URwU∗, or x = wU∗ where E[w∗w] = Rw. That is, x is transmit-

ted in the direction of the eigen vectors of the matrix C = HL−∗

with a power

proportional to the corresponding singular values.

The capacity of the frequency selective fading channel can also be computed

using the frequency domain approach. For each sub-channel p, 0 ≤ p < Nf ,

obtained through the Fourier Transform, Cp is given by:

Cp = Lp−∗

Hp = UpSpVp∗, (2.23)

where Lp

is the Cholesky decomposition of Rzp, Up and Vp are unitary, and

SpSp∗ is diagonal with diagonal elements λpi , 1 ≤ i ≤M .

The capacity of the MIMO wideband channel when it is computed through

the frequency domain approach and the transmitter has a priori knowledge about

the channel state is then given by:

C =1

2

Nf−1∑p=0

log2

(|Hp∗Rx

pHp + Rzp|

|Rzp|

)bits/use, (2.24)

where Rxp = UpRw

pUp∗ is the autocorrelation matrix of the vector of data

of length M transmitted on the pth, 0 ≤ p < Nf , frequency channel, Rwp,

0 ≤ p < Nf , is diagonal,∑Nf

p=1

∑Mm=1 Rw

pm,m = PNf , and

Rwpm,m =

(ν − 1

λpm)+ if λpm > 0

0 if λpm = 0,

(2.25)

for 0 ≤ p < Nf and 1 ≤ m ≤M .

2.4 Simulation Results

In the previous sections the capacity with and without channel state knowledge

at the transmitter was derived for a frequency selective fading MIMO channel

14

with colored noise. In this section, these theoretical results are used to analyze

through simulations the capacity of wideband MIMO communication systems for

various environments and configurations. The simulation models presented in

this section are also used through the dissertation to simulate and analyze the

performance of wideband MIMO communication systems.

2.4.1 Basic Model

In the basic model, the noise is modeled as a white interference. The noise random

variables zi,j are independent in the space and time domain with a correlation

matrix Rz = E[z∗z] = σ2I. The Signal to Noise Ratio (SNR) is defined as

the ratio between the average total power receive at one antenna and the noise

variance σ2. The total receive power is defined as the sum of the signal power on

all multipath components for all channels between every transmit antenna and

one receive antenna. Therefore, as the number of transmit antenna is increased,

the total transmit power from all antennas remains fixed in order to keep the

SNR constant.

The frequency selective fading channel between each transmit and receive

antenna is modeled as an Nc + 1 multipaths channel using an exponential power-

delay profile with complex Gaussian multipath components [Cox94]. Each Ray-

leigh multipath component hm,n,i, 1 ≤ m ≤ M , 1 ≤ n ≤ N ,0 ≤ i ≤ Nc, is

independent and is given by:

hm,n,i =

√1− e−γrms

2(1− e−(Nc+1)γrms)e−

iγrms2 (x+ jy),

where x and y are independent zero mean, unit variance, Gaussian random vari-

ables. The total power of all the multipaths between transmit antenna m and

receive antenna n is equal to one.

15

The root-mean square (rms) delay spread of the frequency selective fading

channel is controlled by the parameter γrms and the number of multipaths Nc+1.

The rms delay spread is a popular parameter to characterize dispersive channels

and is an indication of the energy spread of the channel and the bandwidth over

which the channel can be considered constant in the frequency domain (coherence

bandwidth). A rule of thumb is that if the rms delay spread is more than 10%

of the symbol period, the channel is frequency selective. Typical values of rms

delay spread are in the range of 50 to 200 ns for indoor environments and 1 to 3

µs for outdoor environments [Rap96]. Therefore, communication systems using a

symbol rate above 2 MHz will experience frequency selective fading for all typical

indoor environments.

The rms delay spread is the square root of the second central moment of the

power delay profile and is defined from a single power delay profile which is the

average of consecutive realizations of the multipath channel [Rap96]. The rms

delay spread for the channel model above is given by:

τ rms =√τ2 − τ 2

where τ is the mean excess delay of the channel and is defined by

τ =

∑i α

2i τi∑

i α2i

=

∑Nci=0 ie

−γrms∑Nci=0 e

−γrms,

and

τ2 =

∑i α

2i τ

2i∑

i α2i

=

∑Nci=0 i

2e−γrms∑Nci=0 e

−γrms.

Table 2.1 presents the frequency selective fading channel parameters used in this

dissertation to simulate channels with different rms delay spread. Note that τ rms

is specified as a fraction of the symbol period Ts.

For all the results, the capacity of the MIMO frequency selective fading chan-

nel is defined as the 95% outage capacity. That is, the capacity of the MIMO

16

Table 2.1: Frequency Selective Fading Channel Parametersτ rms Number of multipaths (Nc + 1) γrms

0.25Ts 2 2.630.5Ts 3 1.56Ts 4 0.52

channel was computed for 5000 independent realizations of the MIMO channel

and at least 95% of the recorded capacities were greater than the capacity pre-

sented in the results in this section. Unless mentioned otherwise, the results have

been obtained by simulating a MIMO frequency selective fading channel with

τ rms = 0.5Ts and a 20 dB average SNR at each receive antenna. The channel

capacity was computed using the frequency domain technique with Nf = 16 (the

choice of this technique is explained below). The reported capacity is also for

a system where the transmitter has no knowledge of the current channel state,

unless indicated otherwise.

Figure 2.2 shows the capacity of the frequency selective fading channel as a

function of the number of receive antennas. For this MIMO system, the number

of transmit antennas M was set equal to the number of receive antennas N . The

capacity of a smart antenna array system where the number of transmit antenna

is fixed to one is also included for comparison purposes. The results show that

the MIMO channel capacity for the wideband channel behaves similarly to the

capacity of MIMO narrowband systems reported previously [Fos96, FG98] and

illustrated in Figure 1.1. That is, the capacity of a MIMO system in a frequency

selective fading environment grows linearly with the number of transmit/receive

antennas. This demonstrates the tremendous potential of wideband MIMO sys-

tems in term of capacity increase. Furthermore, the capacity of the smart antenna

array system grows logarithmically as a function of the number of receive anten-

17

1 2 3 4 5 6 7 80

5

10

15

20

25


Cap

acity

(bi

ts/u

se)

Smart Antenna Array (M=1)

MIMO (M=N)

Figure 2.2: Capacity of the MIMO Frequency Selective Fading Channel.

nas. Therefore, although adding more receive antennas provides an improvement

in channel capacity, the enhancement is limited unless the number of transmit

antennas is increased as well.

Figure 2.3 compares the Cumulative Distribution Function (CDF) of a 1 by 1

MIMO system and a 4 by 4 MIMO system for a frequency selective fading chan-

nel. The CDF for each system have similar characteristics. Indeed, for systems

with a single transmit/receive antenna to MIMO systems with 8 transmit/receive

antennas, the mean of the channel capacity distribution is increasing but the vari-

ance always stayed within a range of 0.4 to 0.5 for all the MIMO configurations.

This indicates that for a system with an equal number of transmit and receive

antennas, the channel capacity is similarly distributed around the mean for its

specific antenna configuration.

18

0 2 4 6 8 10 12 140

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Capacity (bits/use)

Pro

babi

lity(

Cap

acity

<A

bsci

ssa)

M=N=1

M=N=4

Figure 2.3: CDF of MIMO Channel Capacity.

Figure 2.4 compares the capacity of the wideband MIMO channel when using

the time domain technique and the frequency domain technique to compute the

channel capacity. For both cases, the number of transmit antennas is set equal

to the number of receive antennas and Nb = Nf = 16. The two techniques yield

similar results and the difference in capacity is less than 1.5%. The capacity differ-

ence can be explained by the fact that the frequency domain capacity calculation

takes into account all the channel taps while the time domain approach truncates

the last NcM rows of the channel matrix. This creates a finite boundary effect

for the time domain channel which reduces the richness of the environment and

prevents the exploitation of all the information available. This is also illustrated

by the fact that the difference between the techniques increases as the number

of transmit/receive antennas increases since a larger portion of the channel is

ignored for the time domain capacity calculation. The corresponding boundary

19

1 2 3 4 5 6 7 80

5

10

15

20

25

Number of Transmit/Receive Antennas

Cap

acity

(bi

ts/u

se)

Time Domain

Frequency Domain

Figure 2.4: Comparison of Techniques for Capacity Calculation.

effect for the frequency domain would be to consider the effect of cyclic prefix

required to compute the FFT. Fortunately, this can be ignored when computing

the theoretical channel capacity.

The capacity of the channel is also affected by Nf and Nb in relation to

τ rms as illustrated in Figure 2.5 for M = N = 4. The results show that for

a channel with τ rms = 0.5Ts, the frequency domain capacity converges to the

maximum for Nf greater than 4. Note that this frequency resolution roughly

corresponds to the 10% factor where the channel can be considered as flat. If

the rms delay spread increases, the number of FFT points required to converge

to the maximal capacity of the channel increases. For example, for τ rms = Ts, it

was observed that Nf = 16 was sufficient to converge to the maximum capacity

value. The capacity of the time domain calculation also converges to the same

value when Nb increases, albeit much more slowly. The computational complexity

20

5 10 15 20 25 309

9.1

9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

10

Number of Trials/Number of FFT Points

Cap

acity

(bi

ts/u

se) Time Domain

Frequency Domain

Figure 2.5: Effect of Nb and Nf on the Capacity.

increases faster for the time domain approach as a function ofNb since the channel

matrix becomes much larger. Indeed, for channels with large τ rms, the required

Nb becomes large enough that the computation time is unreasonably large and

rounding effects start to affect the results, even with floating point calculations.

On the other hand, the FFT used for the frequency domain capacity calculation

keeps the problem complexity manageable and computation time reasonable. For

this reason, the frequency domain capacity calculation is used for the remainder

of this chapter.

The impact of the frequency selective fading channel rms delay spread on

the MIMO channel capacity is shown in Figure 2.6. τ rms = 0Ts corresponds to

a flat fading channel (i.e., a channel with a single multipath). The capacity of

the MIMO system, as for the traditional single antenna system, increases as a

function of the delay spread and the capacity improvement provided by the mul-

21

1 2 3 4 5 6 7 80

5

10

15

20

25


Cap

acity

(bi

ts/u

se)

τrms

=0Ts

τrms

=0.25Ts

τrms

=0.5Ts

τrms

=Ts

Figure 2.6: Effect of τ rms on the Capacity.

tipath diversity remains constant as a function of the number of transmit/receive

antennas. Therefore, for MIMO communication systems in frequency selective

fading environments, the capacity improvements provided by the frequency di-

versity and by the number of transmit/receive antennas are additive. However,

it should be noted that only the absolute capacity improvement due to the fre-

quency selective channel stays constant for different number of transmit/receive

antennas. The relative impact of multipaths on the system capacity decreases as

the number of antennas increases. That is, the MIMO properties provide a much

greater channel capacity increase than the frequency diversity.

The results presented previously are for a MIMO system with an equal number

of transmit and receive antennas. The effect of having more receive antennas

than transmit antennas is illustrated in Figure 2.7. The capacity for the cases

where the number of receive antennas is equal, one more, and two more, than the

22

1 2 3 4 5 6 7 80

5

10

15

20

25

Number of Transmit Antennas

Cap

acity

(bi

ts/u

se)

N=M

N=M+1

N=M+2

Figure 2.7: Capacity with Receive Diversity.

number of transmit antennas are presented. The capacity increases as a function

of the number of receive antennas and the impact is similar for different number

of transmit antennas. The capacity improvement due to increasing the number of

receive antennas is however less than increasing the number of transmit/receive

antennas. Therefore, independently of the number of transmit antennas, the

MIMO system behaves as a smart antenna array system when N > M . The only

difference is the initial capacity for N = M .

The smart antenna array behavior of the MIMO system as a function of the

number of receive antennas is better illustrated in Figure 2.8. It compares the

capacity of the MIMO system when the number of receive antennas is equal to

the number of transmit antennas (N = M) and when the number of transmit

antennas is fixed to 4 (M = 4). The MIMO system capacity exhibits a logarithmic

behavior, similarly to a classic smart antenna array system, as a function of

23

1 2 3 4 5 6 7 80

5

10

15

20

25


Cap

acity

(bi

ts/u

se)

M=N

M=N with Ch. Know.

M=4

M=4 with Ch. Know.

Figure 2.8: Effect of the Number of Receive Antennas on the Capacity for M = 4.

the number of receive antennas when M is fixed to 4 versus the linear capacity

increase of a N = M MIMO system. It is interesting to note that when the

number of receive antennas is less than the number of transmit antennas (N < 4),

the capacity is larger than for theN = M MIMO system. The additional transmit

antennas increase the capacity due to transmit diversity.

Figure 2.8 also illustrates the effect of channel knowledge at the transmitter on

the capacity of the MIMO frequency selective fading channel. For the case where

the number of receive antennas is larger or equal to the number of transmit

antennas, channel knowledge marginally increases the capacity of the system.

This is due to the fact that the rank of the channel matrix is equal to the number

of transmit antennas and minimal improvement can be obtained by changing

the correlation matrix of the transmitted signal. However, when M < N , the

rank is smaller than the number of transmit antennas and channel knowledge

24

1 2 3 4 5 6 7 80

5

10

15

20

25


Cap

acity

(bi

ts/u

se) M=N

M=N with Ch. Know.

N=4

N=4 with Ch. Know.

Figure 2.9: Effect of the Number of Transmit Antennas on the Capacity for

N = 4.

significantly improves the channel capacity since the transmitter energy can be

focused only in the direction of the available eigen vectors.

Figure 2.9 shows the effect of the number of transmit antennas on the channel

capacity of a wideband MIMO system with a fixed number of receive antennas

(N = 4). The logarithmic improvement due to transmit diversity is apparent for

M > 4. Also, the increase in capacity due to channel knowledge is significant only

when the channel matrix rank is smaller than the number of transmit antennas

(i.e., for M > 4). For N < 4, the capacity is larger than for a M = N system

due to the receive diversity provided by the additional receive antennas.

Figure 2.10 compares the effect of the number of transmit antennas and receive

antennas on the MIMO channel capacity when N = 4 and M = 4, respectively.

The following regions of operation can be identified. If the number of variable

25

1 2 3 4 5 6 7 82

4

6

8

10

12

14

Number of Variable Antennas

Cap

acity

(bi

ts/u

se)

M=N

M=4

M=4 with Ch. Know. and N=4 with Ch. Know.

N=4

Figure 2.10: Comparison of the Effect of the Number of Transmit and Receive

Antennas on the MIMO Channel Capacity.

antennas is less than four, for M = 4 the improvement over the MIMO system

with M = N is due to transmit diversity, while for N = 4 it is caused by receive

diversity. Inversely, when the number of variable antennas is greater than four,

the system exhibits transmit diversity for N = 4, and receive diversity for M = 4.

These results show that the capacity improvement due to receive diversity is

larger than for transmit diversity when the transmitter has no knowledge of the

channel state. However, it is interesting to note that with channel knowledge

at the transmitter, transmit and receive diversity are equivalent. Also, when

the number of variable antennas is less than four, the capacity increases linearly,

while when it is greater than four it has a logarithmic characteristic.

The following conclusions can therefore be drawn for MIMO systems in fre-

quency selective fading channels:

26

• Frequency diversity improves the MIMO capacity independently of M and

N ;

• The MIMO capacity increases linearly as a function of min(M,N);

• If M > N the MIMO system exhibits transmit diversity and the capacity

increases logarithmically as a function of M −N ;

• If N > M the MIMO system exhibits receive diversity and the capacity

increases logarithmically as a function of N −M ;

• Without channel knowledge, receive diversity performs better than transmit

diversity;

• With channel knowledge at the transmitter the capacity improvement due

to transmit diversity is the same as for receive diversity.

Figure 2.11 shows the capacity of the MIMO frequency selective channel as

a function of the SNR for M = N = 4. The channel capacity has a slight expo-

nential relationship with the SNR. The results illustrate the impact of channel

knowledge at the transmitter for low SNR even if the channel matrix rank is equal

to the number of transmit antenna. At a 0 dB SNR, the capacity improvement

is 32.5%, while at 20 dB SNR it is only 1.4%. This is due to the fact that at

a high SNR the system noise is mainly due to inter-symbol and inter-channel

interference and water filling does not help since increasing the signal in one

eigen channels greatly affects the performance of another channel. However, at

low SNR the additive noise is dominating and water filling can pour more en-

ergy in the eigen channels with better gains without significantly affecting the

performance of other channels.

27

0 2 4 6 8 10 12 14 16 18 201

2

3

4

5

6

7

8

9

10

11

SNR (dB)

Cap

acity

(bi

ts/u

se)

MIMO

MIMO with Ch. Know.

Figure 2.11: Effect of the SNR on the Capacity for M = N = 4.

2.4.2 Colored Noise Model

In this section, the effects of colored noise on the capacity of a MIMO communi-

cation system in frequency selective fading channels are analyzed using the model

depicted in Figure 2.12. The noise is modeled as a combination of colored noise

and white noise. The colored noise models the interference produced by another

MIMO user or device and is obtained by filtering a white Gaussian noise process

through a MIMO channel similar to the one used for data transmission. Formally,

the interference in this model is given by:

z =√γintvHint +

√1− γintn. (2.26)

The noise vectors n = [nNb · · ·n1], ni = [n1,i · · ·nN,1], and v = [vNb · · ·v1],

vi = [v1,i · · · vMint,1], are independent from each other, independent from the

transmitted data, and each jointly Gaussian with a correlation matrix Rn =

28

x1,i

xM,i

v1

vMint,i

y1,i

yN,i

Ht,Nc

Ht,Ncint

intγ

n1,i

int1 γ−

nN,i

intγ

int1 γ−

Figure 2.12: MIMO Channel Model for Colored Noise.

E[n∗n] = σ2I and Rv = E[v∗v] = σ2I, respectively. The channel interference

matrix Hint is constructed in a similar way to the channel matrix H. The only

difference is that it has Mint inputs and Nc,int + 1 multipaths. If Nc,int = Nc,

the interference channel matrix is modeled using the same τ rms as the channel

matrix. The correlation matrix of the noise is given by

Rz = σ2(γintHint

∗Hint + (1− γint)I). (2.27)

The properties of the colored noise are controlled by the following parameters:

Mint and Nc,int. The balance between colored and white noise in the interference

29

z depends on γint. For γint = 0 the interference is white and for γint = 1 it is

entirely colored. The average SNR at the receiver in the presence of colored noise

is given by the ratio between the average total power receive at one antenna and

the noise variance σ2.

The capacity of a MIMO communication system in a frequency selective fad-

ing environment with colored noise with Nc,int = Nc, Mint = M , and γint = 1, is

shown in Figure 2.13. The effect of having more receive antennas than transmit

antennas is also illustrated. This system emulates a network of identical MIMO

communication devices. The capacity of a system that is interference limited is

greater than when it is noise limited. This is due to the ability of the MIMO

system to partially null out the interference source, even if the number of re-

ceive antennas is equal to the number of transmit antennas. The beamforming

capability of MIMO is further illustrated by the fact than when the number of

receive antennas is increased, the capacity improvement is larger in the presence

of colored noise than with white noise. The improvement is due to a combination

of added receive diversity and beamforming capabilities provided by the extra

degrees of freedom.

Figure 2.14 shows the capacity of the MIMO system in the case where the

interference source is a traditional communication system with Mint = 1. The

interference source is mitigated by the MIMO receiver and the system provides

tremendous capacity. Even for the case where N = M , the MIMO system can

significantly alleviate the interference source. If an additional antenna is added

(N = M+1), the extra degree of receiver freedom is used to more effectively null

the interference and provides diversity and beamforming processing. However,

when a second additional receive antenna is supplied (N = M + 2), the capacity

improvement is less since the extra degree of freedom can not be used to cancel the

30

1 2 3 4 5 6 7 80

5

10

15

20

25

30

35


Cap

acity

(bi

ts/u

se)

N=M

N=M+1

N=M+2

N=M−γint

=1

N=M+1−γint

=1

N=M+2−γint

=1

Figure 2.13: MIMO Channel Capacity in the Presence of Colored Noise.

interference. The second additional antenna therefore mostly provides a diversity

gain.

The impact of the number of receive antennas on the MIMO channel capacity

in the presence of colored noise is illustrated in Figure 2.15. The number of

transmit antennas is fixed to M = 4. In the presence of white noise, as explained

previously, the additional receive antennas provide receive diversity. For the

case where the interference has a single transmit antenna (Mint = 1), the fifth

receive antenna provides beamforming capability, as seen from the large capacity

improvement. However, for the following additional antennas, the capacity is a

logarithmic function, which indicates a diversity improvement. This is due to the

fact that a single extra degree of freedom, provided by the fifth receive antenna, is

sufficient to efficiently create a beam in the direction of an interference source with

a single transmit antenna. However, a MIMO interference source with 4 transmit

31

1 2 3 4 5 6 7 80

5

10

15

20

25

30

35

40

45

50


Cap

acity

(bi

ts/u

se)

N=M

N=M+1

N=M+2

N=M−γint

=1

N=M+1−γint

=1

N=M+2−γint

=1

Figure 2.14: MIMO Channel Capacity in the Presence of Colored Noise with

Mint = 1.

antennas (Mint = M = 4) has a more complex spatial signature. Therefore, each

additional antenna provides extra beamforming and diversity capabilities. This

is indicated by the linear improvement in capacity as a function of the number of

receive antenna. The capacity for Mint = 4 initially improves linearly and then

converges logarithmically to the capacity of the system with Mint = 1, since each

extra receive antenna provides less interference cancellation than the previous

one and more diversity improvement.

For a MIMO system with a given number of receive antennas, a designer

might wonder if he should limit the number of transmit antennas to maximize

the channel capacity. Figure 2.16 illustrates the capacity tradeoffs as a function of

the number of transmit antennas for different system configurations. The results

presented have been obtained with the number of receive antennas fixed to N = 4

32

4 5 6 7 88

10

12

14

16

18

20

22

24

26

28


Cap

acity

(bi

ts/u

se)

White Noise

Mint

=M−γint

=1

Mint

=1−γint

=1

Figure 2.15: Effect of the Number of Receive Antennas on the Capacity in the

Presence of Colored Noise for M = 4.

and the number of transmit antennas limited to M = 4. In the presence of white

noise, as was explained previously, the MIMO capacity monotonously increases

with the number of transmit antennas and is therefore maximized for N = 4.

The MIMO communication system has a similar behavior if the interference is

colored and has a fixed number of transmit antennas (Mint = 1 or Mint = 4).

In a network where all MIMO users have the same number of transmit antennas

(Mint = M), the capacity is roughly similar for M = 2 to 4 with a maximum at

M = 3. Therefore, the following conclusion can be drawn: if the designer has

no control on the nature of the interference source (i.e., white or fixed number

of transmit antennas) he should used as many transmit antennas as possible; on

the other hand, if the system controls the interference source characteristics, the

number of transmit antennas should be limited in all the MIMO communication

33

1 2 3 42

4

6

8

10

12

14

16

18

20


Cap

acity

(bi

ts/u

se)

White Noise

Mint

=M−γint

=1

Mint

=1−γint

=1

Mint

=4−γint

=1

Figure 2.16: Effect of the Number of Transmit Antennas on the Capacity in the

Presence of Colored Noise for N = 4.

devices in order to allow a better interference cancellation at the receivers and

increase the total throughput.

The parameters γint, Mint, and Nc,int, affect the properties of the colored noise.

Figures 2.17 to 2.19 show the impact of these parameters on the capacity of the

MIMO channel in a frequency selective fading environment. The results presented

in Figure 2.17 show the capacity increase that occurs when the interference varies

from white noise (γint = 0) to fully colored noise (γint = 0). The effect of colored

noise starts to be dominant when γint = 0.5, that is, when half of the interfer-

ence power is colored. Figure 2.18 clearly shows the effect of the complexity of

the interference spatial signature on the capacity of the MIMO communication

system. For Mint = 1, the interference has a distinct spatial direction and can

be mitigated by the receiver, even if M = N . However, as the number of trans-

34

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 19

10

11

12

13

14

15

16

17

18

19

γint

Cap

acity

(bi

ts/u

se)

Mint

=M

Mint

=M

with Ch. Know.

Mint

=1

Mint

=1

with Ch. Know.

Figure 2.17: Effect of γint on the Capacity for M = N = 4.

mit antennas at the interference source increases and goes above the number of

receive antennas, the capacity decreases since the receiver has more difficulty to

cancel the complex spatial signature of the interference. Eventually, the receiver

can not take advantage of the structure in the noise and the capacity converges to

the capacity of a MIMO communication system where the noise has no structure

(i.e., capacity in the presence of white noise). Figure 2.19 shows that the char-

acteristics of the multipath channel has a limited impact on the capacity of the

wideband MIMO communication system. This is fortunate since this parameter

is not under the control of the designer. In fact, in most systems the multipath

characteristics of the interference channel and data channel are similar.

Figure 2.17 and 2.18 also illustrate the impact of channel state knowledge

(knowledge of both the transmission channel and the interference characteristics)

at the transmitter on the capacity of the MIMO system. The impact of channel

35

1 2 3 4 5 6 7 89

10

11

12

13

14

15

16

17

18

19

Mint

Cap

acity

(bi

ts/u

se)

White Noise

γint

=1

γint

=1 with Ch. Know.

Figure 2.18: Effect of Mint on the Capacity for M = N = 4.

1 2 3 4 5 6 7 80

5

10

15

20

25


Cap

acity

(bi

ts/u

se)

White Noise

γint

=1−Ncint

=Nc

γint

=1−Ncint

=0

Figure 2.19: Effect of Nc,int on the Capacity for M = N = 4.

36

knowledge is negligible in colored noise. Indeed, the improvement due to channel

knowledge decreases as the noise structure increases (i.e., when γint increases and

Mint decreases).

2.4.3 Channel Correlation and Ricean Distribution Model

The previous models assumed an ideal MIMO frequency selective Rayleigh fad-

ing channel. In particular, it was assumed that the Rayleigh fading multipaths

are independent for each pair of transmit/receive antennas. A non-line of sight

propagation was also assumed, resulting in a purely Rayleigh fading distribution.

In a real environment these assumptions might not hold. The impact of channel

correlation and line-of-sight propagation on the capacity of a wideband MIMO

channel is analyzed in this section.

The signals received at two antennas exhibit a level of correlation depending

on the distance between the antennas. If the antennas are close, the channels from

a transmitter to both antennas are similar and the signal received on each antenna

are highly correlated. For two distant antennas, the channels are different and

the received signals independent. Using the reciprocity property of the wireless

channel, a similar argument can be made for the correlation between two signals

received at a single antenna but transmitted from two different antennas. Clarke’s

model [Cla68, St96] is often used to characterize the correlation between the

envelope of signals received at two antennas and is given by:

Φ(d) = J0(2πd/λ). (2.28)

J0(x) is the zero-order Bessel function of the first kind, d is the distance be-

tween the antennas, and λ is the wavelength. Figure 2.20 shows the normalized

auto-correlation of the signal envelope as a function of the normalized distance

d/λ. Signals received at antennas separated by more than 0.4λ are essentially

37

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−0.5

0

0.5

1

Normalized distance (d/λ)

Nor

mal

ized

aut

o−co

rrel

atio

n

Figure 2.20: Autocorrelation of the Signal Envelope.

independent while for antennas separated by more than λ/4, the correlation is

less than 0.5. For a 2.4 GHz operating frequency, these correspond to an antenna

separation of 5 cm and 3.1 cm, respectively.

The following assumptions are made for the channel correlation model used

in the simulations:

• The channel coefficients for different multipaths are independent;

• The channel coefficients correlation between two transmit antennas is given

by α∆mt , where ∆m is the absolute value of the index separation between

the two transmit antennas and αt, 0 ≤ αt ≤ 1, is the transmit correlation

factor;

• The channel coefficients correlation between two receive antennas is given

by α∆nr , where ∆n is the absolute value of the index separation between the

38

two receive antennas and αr, 0 ≤ αr ≤ 1, is the receive correlation factor;

• The correlation for the channels between two pairs of transmit/receive an-

tennas is given by the product of the transmit and receive correlation values

(i.e., α∆mt α∆n

r ).

The channel matrix corresponding to this model is built as follows. Let define

the channel vector hi = [h1,1,i · · ·hM,1,i · · ·h1,N,i · · ·hM,N,i] for multipath i. The

coefficients hm,n,i are independently generated using the exponential power-delay

profile model presented in Section 2.4.1. Let also define the correlated chan-

nel vector hci = [hc1,1,i · · ·hcM,1,i · · ·hc1,N,i · · ·hcM,N,i] for multipath i. The auto-

correlation matrix of the channel vector hci is given by:

Rhci

= E[hc∗ih

ci]

=1− e−γrms

(1− e−(Nc+1)γrms)e−iγrmsRc

where

Rc =

α0tα

0r · · · αM−1

t α0r · · · α0

tαN−1r · · · αM−1

t αN−1r

.... . .

.... . .

.... . .

...

αM−1t α0

r · · · α0tα

0r · · · αM−1

t αN−1r · · · α0

tαN−1r

.... . .

.... . .

.... . .

...

α0tα

N−1r · · · αM−1

t αN−1r · · · α0

tα0r · · · αM−1

t α0r

.... . .

.... . .

.... . .

...

αM−1t αN−1

r · · · α0tα

N−1r · · · αM−1

t α0r · · · α0

tα0r

.

The correlated channel vector for multipath i is:

hci = hiT

c,

39

where Tc is the Cholesky decomposition of Rc (i.e., Rc = Tc∗Tc and Tc is upper

triangular). Since

E[h∗ihi] =1− e−γrms

(1− e−(Nc+1)γrms)e−iγrmsI,

the correlation matrix of hci is as desired. The elements of hc

i are then used to

fill up the matrix Ht for each multipath i, 0 ≤ i ≤ Nc.

In the presence of a line-of-sight propagation path, a dominant stationary sig-

nal component is present in the first multipath detected at the receiver. This has

the effect of adding a deterministic signal to the random multipath components

arriving from different angles at the receiver. The resulting signal is modeled

using a Ricean fading distribution [Rap96]. It is assumed that only the first mul-

tipath is modeled as Ricean. This is justified by the fact that later multipaths

are composed of several reflected signals without a strong dominant deterministic

signal.

For the line-of-sight propagation model, the multipaths for i ≥ 1 are generated

using the basic model in Section 2.4.1, or the correlation model presented in this

section. The first Ricean multipath component hm,n,0, 1 ≤ m ≤ M , 1 ≤ n ≤ N ,

is given by:

hm,n,0 =

√K(1− e−γrms)

(1 +K)(1− e−(Nc+1)γrms)+

1√1 +K

h′m,n,0,

where h′m,n,0 is obtained using the basic or the correlation multipath model. The

Ricean multipath is characterized by the parameter K, which is the ratio between

the deterministic and the Rayleigh signal power. The Rayleigh component power

determines the variance of the multipath. Note that the total power of the first

multipath is the same as for the other models and therefore the rms delay spread

of the multipath profile is unchanged by the Ricean distribution.

40

1 2 3 4 5 6 7 80

5

10

15

20

25


Cap

acity

(bi

ts/u

se)

α=0

α=0.5

α=0.75

α=1

Figure 2.21: MIMO Channel Capacity in the Presence of Correlation.

Figure 2.21 shows the capacity of the MIMO frequency selective fading channel

in the presence of channel correlation. For all the results presented in this section,

it is assumed that the envelope correlations at the transmitter and receiver are

equal (i.e. αt = αr = α). Channel correlation decreases the channel capacity,

however it is interesting to note that for channel correlation smaller than 0.75,

the linear increase of the capacity as a function of the number of antennas is

maintained. Only the slope of the linear relation decreases as a function of the

channel correlation. For a fully correlated channel (α = 1), the capacity increases

as a function of the number of transmit/receive antennas, but in a logarithmic

relationship. In this regime, the receiver operates as a beamformer and improves

the capacity by increasing the receive SNR (this can be simply accomplished by

using an Alamoutti space-time code).

The impact of the channel correlation factor on the MIMO channel capacity

41

is illustrated in Figure 2.22 for a system with four transmit and receive antennas

(M = N = 4). The capacity degradation has an exponential relation with

the channel correlation factor. For correlation values below 0.25, the capacity

degradation is negligible. Antenna spacing in MIMO systems is a huge concern

since it affects the device size. This result shows that antenna could be as close

as 0.3λ (3.75 cm at 2.4 GHz) without significantly degrading the MIMO system

performance. Even if the correlation increases to 0.5, the capacity decreases by

less than 10%. However, for correlation factors above 0.5, the capacity rapidly

degrades. Channel knowledge in the presence of channel correlation significantly

improves the capacity. The absolute improvement is larger as the correlation

factor increases and since the capacity also decreases, the relative improvement

becomes important, varying from 1% when α = 0 to 35% for α = 1. When

α = 1, channel knowledge allows the MIMO system to perform beamforming at

both the transmitter and the receiver.

Figure 2.23 shows the relative capacity degradation of a MIMO system with

M = N = 4 in a frequency selective environment for various values of channel rms

delay spread. It is clear that for α ≤ 0.9, flat fading channels (τ rms = 0Ts) and

frequency selective fading channels are equally affected by channel correlation.

That is, multipath diversity does not mitigate the effects of channel correlation.

As shown in Figure 2.6, the capacity of a wideband MIMO system is mainly de-

pendent on the number of transmit/receive antennas. Channel correlation affects

the independence of the eigen channels and equally reduces the effective MIMO

capacity of each multipath. Therefore, even if the multipaths are independent,

the frequency diversity does not help to improve the MIMO properties of the sys-

tem and the channel capacity decreases. The multipath diversity helps to reduce

the capacity degradation only for fully correlated channels (α = 1).

42

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 12

3

4

5

6

7

8

9

10

11

α

Cap

acity

(bi

ts/u

se)

Without Ch. Know.

With Ch. Know.

Figure 2.22: Effect of Correlation Factor on the Capacity for M = N = 4.

Figure 2.24 shows the impact of a line-of-sight propagation path on the MIMO

channel capacity in frequency selective fading environments. In a Ricean fading

channel, the capacity still linearly increases as a function of the number of trans-

mit/receive antennas. The ratio between the deterministic and random received

power (K) affects the slope of the linear relation which decreases as K increases.

The deterministic signal component increases with K, which has the effect of

increasing the correlation between the channel coefficients. Therefore, K has a

similar effect on the MIMO channel capacity as the channel correlation α. It is

interesting to note that for a small number of antennas, the capacity might be

greater for large values of K. This is due to the fact that, although it degrades

the MIMO properties of the signal, the deterministic component of the received

signal decreases the dynamic range of the received signal and therefore increases

the outage capacity.

43

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

10

20

30

40

50

60

70

80

α

Cap

acity

Dec

reas

e in

Per

cent

age

τrms

=0Ts

τrms

=0.25Ts

τrms

=0.5Ts

τrms

=Ts

Figure 2.23: Effect of τ rms on the Relative Degradation of the Capacity in the

Presence of Correlation for M = N = 4.

The impact of K on the wideband MIMO channel capacity is illustrated in

Figure 2.25 for M = N = 4. If the deterministic signal component is less than

the random signal power (i.e., K < 0 dB), the capacity degradation due to the

line-of-sight propagation environment is negligible. For some values of K, the

MIMO outage capacity is even slightly better in a Ricean environment. The

capacity decreases due to the line-of-sight propagation is much less severe than

for the channel correlation as seen by comparing with Figure 2.22. For example,

if the line of sight signal power is 10 times larger than the random component,

the channel capacity is still acceptable. The results presented in Figure 2.25

also show that channel knowledge at the transmitter provides a larger capacity

improvement as K increases.

Figure 2.26 shows the relative capacity degradation as a function of K for

44

1 2 3 4 5 6 7 80

5

10

15

20

25


Cap

acity

(bi

ts/u

se)

Rayleigh

Ricean−K=0 dB

Ricean−K=10 dB

Figure 2.24: MIMO Channel Capacity for a Ricean Fading Channel.

a MIMO system with M = N = 4 in a frequency selective environment with

various values of channel rms delay spread. Unlike channel correlation, the value

of τ rms has an impact on the influence of K on the channel capacity degradation.

For example, in a flat fading environment, K = 0 dB results in a 5% MIMO

capacity degradation, while for τ rms = Ts, K increases to 15 dB before a 5%

degradation occurs. This is explained by the fact that line of sight propagation is

only present in the first multipath. Therefore, as K increases the MIMO property

of the first multipath degrades but other multipaths are unaffected. Therefore,

for large rms delay spread, multipath diversity helps to mitigate the effect of the

Ricean fading channel in the first multipath and preserves much of the MIMO

channel properties.

45

−20 −15 −10 −5 0 5 10 15 208.2

8.4

8.6

8.8

9

9.2

9.4

9.6

9.8

10

10.2

K (dB)

Cap

acity

(bi

ts/u

se) Rayleigh

Ricean

Ricean with Ch. Know.

Figure 2.25: Effect of K on the Capacity for a Ricean Fading Channel for

M = N = 4.

2.4.4 Channel Knowledge

The simulation results presented in the previous sections have shown that channel

knowledge at the transmitter has various impacts on the MIMO channel capacity

depending on the system configuration and the propagation environment. Fig-

ure 2.27 shows the CDF of capacity improvement provided by channel knowledge

at the transmitter for different cases. For each channel instance, the MIMO ca-

pacity for the frequency selective fading channel was computed with and without

channel knowledge at the transmitter. The relative improvement in percentage

was noted and the CDF for the environment and configuration compiled. Chan-

nel knowledge at the transmitter has a significant impact on the MIMO channel

capacity in the following cases:

46

−20 −15 −10 −5 0 5 10 15 200

5

10

15

20

25

30

35

40

45

50

K (dB)

Cap

acity

Dec

reas

e in

Per

cent

age

τrms

=0Ts

τrms

=0.25Ts

τrms

=0.5Ts

τrms

=Ts

Figure 2.26: Effect of τ rms on the Relative Degradation of the Capacity for a

Ricean Fading Channel for M = N = 4.

• Low signal to noise ratio;

• Number of transmit antenna greater than the number of receive antennas

(M > N);

• Large channel correlation (e.g., α at transmitter and receiver above 0.5);

• Large Ricean parameter K (e.g., K > 10 dB for τ rms = 0.5Ts).

47

0 1 2 3 4 5 6 7 8 9 100

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Capacity Improvement in Percentage

Pro

babi

lity(

Cap

acity

<A

bsci

ssa)

Basic

α=0.75

Ricean−K=10 dB

γint

=1

SNR=10 dB

M=4 and N=3

Figure 2.27: CDF of Capacity Improvement with Channel Knowledge at the

Transmitter.

48

CHAPTER 3

Smart Antenna Array Space-Time System

In this chapter, field measurements and simulation results for a smart antenna

array system with decision feedback equalization are presented. These results

illustrate, through a real system, the potential and feasibility of wideband com-

munications space-time systems. A smart antenna array system with decision

feedback equalization is a special case of a MIMO communication system where

a single transmit antenna is used resulting in a SIMO communication channel

instead of the MIMO link. A highly versatile QAM testbed system using a DFE

and a smart antenna array was designed around the DiverQAM chip fabricated

at UCLA [PS99]. Extensive field measurements were obtained using the testbed

and compared with simulation results.

3.1 Testbed Overview

The testbed developed for this study consists of a mix of commercially available

components, as well as elements designed and fabricated at UCLA. It operates

at a carrier frequency of 2.44 GHz in the middle of the ISM band. It is highly

versatile, and when the receiver baseband processing block is interfaced with

a PC, it provides an ideal environment for the study of QAM systems using

equalization and smart antenna array.

The testbed system consists of a transmitter and receiver section. The trans-

49

4 ChannelRF-to-IF

Down-Converter

RF inputFc= 2.44 GHz

IF inputFc= 43.75 MHz

4 Channel IFDown-Converter

IF inputFc= 7.5 MHz

A/D

UCLA DesignedBaseband

Demodulator ASIC(DIVERQAM)

10 bit samplingFs= 30 MHz

Controller/PCInterface

4

4

4

Figure 3.1: Block Diagram of the Receiver.

mitter is composed of a 2.4 GHz RF/IF front-end coupled with a commercially

available QAM modulator unit. Random data are generated in the modulator

unit and are modulated using any one of 4-, 16- or 64-QAM constellation at a

symbol rate of 5 Mbaud using a square-root raised-cosine pulse shape (α = 0.15).

The baseband signal is then up-converted and amplified to produce a RF signal

at a carrier frequency of 2.44 GHz. The transmit antenna is a homemade 14λ

sleeve monopole antenna with a nearly omni-directional beam pattern.

Figure 3.1 illustrates a system level block diagram of the receiver section. The

2.44 GHz RF signal is received using a 27.94 cm long printed circuit antenna array

designed at UCLA [JR97]. The antenna consists of four tab-monopole elements

spaced 12λ apart at 2.48 GHz (6.05 cm). Each tab-monopole element has a nearly

omni-directional azimuth pattern. The RF signals from the four antennas are

individually amplified and down-converted to an IF frequency of 43.75 MHz in

the RF-to-IF receiver unit. Each of the four RF down-converters uses a two-stage

down-conversion (dual-IF) architecture. A Variable Gain Amplifier (VGA) that

provides up to 40 dB of gain is also inserted in each of the four RF-to-IF down

50

conversion paths. The gain of the VGA amplifiers is set using an external voltage.

The four 43.75 MHz IF signals are then down-converted to a frequency of

7.5 MHz. A ten bit A/D operating at 30 MHz is used to convert the analog

baseband signal to an over-sampled digital signal. Prior to the A/D converter, a

second VGA amplifies the signal to properly load the A/D converter. This VGA

provides up to 20 dB of gain and is controlled by a signal sent from the automatic

gain control (AGC) loop in the DiverQAM baseband demodulator.

The four digital signals after the A/D’s are demodulated in the DiverQAM

ASIC. Through a micro-controller interface, a PC is used to reconfigure the base-

band demodulator and to monitor real-time information regarding the state of

the control and recovery loops, the equalizer and antenna array coefficients, the

scatter plot of the slicer input, and the slicer signal-to-noise ratio (SNR). A high

degree of programmability and observability is critical in the evaluation of wire-

less communication systems.

The DiverQAM block diagram is shown in Figure 3.2. The four 7.5 MHz

digital IF signals are first down-converted to baseband and decimated to 2 ×

fsymb. The variable decimator allows the demodulator to continuously support

symbol rate from 1 to 7.5 Mbaud. Furthermore, the variable decimator provides

a means to resample the signal with proper clock frequency and sampling phase.

For all reported experiments, the symbol rate was set to 5 Mbaud due to the

inability of the transmitter to handle variable symbol rates. The smart antenna

array provides the spatial processing and combines the signals from the four

antennas. The number of elements used in the antenna array can be selected

from one to four. The antenna array coefficients are updated once per symbol

using either the decision directed sign-LMS algorithm or the Constant Modulus

Algorithm (CMA) [Hay96]. The error signal used to update the smart antenna

51

Nyquist 2

Error

Logic

FFF

e(n)

FBF

e(n)

e(n)

Adaptive Decision

Feedback Equalizer

Antenna

Array

Taps

e(n)

Adaptive

Antenna Array

Carrier

Recovery

Timing

Recovery

AGC

2/jne

10 Variable

Decimator

10 Variable

Decimator

10 Variable

Decimator

10 Variable

Decimator

Figure 3.2: Block Diagram of the DiverQAM Space-Time Baseband Receiver.

array coefficients is provided by the slicer, and is the same as the error signal

used to adapt the equalizer coefficients. Note that both adaptive algorithms are

blind algorithms, however the receiver can also use, when available, a training

sequence to produce the error signal for the sign-LMS algorithm.

The combined signal from the smart antenna array is filtered through a square-

root raised-cosine Nyquist filter before entering the DFE which provides the time

processing of the signal. The Feedforward Filter (FFF) of the DFE equalizer can

be configured to operate in T- or T/2-spaced mode. For both modes of operation,

the feedforward filter can be programmed to contain from 1 to 8 taps. After the

feedforward section of the DFE, the signal is decimated by two and mixed with

a signal provided by the carrier recovery loop. The derotated signal is finally

combined with the output of the feedback filter section of the DFE to obtain the

demodulator soft-decision. The Feedback Filter (FBF) consists of 0, 8 or 16 taps.

52

For all the measurements using the DFE (i.e., more than one feedforward filter

tap), the feedback filter length was set to 8 taps. Finally, the synchronization

section creates the control signals for three control loops: carrier, symbol timing,

and AGC. Note that both the carrier and symbol timing recovery loops are fully

implemented in the digital baseband demodulator. More details on the testbed

can be found in [FDP00, FD02].

3.2 Test Environment

Using the space-time testbed described in the previous section, an extensive set

of 59,262 measurements was collected in typical working environments on the 5th

floor of the Engineering IV building at UCLA. For all the measurements, the

receiver section of the testbed was fixed and the transmitter section was put on

a cart and moved to gather measurements. For each cart location, measurements

were taken for seven distinct antenna placements separated by one wavelength.

These antenna spacing were chosen to be greater than λ/2 in order to ensure

independent channel impulse responses for any two antenna placements. The

measurement scenario assumed a base station to mobile wireless LAN environ-

ment. Therefore, the transmit antenna was placed above major obstacles at a

height of 1.7 m and the receive antenna was located at desktop level, approxi-

mately 1.2 m above the floor.

Figure 3.3 shows the floor plan of the Wireless Integrated Systems Laboratory

where a first set of measurements was taken. This environment characterizes the

performance of the testbed for wireless communications within a small room and

for transmissions through one or more walls. For the measurements conducted

in this environment, the receiver was fixed at a location inside room 54-116 as

indicated in Figure 3.3 while the transmitter unit was moved around on the floor

53

RX Location��

��

��

��

��

��

��

��

��

3 m 9.2

m

9.2 m9.7 m

2.4 m

2.7

m

Hallway

Room 54-114

Room 54-116

Paths along which TX wasmoved in each environment

Figure 3.3: Partial Plan of the 5th Floor of the UCLA Engineering IV Building.

as indicated by the dashed lines. Measurements were taken for three different

scenarios: (1) transmission within room 54-116; (2) transmission between rooms

54-114 and 54-116; (3) transmission between the hallway and room 54-116. The

transmit power level for 4-QAM transmission was set to 0 dBm for the first

scenario and to 10 dBm for the second and third scenarios. Measurements for 301

different antenna locations were taken for the first scenario, 231 antenna locations

for the second scenario, and 203 antenna locations for the third scenario.

The performance of the testbed in a typical open space cubicle environment

was also characterized. Figure 3.4 shows the floor plan of the student cubicle area

on the 5th floor of the UCLA Engineering IV building where the second set of

measurements was taken. The height of the cubicle walls is 1.65 m. Measurements

54

��

��

��

��

��

��

��

��

15.1 m

14.7

m

16.5 m

32.5 m

26.8 m

11.6

m

12.6 m

39.4 m

26.3 m

ReceiverLocation

Path along whichtransmitter wasmoved for small

range transmission

Path along whichtransmitter wasmoved for long

range transmission

CubiclesWall

Figure 3.4: Plan of the Cubicle Area on the 5th Floor of the UCLA Engineering

IV Building.

were grouped into two communication scenarios. The first set of measurements

shows the performance of the testbed for a communication range of 4 to 19 m,

while the second set of measurements characterizes the performance for long range

transmissions up to 33 m. Measurements for 224 antenna locations were taken for

the first scenario, and 203 antenna locations for the second measurement scenario.

Impulse response measurements were carried out using a vector network an-

alyzer in order to characterize the different wireless environments [RST91]. The

channel measurements provide a frame of reference with previous studies, and

55

Table 3.1: Environment Description

EnvironmentNumber

DescriptionTransmit

PowerWalls

TransmissionRange

Average rmsdelay spread

(τ rms)

1Within Room

54-1160 dBm 0 3 to 9 m

24.8 ns (0.12Tsymb)

2Between Rooms

54-114 and 54-11610 dBm 1 10 to 16 m

35.4 ns (0.18Tsymb)

3From Hallway to

Room 54-11610 dBm 1 or 2 6 to 20 m

31.2 ns (0.16Tsymb)

4Small Range inCubicle Area

10 dBm 0 4 to 19 m38.2 ns (0.19

Tsymb)

5Long Range inCubicle Area

10 dBm 0 or 1 17 to 33 m50.5 ns (0.25

Tsymb)

for the simulation results presented in Section 3.4. Table 3.1 summarizes the

different environments tested and their characteristics. Note that at a 5 MBaud

symbol rate, the signal bandwidth (5 MHz) is on the same order as the channel

coherence bandwidth (1/2πτrms).

For each antenna location, measurements using 4-, 16- and 64-QAM constel-

lations using 17 different receiver configurations were gathered. Thus a total of

59 262 measurements were collected in the five environments. The PC automati-

cally reconfigured the DiverQAM demodulator with the proper constellation size

and receiver configuration, and recorded the corresponding slicer-SNR. For each

of the five environments and 51 receiver configurations, a Cumulative Distribu-

tion Function was generated and the SNR performance at 5% outage was noted.

Each of the CDF consists of a minimum of 203 measurements, therefore using

outage figures below 5% would be prone to statistical uncertainty. The receiver

configurations were selected to characterize both the space (smart antenna array)

and time (DFE) performance of the testbed. Table 3.2 summarizes the different

receiver configurations that were tested for each QAM constellation (note that

56

Table 3.2: Receiver Configurations

Equalizer ModeNumber of DFE

Feedforward Filter TapsNumber of Antenna

Array ElementsT 4 1T 8 1

T/2 1 1T/2 2 1T/2 4 1T/2 6 1T/2 8 1T/2 1 2T/2 2 2T/2 4 2T/2 6 2T/2 8 2T/2 1 4T/2 2 4T/2 4 4T/2 6 4T/2 8 4

when more than one DFE feedforward filter taps is enabled, 8 DFE feedback

filters taps are always enabled).

The measurement procedure was the following. During the initialization

phase, a single antenna was enabled along with the carrier and timing track-

ing loops. Subsequently, if adaptive equalization was used in the configuration,

the coefficient update algorithm for the DFE was enabled with the CMA al-

gorithm. After several symbols, the adaptive algorithm was switched to the

decision-directed sign-LMS algorithm and the smart antenna array was enabled.

For configurations without adaptive equalization, the smart antenna array was

first enabled with the CMA algorithm and then switched to the decision-directed

sign-LMS algorithm. This initialization procedure was found to yield a higher lock

probability. After the initialization phase, if the Bit Error Rate Tester (BERT)

57

was locked on the incoming sequence, the SNR was recorded. Otherwise, the ini-

tialization procedure was repeated up to four times. If after the fifth acquisition

attempt, the demodulator had not been able to acquire the channel correctly,

the measurement was declared as “unlocked”. For measurements with the smart

antenna array, the procedure was repeated for each receive antenna used as the

initial acquisition path before declaring the measurement “unlocked”. It is impor-

tant to record these unsuccessful measurements since they are part of the system

outage. It is reasonable to assume that the SNR of an “unlocked” transmission

is lower than the smallest SNR measured. An arbitrary 0 dB SNR was therefore

assigned to these measurements. Finally, the measurements were taken during

normal utilization of the environment. That is, people were moving around while

measurements were gathered, resulting in a slowly varying channel. This reflects

an expected utilization of high-speed wireless communication systems.

3.3 Field Measurements

The general performance results obtained from the measurements is first dis-

cussed in this section. The specific impact of spatial processing (smart antenna

array) and time processing (DFE) on the system performance is then explored.

Figure 3.5 shows the 30 Mbps (64-QAM at 5 Mbaud) slicer-SNR CDF for the

testbed operating with 4 FFF DFE-taps for small range transmissions in the cu-

bicle area. The CDFs underscore the tremendous impact that spatial processing

has on the overall system performance. For example, at a 10% outage, the system

using a four element array delivers a Bit Error Rate (BER) of 5× 10−7 (26.8 dB

SNR). Decreasing the number of antenna to two results in a BER of 5×10−4 (23.1

dB SNR) while using only equalization does not provide an acceptable system

performance. These measurements show that a system transmitting 10 mW of

58

15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR(DB)

Pro

babi

lity

(Slic

er−

SN

R <

Abs

ciss

a)

1 Antenna

2 Antennas

4 Antennas

Figure 3.5: 30 Mbps Slicer-SNR with 4 FFF DFE Taps for Small Range Trans-

mission in Cubicle Area.

power, operating in an open cubicle area, and using a 4 tap FFF DFE and a 4 ele-

ment smart antenna array, can support a 19 m cell radius and deliver an uncoded

BER of 6 × 10−4 (23 dB SNR) at a 1% outage for 30 Mbps transmissions. The

saturation of the CDF at high SNR is due to a system SNR ceil of approximately

32 dB. This is a manifestation of the noise floor introduced by the quantization

effects and the residual errors of the adaptive algorithm and recovery loops in a

non-stationary environment.

Table 3.3 presents the 5% outage slicer SNR for all 4-QAM receiver configura-

tions and environments. These results are representative of the observed perfor-

mance of the smart antenna array space-time prototype since the field measure-

ments indicated that for a given channel and receiver configuration, the measured

SNR for different QAM constellations only change by the difference in the average

59

Table 3.3: 4-QAM 5% Outage SNR

EqualizerMode

Numberof

Antenna

WithinRoom54-116

FromRoom

54-114 to54-116

FromHallwayto Room54-116

SmallRangeCubicle

Area

LongRangeCubicle

AreaT - 4 Taps 1 17.4 dB 15.4 dB 12.4 dB 14.5 dB 0.0 dBT - 8 Taps 1 17.8 dB 15.7 dB 11.9 dB 15.8 dB 0.0 dB

T/2 - 1 Tap 1 6.5 dB 5.1 dB 5.5 dB 5.7 dB 0.0 dBT/2 - 2 Taps 1 11.1 dB 9.5 dB 7.7 dB 6.8 dB 0.0 dBT/2 - 4 Taps 1 17.3 dB 15.4 dB 11.9 dB 15.7 dB 7.5 dBT/2 - 6 Taps 1 17.4 dB 16.1 dB 13.1 dB 15.8 dB 8.4 dBT/2 - 8 Taps 1 16.8 dB 15.4 dB 12.2 dB 16.3 dB 8.1 dBT/2 - 1 Tap 2 13.9 dB 12.7 dB 12.3 dB 12.7 dB 9.4 dBT/2 - 2 Taps 2 12.4 dB 11.5 dB 12.2 dB 12.6 dB 10.0 dBT/2 - 4 Taps 2 24.7 dB 23.7 dB 18.8 dB 20.9 dB 13.3 dBT/2 - 6 Taps 2 24.9 dB 23.4 dB 19.0 dB 21.5 dB 12.5 dBT/2 - 8 Taps 2 25.1 dB 23.6 dB 18.9 dB 21.7 dB 13.3 dBT/2 - 1 Tap 4 22.6 dB 21.0 dB 17.9 dB 19.2 dB 14.9 dBT/2 - 2 Taps 4 14.8 dB 14.3 dB 13.6 dB 12.1 dB 12.2 dBT/2 - 4 Taps 4 26.8 dB 25.7 dB 22.7 dB 24.7 dB 17.3 dBT/2 - 6 Taps 4 27.4 dB 26.3 dB 22.9 dB 24.6 dB 17.0 dBT/2 - 8 Taps 4 27.2 dB 26.2 dB 22.8 dB 24.5 dB 17.0 dB

transmit power level (i.e., approximately 1 dB from 4-QAM to 16- and 64-QAM

constellations). However, a higher percentage of “unlocked” measurements was

observed with 16- and 64-QAM since, for the same SNR, 4-QAM has a lower BER

than 16- and 64-QAM; therefore, for 4-QAM the blind algorithms have a larger

probability of converging and locking than for larger QAM constellations. This

observation suggests a strategy where 4-QAM is initially used for frame synchro-

nization and training. Then, depending on the measured SNR, the configuration

can be switched to a higher QAM constellation for higher throughput.

It is also interesting to observe that the impact of the different receiver con-

figurations on the testbed’s performance follows similar trends in all five environ-

60

Table 3.4: Comparison of 5% Outage SNR for a 4 FFF Taps DFE

EnvironmentWithinRoom54-116

FromRoom

54-114 to54-116

FromHallwayto Room54-116

SmallRangeCubicle

Area

LongRangeCubicle

Area1 Antenna 17.3 dB 15.4 dB 11.9 dB 15.7 dB 7.5 dB4 Antennas 26.8 dB 25.7 dB 22.7 dB 24.7 dB 17.3 dB

Improvement 9.5 dB 10.3 dB 10.8 dB 9 dB 9.8 dB

ments. For example, Table 3.4 compares the 5% outage SNR for a system using

a DFE with 4 FFF-taps and either 1 antenna or a 4 element antenna array. This

indicates that the results presented in this chapter can be utilized with confi-

dence to predict the performance of high speed QAM wireless systems employing

equalization and smart antenna array technology.

3.3.1 Time Processing Performance

In the next two sections, only CDF’s for the 4-QAM mode operating between

room 54-114 and room 54-116 (environment 2) are presented. This is justified

by the discussion in the previous section which demonstrated that the field mea-

surements for a given constellation size and environment are representative of the

system performance trends with different constellation sizes and environments.

Figure 3.6 compares the system performance for the T-spaced equalizer mode and

the T/2-spaced equalizer mode when 4 FFF taps are used. On average, the T/2-

spaced equalizer provides a better performance. However, for low SNR’s, it was

observed that the T/2- spaced equalizer has a little more difficulty to acquire the

channel, which explains the fact that the T-spaced equalizer performs slightly

better for low outage probabilities (below 3%). However, this phenomenon is

expected to disappear when a training sequence is available, and it is therefore

61

10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR(DB)

Pro

babi

lity

(Slic

er−

SN

R<

Abs

ciss

a)

T−Spaced Equalizer

T/2−Spaced Equalizer

Figure 3.6: Comparison of T- and T/2-Spaced Equalizer with 4 FFF Taps.

preferable to use a T/2-spaced equalizer for a high-speed wireless system. Similar

results were obtained when the size of the FFF was increased to 8-taps.

The results presented in Table 3.3 show that the DFE time processing has a

significant impact on the performance of the system. However, it would be in-

teresting to determine the equalizer length that is required for indoor communi-

cations with this system. The DiverQAM demodulator offers complete flexibility

in the selection of the number of FFF taps while only a subset of possible FBF

lengths is available. This is due to the fact that the hardware cost for the FFF

is much higher than the FBF since the number of bits required in the datapath

is larger for the FFF. The number of FBF taps was therefore fixed at 8 taps.

Figure 3.7 shows the CDF’s of the slicer-SNR for different number of FFF equal-

izer taps when a single antenna is used. The CDF curves illustrate the impact of

the equalizer on the system performance and clearly show that 4 FFF taps are

62

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR(DB)

Pro

babi

lity

(Slic

er−

SN

R<

Abs

ciss

a)

1 FFF Tap

2 FFF Taps

4 FFF Taps

6 FFF Taps

8 FFFTaps

Figure 3.7: Slicer-SNR CDF for Different Equalizer Sizes with a Single Antenna.

sufficient to mitigate most of the ISI introduced by the channel.

Figure 3.8 presents the slicer-SNR CDF’s for different equalizer sizes when

a 4 element smart antenna array is employed. Unfortunately, it was observed

that the system with 2 FFF taps was highly unstable when the antenna array

was larger than one. This might be due to the fact that the error surface of

the decision-directed LMS algorithm could have several close local minima for

this configuration. Thus, when some decision errors are made, the algorithm

“jumps” from one local minimum to another. Therefore, the recorded SNR does

not reflect the minimum SNR that can be achieved using this configuration. This

problem is expected to be resolved by using a training sequence for adaptation.

The instability explains the fact that the 2 FFF tap equalizer performs worst

than the no equalizer (1-tap) case and therefore a 4-tap FFF equalizer is still

required to remove the ISI even when a 4 element smart antenna array is used.

63

10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR(DB)

Pro

babi

lity

(Slic

er−

SN

R<

Abs

ciss

a)

2 FFF Taps

1 FFF Tap

4 FFF Taps

6 FFF Taps 8 FFFTaps

Figure 3.8: Slicer-SNR CDF for Different Equalizer Sizes with a Four Elements

Smart Antenna Array.

Similar results were also observed for a 2 element smart antenna array. The

results presented in this section clearly indicate that a 4 tap FFF DFE with 8

FBF taps is able to remove most of the ISI for indoor wireless communications.

3.3.2 Spatial Processing Performance

In this section the impact of spatial processing on the performance of the testbed

is quantified. Figure 3.9 shows the impact of the array size when a DFE with 4

FFF taps is used. It is clear that the spatial processing provided by the smart

antenna array results in a significant performance improvement. For example,

using 2 antennas instead of one antenna gives an SNR increase of 7.1 dB at 10%

outage and 8.3 dB at 5% outage. On the other hand, doubling the array size

from 2 to 4 antennas only provides a SNR increase of approximately 2 dB for an

64

10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR(DB)

Pro

babi

lity(

Slic

er−

SN

R<

Abs

ciss

a)

1 Antenna

2 Antennas

4 Antennas

Figure 3.9: Slicer-SNR CDF when Using a 4 FFF Taps DFE.

outage in the range of 5% to 10%. However, this is enough to provide a reliable

transmission for larger QAM constellations (see Table 3.3) or reduce the BER by

an order of magnitude. Furthermore, for smaller outage the SNR improvement is

more significant when using 4 antennas. For example, at a 2% outage probability,

the SNR improves by 8.3 dB when using 2 antennas instead of one antenna, and

doubling the number of antennas to 4 provides an additional improvement of 5.2

dB.

Figure 3.10 compares the system performance when only the smart antenna

array is used. The CDF for the receiver configuration using only a 4 taps FFF

DFE is also included for comparison purposes. The impact of using a 2 elements

antenna array, without the DFE, instead of a single antenna is similar to the

improvement observed with the same smart antenna array configurations when

the DFE was employed in combination with the antenna array (Figure 3.9).

65

0 5 10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR(DB)

Pro

babi

lity(

Slic

er−

SN

R<

Abs

ciss

a)

1 Antenna − 1 FFF Tap

2 Antennas − 1 FFF Tap

1 Antenna − 4 FFF Taps

4 Antennas −1 FFF Tap

Figure 3.10: Slicer-SNR CDF Using Smart Antenna Array Only.

However, using a 4 elements smart antenna array without the DFE results in a

larger performance improvement. For example, doubling the number of antennas

from 2 to 4 gives an SNR improvement of 8.5 dB at an outage of 10%. It is

also interesting to observe that a 4 elements smart antenna array without a DFE

significantly outperforms a DFE-only system at low outage probabilities. For

example, the 10% outage SNR improves by 5.3 dB when using a 4 elements array

instead of a DFE with 4 FFF taps.

The results show that the SNR gain due to the smart antenna array is larger

than the gain provided in the main lobe of the array pattern of a beamformer (3

dB gain with 2 antennas and 6 dB gain with four antennas). This is an indication

that for indoor wireless communications a smart antenna array takes advantage of

diversity and mitigates the impact of multipath propagation by canceling strong

multipath components. The spatial filtering of multipath components is con-

66

firmed by comparing the results presented in Figure 3.9 and Figure 3.10. For

systems using 1, 2 and 4 elements smart antenna array, the 10% outage SNR was

improved by 10.8 dB, 10.3 dB and 4.2 dB, respectively, when a 4 FFF tap DFE

was added to the system. This shows that the 4 elements smart antenna array is

able to take advantage of the extra degrees of freedom to cancel strong multipath

components since the impact of adding a DFE to the system is not as significant.


The field measurements clearly demonstrated the tremendous impact that time

and space processing have on the performance of high speed wireless systems.

However, it is important to compare the field measurements with system simula-

tions in order to:

• Validate the field measurements;

• Verify the accuracy of simulation results to predict the performance of a

space-time system;

• Develop appropriate spatial channel models.

A complete simulation model of the testbed operating in the QPSK mode was

therefore setup. The simulation model includes the following components: squa-

re-root raised-cosine filters, 10 bits D/A and A/D converters, low pass RF filters,

AGC control loops, carrier and timing recovery, smart antenna array and DFE,

adaptive algorithms, and slicer error quantization. The simulation parameters

were set to model the channel for transmissions from Room 54-116 to Room

54-114.

67

A set of 200 channels was statistically generated according to an exponential

power-delay profile with complex Gaussian multipath components h(n) [Cox94].

The channels generated using this model give 200 independent realizations of the

impulse response from the transmitter to the kth antenna at the receiver:

ck(n) =99∑i=0

hk(i)δ(n− i), (3.1)

where

hk(n) = Ae−γn(x+ jy), A =

√1− e−2γ

2, γ =

Tsample

2τ rms, and {x, y} ∼ N(0, 1).

x and y, and therefore hk(n), are independent for each multipath and receive

antenna. The channel sampling frequency was set to 100 MHz and each channel

consisted of 100 multipaths. Therefore, the channel impulse response spanned

5 symbols. The ensemble average rms-delay-spread, τ rms, was set to 35 ns and

the channels were conditioned such that the ensemble average received SNR was

equal to 27 dB.

Figure 3.11 shows the simulation results for a 4-tap FFF DFE and different

antenna array sizes. The field measurements are also included in the figure for

comparison purposes. The simulation results reinforce the earlier conclusions re-

garding the importance of the smart antenna array. The simulations also provide

a reasonably good estimation of the system performance that can be expected by

employing a smart antenna array. For example, for a 10% outage the simulations

predict an improvement of 6.5 dB when increasing the array size from 1 to 2

antennas and an improvement of 3.5 dB when doubling the number of antennas

from 2 to 4 antennas. For the same configurations, the actual field measurements

indicated improvements of 7.2 dB and 2.4 dB, respectively.

68

10 15 20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR(dB)

Pro

babi

lity(

Slic

er−

SN

R<

Abs

ciss

a)

Field MeasurementsSimulation

1 Antenna

2 Antennas

4 Antennas

Figure 3.11: Simulated Slicer-SNR for Different Array Sizes and a 4-Taps FFF

DFE.

Figure 3.12 compares the simulated performance of a receiver with different

DFE sizes using a four elements smart antenna array. The simulation results

confirm the importance of adaptive equalization when a smart antenna array is

employed. They also support the previous conclusion that a 4-tap FFF DFE is

sufficient to mitigate the ISI. On the other hand, the simulations overestimate the

performance of the 1 FFF tap DFE by approximately 2.5 dB at a 10% outage.

This might indicate that although the model that was used matches the τ rms,

the multipath distribution is not the same as the actual channel distribution.

Generating channels using cluster of arrivals model [SV87] or ray-tracing tech-

niques [Rap96] could be more appropriate for accurate simulations but require

careful modeling of the environment.

69

20 25 30 350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR(DB)

Pro

babi

lity(

Slic

er−

SN

R<

Abs

ciss

a)

1 Tap

2 Taps

4, 6 and 8 Taps

Figure 3.12: Simulated Slicer-SNR for Different DFE Sizes.

3.5 Discussion

In this chapter field measurements conducted with a highly flexible prototype

unit for high speed wireless communications using adaptive equalization and

smart antenna array were reported. The motivation behind this work was to

characterize the performance of smart antenna array space-time systems under

various receiver configurations and environment conditions.

The results clearly showed the potential and feasibility of wideband commu-

nications space-time systems. The smart antenna array DFE system used for the

testbed is a single input multiple output communication system. The results ob-

tained with this system motivate further research in the area of more promising

MIMO wideband communications systems. Furthermore, the presented simula-

tions reinforced the confidence in simulation results as a predictor of the perfor-

70

mance of MIMO wideband systems in real environments.

71

CHAPTER 4

MIMO Equalization

The field measurements and simulation results presented in Chapter 3 clearly

demonstrated the potential of space-time systems under real channel conditions.

It therefore motivates further research activities in this area. Particularly, in

order to increase the capacity of the system, as shown in Chapter 2, the number

of transmit antenna should be increased to obtain a MIMO transmission link.

Equalization is a technique that has been developed to mitigate the ISI created

by multipath propagation in frequency selective fading channels [Rap96, Pro95].

The concept of equalization has been extended to MIMO communication channels

and multi-variable equalization techniques have been proposed [YR94, TAS95,

AS99, AS00, MHC99, CC99]. In this chapter a new framework for the analysis

of a modified MIMO Minimum Mean Squared Error (MMSE) DFE is proposed.

The modified MIMO DFE structure introduced here assumes that decisions for

each data streams are made sequentially and that current decisions of detected

data streams are used to compute the current estimate of a given data stream.

This structure therefore performs an operation similar to interference cancella-

tion and is an extension for the frequency selective fading channel of the MMSE

equivalent of the VBLAST algorithm [FGV99, GFV99, WFG98]. The perfor-

mance of a receiver using finite length feedforward and feedback matrix filters

with interference cancellation under ideal conditions (perfectly estimated chan-

nel and no feedback errors) can be determined using the new MIMO MMSE DFE

72

problem formulation that is proposed in this dissertation. This type of modified

MIMO DFE structure has been studied in [AS00]. However, the new framework

introduced in this chapter allows, in Chapter 7, the derivation of novel adaptive

algorithms for the modified MIMO DFE based on the stochastic gradient algo-

rithm (e.g., LMS algorithm) and the least square solution (i.e., RLS algorithm).

4.1 MIMO Equalization Model

Figure 4.1 shows a discrete-time model for the MIMO channel with M transmit

antennas and N receive antennas. A sequence of symbols {dm(i)} is transmitted

from each antenna, where the symbol sequences are assumed to be IID sequences

(both in time and space) and drawn from a QAM constellation with power σ2d.

The symbols are transmitted over the MIMO dispersive channel consisting of

M · N Finite Impulse Response (FIR) channels, Cm,n(z), of order Nc (i.e., each

channel consists of Nc + 1 multipaths). Each FIR channel Cm,n(z) connects the

transmit antenna m to the receive antenna n.

As shown in Figure 4.1, at any time instant i, the signal received at each of

the N antennas consists of a linear combination of the current symbols dm(i),

m = 1, . . . ,M , and of the previous symbols of each sequence {dm(j), j < i}.

Therefore, the channel introduces ISI and CCI. Furthermore, the MIMO channel

output at each antenna n is corrupted by additive noise vn(i), n = 1, . . . , N .

Let define the data vector d(i) = [d1(i) · · ·dM(i)], the transmit data vector

dt(i) = [d(i) · · ·d(i−Nc)], the received data vector u(i) = [u1(i) · · ·uN(i)], and

the noise vector v(i) = [v1(i) · · · vN(i)]. Additionally, let expand the channel

impulse response between transmit antenna m and receive antenna n as follows:

Cm,n(z) = cm,n(0) + cm,n(1)z−1 + · · · + cm,n(Nc)z−Nc. The MIMO channel can

73

C11(z)

C1N (z)

CMN(z)

C M1(z)

d1(i)

dM(i)

v1(i)

vN(i)

u1(i)

uN(i)

Figure 4.1: MIMO Channel Model.

then be expressed in the following matrix form:

Ct =

c1,1(0) · · · c1,N(0)...

. . ....

cM,1(0) · · · cM,N(0)...

. . ....

c1,1(Nc) · · · c1,N(Nc)...

. . ....

cM,1(Nc) · · · cM,N(Nc)

.

Then, the output of the channel at any time instant i is given by:

u(i) = dt(i)Ct + v(i). (4.1)

The purpose of the receiver is to process the current and previously received

data vectors, the previous data decisions, and the current available data decisions

in order to obtain an estimate of a delayed version of the data vector d(i−∆)

74

denoted by d(i−∆) = [d1(i − ∆) · · · dM(i − ∆)]. The delay ∆ is a parameter

chosen by the designer. The data decisions are obtained by applying the slicer

function to the vector d(i−∆) that maps each of the data estimates to the

closest QAM constellation point (in order to facilitate the theoretical analysis, the

decisions are assumed to be correct). This proposed modified DFE differs from

a conventional DFE structure by using the current decisions that are available

to obtain the data estimate. In the remainder of this dissertation, the modified

MIMO DFE is referred as a MIMO DFE receiver with cancellation since using

current decisions in the feedback in effect cancels the interference caused by these

transmitted data. Note that in order to have a “causal” receiver, the data must

be detected in a given order to be able to use current data decisions to compute

a data estimate. Let assume that higher-indexed data streams are detected first.

Then, the current decisions from data streams m+ 1 to M are used to obtain the

data estimate for stream m.

Figure 4.2 shows the structure of the MIMO DFE receiver with cancellation

that is considered. It can be observe that the MIMO DFE is a matrix coun-

terpart of the scalar DFE where the scalar delay line, the taps and the decision

are replaced, respectively, by a vector delay line, matrix taps and a decision

vector. Suppose that the MIMO DFE receiver with cancellation consists of Nf

feedforward delays and Q feedback delays. The N ×M feedforward tap matri-

ces are denoted by Fj, 0 ≤ j ≤ Nf , and the M ×M feedback tap matrices by

Bj, 0 ≤ j ≤ Q. To satisfy the “causal” constraint, B0 must be strictly lower

triangular (i.e., b0,i,j = 0 for i ≤ j).

Let introduce the data vector d = [d(i) · · ·d(i−Nc −Nf )], the receive vector

u = [u(i) · · ·u(i−Nf )], the noise vector v = [v(i) · · ·v(i−Nf)], and the feed-

back vector df = [d(i−∆) · · ·d(i−∆−Q)]. Let also define the channel matrix

75

Z-1u(i)N

F0 F1

Z-1

FNf

Z-1

B1

Z-1

BQ

M Md(i- )d(i- )

B0

M

Figure 4.2: Architecture of the MIMO DFE Receiver with Cancellation.

C = [C0 · · ·CNf] where

Cj =

0Mj×N

Ct

0M(Nf−j)×N

(4.2)

((·)k×l denotes a matrix with k rows and l columns). It can then be verify that:

u = dC + v. (4.3)

The elements of the noise vector v are zero-mean random variables with an

average variance σ2v and are uncorrelated with the transmitted data sequence d

(E[v∗d] = 0). The auto-correlation of v is given by Rv = E[v∗v] and the average

value of the diagonal elements of Rv is σ2v .

76

Then, let define the feedforward and feedback coefficient matrix by

F =

F0

· · ·

FNf

, and B =

B0

· · ·

BQ

, (4.4)

respectively. The data estimate vector d(i−∆) of the transmitted data d(i−∆)

for the MIMO DFE receiver with cancellation is then given by:

d(i−∆) = −dfB + uF

= [df u]

−BF

= yW.

(4.5)

The MIMO DFE with cancellation receiver error vector is e(i−∆) = d(i−∆)−

d(i−∆) = d(i−∆)− yW.

4.2 MMSE Solution for the MIMO DFE Receiver with

Cancellation

Finding an optimal solution for an architecture is important in several key as-

pects. First, it provides a mean to quickly analyze the receivers performance in

different environments and configurations as will be shown in Section 4.3. Second,

it supplies the optimal solution against which the Mean Squared Error (MSE)

performance of the adaptive algorithms can be compared.

The objective of the MMSE solution is to select the unknown entries of W

that minimize the covariance matrix of the error vector e. In order to solve

this problem the innovations of the vector y [KSH00] is used. Let denote the

innovation vector by x = [x1, . . . , xK ] and the receiver input vector by y =

77

[df u] = [y1, . . . , yK], where K = M(Q+ 1) +N(NF + 1). The innovation vector

needs to be white (i.e., Rx = E[x∗x] = I, where (·)∗ denotes the Hermitian

transpose operation) and xl is a linear combination of {yl, . . . , yK}. Next, let

Ry = L∗L, where L is lower triangular (i.e., L is the Cholesky decomposition of

Ry). Then, if x = yL−1

the innovation vector x has the desired properties.

The MIMO receiver finds the estimate dm(i−∆) using a linear combination of

{ym+1, . . . , yK}. Since {xm+1, . . . , xK} spans the same space as {ym+1, . . . , yK}

(this follows from the properties of the innovation vector [KSH00]), dm(i − ∆)

can also be written as:

dm(i−∆) =K∑

l=m+1

xlwxl,m m = 1, . . . ,M. (4.6)

Note that wxl,m = 0 for l = 1, . . . ,m.

The orthogonality principle states that the MMSE solution is such that for

m = 1, . . . ,M and m < l ≤ K:

E[x∗l em(i−∆)] = E[x∗l (dm(i−∆)− dm(i−∆))]

= E[x∗l (dm(i−∆)−K∑

k=m+1

xkwxk,m)]

= E[x∗l dm(i−∆)− wxl,m]

= 0.

(4.7)

The properties of the innovation vector x and the orthogonality principle then

lead to the following solution:

[Wx

]l,m

=

[Rd(i−∆)x

]l,m

m < l ≤ K,

0 1 ≤ l ≤ m,

(4.8)

([A]i,j denotes the element in row i and column j of the matrix A), for m =

1, . . . ,M , or

Wx ={Rd(i−∆)x

}strictly-lower

, (4.9)

78

where the notation Rd(i−∆)x = E[x∗d(i−∆)] is employed. The data estimates

for the MIMO DFE receiver with cancellation are therefore given by:

d(i−∆) = xWx

= x{Rd(i−∆)x

}strictly-lower

= yL−1{

L−∗

Rd(i−∆)y

}strictly-lower

,

(4.10)

and the optimal coefficient matrix is

Wo = L−1{

L−∗

Rd(i−∆)y

}strictly-lower

. (4.11)

The exact expression of the correlation matrices Ry and Rd(i−∆)y can be

derived from the problem formulation as follows. The autocorrelation matrix of

y is given by:

Ry = E[y∗y] =

RdfRdfu

∗

Rdfu Ru

(4.12)

where:

Rdf= σ2

dIM(Q+1) (4.13)

Rdfu = C∗Rdfd (4.14)

Rdfd =

0M∆×M(Q+1)

σ2dIM(Q+1)

0M(Nf+Nc−∆−Q)×M(Q+1)

(4.15)

Ru = C∗RdC + Rv (4.16)

Rd = σ2dIM(Nf+Nc+1). (4.17)

σ2d is the transmit signal variance and is set to P/M to maintain a constant

transmit power, regardless of the number of transmit antenna. Rv is the noise

79

vector correlation matrix. The cross correlation matrix Rd(i−∆)y is given by:

Rd(i−∆)y = E[y∗d(i−∆)] =

Rd(i−∆)df

Rd(i−∆)u

, (4.18)

where:

Rd(i−∆)df=

σ2dIM

0MQ×M

(4.19)

Rd(i−∆)u = C∗Rd(i−∆)d =

0M∆×M

σ2dIM

0M(Nf+Nc−∆)×M

. (4.20)

Let define the MMSE vector J = [J1 · · ·JM ] where Jm = E[em(i−∆)∗em(i−∆)].

J for the MIMO DFE receiver with cancellation is given by:

J = diag(Re(i−∆))

= diag(E[e(i−∆)∗e(i−∆)

])= diag

(E[(

d− d(i−∆))∗(

d− d(i−∆))])

= diag(Rd −Wo∗Rd(i−∆)y −Rd(i−∆)y

∗Wo + Wo∗RyWo).

(4.21)

diag(A) is defined as the vector of diagonal elements of A (i.e., diag(A) =

[a0,0 · · ·ai,i]).

Note that using a similar technique the solution for a traditional structure

of the MIMO DFE receiver without cancellation can be obtained. The optimal

coefficient matrix is given by:

Wo = L−1{

L−∗

Rd(i−∆)y

}lower(M)

, (4.22)

where

[{A}

lower(M)

]l,m

=

[A]l,m M < l ≤ K,

0 1 ≤ l ≤M.

(4.23)

The MMSE for this structure is still given by Equation 4.21.

80


The performance of the optimal MIMO MMSE DFE receiver with cancellation

is evaluated through simulations for different configurations and environments in

this section. Unless specified otherwise, the simulation channel conditions are

the following. The SNR was fixed to 20 dB. The SNR is defined as ρ = Pavg/σ2v

where Pavg denotes the average total energy received at one antenna (i.e., average

energy received from all transmit antennas and multipaths). σ2v is the average

noise energy at one antenna and the noise is assumed to be white (Rv = σ2vI). The

channel impulse response was generated using the Rayleigh fading exponential

power delay profile specified in Section 2.4.1 for τ rms = 0.5Ts (Nc = 2 and γrms =

1.56). The number of feedforward taps in the MIMO DFE was set to 4 (Nf = 3)

and the number of feedback filter taps to 3 (Q = 2). The estimation delay ∆ was

fixed to 3.

4.3.1 Frequency Selective Fading Channels

The slicer-SNR CDF of the MIMO MMSE DFE receiver with cancellation in

a Rayleigh frequency selective fading environment is shown in Figure 4.3 for

different MIMO antenna configurations. The CDF is generated using 1000 (2000

for M = N = 1) independent MIMO channel realizations. For each channel

realization, the optimal solution is computed and the MMSE Jm for each data

stream m, 1 ≤ m ≤ M , calculated using equation 4.21. The slicer-SNR Sm for

stream m is then given by:

Sm =E[d∗m(i−∆)dm(i−∆)]

E[e∗m(i−∆)em(i−∆)]

=1/M − Jm

Jm,

(4.24)

81

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

M=1−N=1

M=2−N=2

M=4−N=4

Figure 4.3: Slicer-SNR of MIMO DFE Receiver with Cancellation.

since

E[d∗m(i−∆)dm(i−∆)] = 1/M

= E[(dm(i−∆) + em(i−∆)

)∗(dm(i−∆) + em(i−∆)

)]

= E[d∗m(i−∆)dm(i−∆)] + E[e∗m(i−∆)em(i−∆)]

= E[d∗m(i−∆)dm(i−∆)] + Jm,

and E[d∗m(i − ∆)em(i − ∆)] = 0 using the orthogonality principle. The slicer-

SNR for each data stream and channel realization is recorded and used to plot

the CDF. The CDF indicates for each SNR on the abscissa the probability of

observing a data stream with a lower slicer-SNR. For example, for a MIMO

system with 4 transmit and receive antennas (M = N = 4), for a target SNR

of 14 dB the system outage is 10%. That is, there is is a 10% probability of

82

observing a slicer-SNR lower than the required SNR of 14 dB.

During the derivation of the optimal solution for the MIMO DFE with cancel-

lation it was assumed, without loss of generality, that data streams are detected

in reverse order from stream M to 1 and the current decisions from data streams

m + 1 to M are used to compute the data estimate for stream m. If a different

detecting order is preferred, the channel matrix rows can be appropriately re-

ordered such that the transmit antennas match the desired detection order. That

is, the data stream that needs to be detected first should correspond to transmit

antenna M and the last detected data stream corresponds to transmit antenna 1.

Myopic data detection ordering [FGV99] was used to obtain the results presented

in this section, unless mentioned otherwise. Myopic data detection is optimal in

the max-min sense: for each channel realization the algorithm maximizes the

worst data stream slicer-SNR. The myopic ordering algorithm is summarized as

follows: the data stream with the largest SNR, given that previously detected

current data are available to all undetected data stream, is detected next. The

first detected data stream is therefore the one with the largest SNR without can-

cellation. The second detected stream, is the one with the largest SNR given that

the first data stream is available for detection. This process continues to the last

detected data stream.

For the same amount of transmit power, the transmission data rate of the

MIMO system presented in this chapter increases linearly as a function of the

number of transmit antennas. The results presented in Figure 4.3 show an inter-

esting phenomenon for this MIMO receiver. As the number of transmit/receive

antennas increases, the outage probability is also improving. For example, for

a 10 dB target SNR (10−3 bit error rate for 4-QAM), the outage probability is

0.05%, 1.4% and 4.9% for a MIMO system with 4, 2, and 1 transmit/receive an-

83

0 5 10 1510

−3

10−2

10−1

100

SNR (dB)

BE

R

Theoretical − Without Cancellation

Theoretical − Cancellation

Optimal Feedback− Without Cancellation

Decision Feedback − Without Cancellation

Optimal Feedback− Cancellation

Decision Feedback −Cancellation

Figure 4.4: BER Performance of MIMO DFE Receiver with Cancellation.

tennas, respectively. Alternatively, using 4 transmit and receive antennas while

keeping the total transmit power level constant results in a 3.3 dB SNR improve-

ment at a 5% outage compared to the SISO system performance. Therefore, not

only the raw transmission data rate has been increased by 4 but the receive signal

quality has improved. These results illustrate the high data throughput that can

be expected with space-time communication systems.

Figure 4.4 shows the bit error rate performance of the MIMO DFE receiver

with cancellation in wideband channels. The BER was obtained by simulating

500 independent realizations of the Rayleigh frequency selective fading MIMO

channel with four transmit and receive antennas (M = N = 4). For each channel

realization, the noise level was varied to obtain different channel SNR. For each

channel realization and SNR, the MIMO system was simulated for 2500 4-QAM

symbols and the BER computed for each data stream. The BER in Figure 4.4 is

84

calculated by averaging the BER for all data streams and channel realizations for

a given SNR. The results compare the performance of the receiver for a config-

uration with and without cancellation. The performance of the optimal MIMO

system where correct decisions are used in the feedback and cancellation section

of the DFE (optimal receiver) is also compared with the performance of the sys-

tem when actual decisions, including errors, are feedback. The theoretical BER

of the MIMO system assuming a Gaussian error distribution is also shown. The

theoretical BER for stream m is obtained using the following equation [Pro95]:

PQm = 2

(1− 1√

Q

)erfc

(√Sm

2(Q− 1)

)[1− 1

2

(1− 1√

Q

)erfc

(√3Sm

2(Q− 1)

)]≈ log2(Q)PQ

m,B,

where PQm is the Symbol Error Rate (SER) for stream m and a Q points QAM

constellation (for the simulated case Q = 4), erfc is the complementary error

function, and PQm,B is the BER.

The optimal BER performance of the MIMO system with and without can-

cellation closely agrees with the theoretical BER. The maximum discrepancy is

below 0.4 dB for BER values below 0.1. The difference can be attributed to the

fact that the noise is non-Gaussian, as assumed for the theoretical calculation,

and the approximation used to obtain the BER from the SER. This result shows

that the theoretical BER can be used with confidence to quickly estimate the

BER performance of the optimal MIMO DFE receiver with cancellation.

The performance improvement of the optimal MIMO DFE receiver with can-

cellation compared to the receiver without cancellation increases as the BER

decreases. For example, for a BER of 0.1 the improvement is 1.4 dB while for a

BER of 0.01 the improvement is 3.5 dB. This underlines the importance of using

85

interference cancellation in MIMO systems. Optimal error free decision feedback

can be obtained by using forward error correction before feeding back the data

in the cancellation section of the receiver. However, this introduces a large la-

tency and requires multiple iterations similar to Turbo decoding. An alternative

approach is to feedback directly the decisions, including errors. For the MIMO

DFE with cancellation, the performance degradation due to error feedback de-

creases as the target BER Increases: from 2.2 dB to 1.6 dB for a BER from 0.1 to

0.01. For the MIMO DFE without cancellation, the performance is also degraded

due to the DFE feedback section and is constant to approximately 1.2 dB for all

SNR’s. The degradation is larger for the MIMO DFE receiver with cancellation

since it also relies on data feedback for cancellation in addition to MIMO ISI

suppression. For large BER, the system with cancellation even performs worst

than without cancellation. However, as the SNR increases, it outperforms the

system without cancellation: for SNRs larger than 8.4 dB it is preferable to use

a MIMO DFE receiver with cancellation and decision feedback than a receiver

without cancellation and error free data feedback.

The slicer-SNR CDF for a MIMO system with four transmit antennas and

receive diversity is shown in Figure 4.5. The outage performance of the system

improves, as expected, when the number of receive antennas increases. For ex-

ample, for a 10% outage probability the slicer-SNR improves by 2.5 dB when

the number of receive antennas increases from N = 4 to N = 5 and by 1.6 dB

when the number of antennas further increases to N = 6. The CDF illustrates

the receive diversity behavior of the MIMO DFE receiver with cancellation as

predicted in Section 2.4.1. The raw data throughput of the system is constant as

the number of receive antennas increases and the SNR improves logarithmically

as a function of the number of antennas. The SNR improvement allows for a user

throughput increase, after taking into account the channel coding.

86

8 10 12 14 16 18 20 22 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

M=4−N=4

M=4−N=5

M=4−N=6

Figure 4.5: CDF with Receive Diversity for M = 4.

The slicer-SNR CDFs for different rms delay spread of the frequency selective

fading channel are given in Figure 4.6 for a MIMO system with four transmit

and receive antennas (M = N = 4). The channel parameters for the different

τ rms are given in Table 2.1. To accommodate for the largest excess delay, the

parameters of the MIMO DFE receiver with cancellation have been changed for

this simulation to Nf = 4, Q = 3, and ∆ = 4. The results show that the MIMO

DFE with cancellation is able to take advantage of the frequency diversity of the

wideband channel to improve the slicer-SNR for low outages. For higher SNR

the performance deteriorates due to the added ISI. However, the throughput of a

system is dominated by the worst case performance at low outage. Therefore, the

improved SNR at low outage will provide a higher user throughput as predicted

by the capacity results presented in Section 2.4.1.

Optimal myopic ordering needs M detection steps. Each detection step m,

87

6 8 10 12 14 16 18 20 22 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

τrms

=0Ts

τrms

=0.25Ts

τrms

=0.5Ts

τrms

=Ts

Figure 4.6: CDF for Different τ rms for M = N = 4.

m = 1, . . . ,M , requires the computation of a new martix Ry, a Cholesky cal-

culation and matrix inversion of L, and computation of the optimal coefficient

matrix W and the MMSE for the remaining M −m data streams. Myopic de-

tection ordering is therefore computationally intensive. A sub-optimal detection

ordering algorithm is introduced to decrease the computational complexity. For

the sub-optimal algorithm, the MMSE vector for the M data streams is first

computed for the MIMO DFE receiver without cancellation. The data streams

are then detected in increasing order of their MMSE. The sub-optimum ordering

only requires 2 detection steps, regardless of the number of transmit antennas.

Figure 4.7 shows the slicer-SNR performance of the optimal and sub-optimal or-

dering algorithms for a MIMO system with four transmit and receive antennas.

For comparison purpose, the performance for the MIMO DFE without cancella-

tion and of the MIMO DFE with cancellation in a random order (i.e., non-ordered

88

6 8 10 12 14 16 18 20 22 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

No Cancellation

Non−OrderedCancellation

Sub−OptimallyOrdered Cancellation

Optimally Ordered Cancellation

Figure 4.7: CDF for Various MIMO DFE Algorithms with M = N = 4.

cancellation) are also included. The CDFs show that the sub-optimum system

performs almost as well as the optimal ordering system. Although, using cancel-

lation in the receiver significantly improves the MIMO system performance, using

a correct detection order is critical to obtain the maximum performance from the

receiver. For example, at a 5% outage, a 1.9 dB SNR improvement is obtained

by using non ordered cancellation, while if sub-optimum ordering is used a 4.5

dB SNR improvement is observed. Sub-optimal ordering decreases the optimal

SNR by only 0.1 dB at a 5% outage.

The impact of cancellation and ordering increases as the number of transmit

and receive antennas increases as illustrated in Figure 4.8 for a MIMO system with

eight transmit and receive antennas. At a 5% outage, a 2.3 dB SNR improvement

is now obtained by using non ordered cancellation, while if sub-optimum ordering

is used, a 5 dB SNR improvement is observed for M = N = 8. The sub-optimum

89

6 8 10 12 14 16 18 20 22 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

No Cancellation

Non−Ordered Cancellation

Sub−Optimally Ordered Cancellation

Optimally OrderedCancellation

Figure 4.8: CDF for Various MIMO DFE Algorithms with M = N = 8.

ordering algorithm still performs similarly to the optimal myopic ordering algo-

rithm. The maximum performance degradation due to the sub-optimal ordering

algorithm remains smaller than 0.5 dB, which is acceptable due to the significant

computational complexity diminution benefit.

Figure 4.9 illustrates the capacity that could be achieved with an uncoded

communication system using a MIMO DFE receiver with cancellation. To obtain

the achievable capacity, the system was simulated using the optimal receiver co-

efficients and detection ordering but using actual decisions in the DFE feedback

section. For each number of receive antennas N , the number of transmit anten-

nas M and the QAM constellation size Q was varied to find the system capacity.

For each combination of N , M , and Q the MIMO communication system was

simulated for 2000 independent channel realizations. For each channel realiza-

tion, a block of 100 M−vector Q−QAM symbols was transmitted from the M

90

1 2 3 4 5 6 7 82

4

6

8

10

12

14

16

18

20

22

Number of Receiving Antennas

Cap

acity

(bi

ts/u

se)

MIMO System

Smart Antenna Array

Figure 4.9: Simulated Capacity for Uncoded MIMO DFE with Cancellation Sys-

tem.

transmit antennas. The achievable capacity was computed using an optimization

algorithm involving increasing the number of transmit antennas M and the QAM

constellation size Q. M was restricted to be smaller or equal to N and Q to a

power of two. The achievable MIMO capacity for a given number of transmit

antenna N is defined as the maximum value of M log2Q bits/use for which the

block error rate over the 2000 channel realizations is smaller than 5%. A block er-

ror for a channel realization occurs if one or more of the 100 M−vector Q−QAM

symbols is received in error. For comparison purpose, the achievable capacity of

a smart antenna array system is also included. For this system, for a given num-

ber of receive antennas N , only the constellation size Q was allowed to increase

and the number of transmit antenna M was fixed to 1. The simulation results

show that even without any form of coding there is a tremendous capacity im-

91

provement that can be achieved using the MIMO DFE receiver with cancellation

space-time architecture. Furthermore, a linear capacity increase was obtained for

the space-time system and a logarithmic capacity increase for the smart antenna

array system, as predicted by theory. Note that for this avaluation, the capacity

is restricted to specific integer values, which explains the uneven aspect of the ca-

pacity curves. The channel parameters are the same as the ones used to compute

the theoretical frequency selective fading MIMO channel capacity presented in

Figure 2.2. Although the theoretical channel capacity indicates error-free trans-

mission, which is not the case for the simulated capacity, it can be observed that

the simulated capacity is close to the theoretical channel capacity. This demon-

strates the potential of the MIMO DFE with cancellation architecture to achieve

high throughput in MIMO frequency selective fading channels. This result should

be confirmed by simulating this system using Convolutional or Turbo coding of

the serial data streams before multiplexing on the transmit antennas.

4.3.2 Correlated and Ricean Fading Channels

In the previous section, an ideal MIMO Rayleigh frequency selective fading chan-

nels was assumed. In this section, the impact of channel impairments, such

as antenna correlation and line-of-sight propagation, on the performance of the

MIMO DFE receiver with cancellation is analyzed.

Figure 4.10 shows the performance of the MIMO system in the presence of

correlation between the propagation channels for the different pairs of trans-

mit/receive antennas. The correlation model presented in Section 2.4.3 is used

and the envelope correlations at the transmitter and receiver are assumed to be

equal (i.e. αt = αr = α). The performance degrades as the channel correlation

increases. However, for α ≤ 0.25, the slice-SNR reduction is relatively small (be-

92

6 8 10 12 14 16 18 20 22 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

α=0

α=0.25

α=0.5

α=0.75

Figure 4.10: CDF for Correlated Channels for M = N = 4.

low 0.4 dB). Even for a channel correlation of 0.5, the performance loss is still

acceptable. At a 10% outage the slicer-SNR decreases by 2 dB for α = 0.5 versus

an independent fading channel (α = 0). It is interesting to note that even for

large correlation values up to α = 0.75, although there is a large performance

degradation, the MIMO system is still functional. The performance results for

the MIMO DFE receiver with cancellation in the presence of channel correlation

are also in agreement with the theoretical channel capacity results presented in

Section 2.4.3.

Line-of-sight propagation is modeled using the MIMO Ricean channel model

presented in Section 2.4.3. The performance of the MIMO DFE receiver with can-

cellation for an independent Rice fading distribution is illustrated in Figure 4.11.

For a MIMO channel where half the power is received from the line-of-sight prop-

agation path (K = 0 dB), the performance loss for the MIMO DFE with can-

93

6 8 10 12 14 16 18 20 22 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

Rayleigh

K=0 dB

K=6 dB

K=10 dB

Figure 4.11: CDF for Ricean Channel for M = N = 4.

cellation receiver is less than 0.7 dB. For larger Ricean factors K, the 5% outage

slicer-SNR decreases by 2 dB and 2.9 dB for K=6 dB and K=10 dB, respectively.

The receiver is therefore able to perform relatively well even if 90% percent of

the received power in the first multipath is from the line-of-sight propagation.

4.3.3 Colored Noise

In the previous simulations, additive white Gaussian noise was assumed (Rv =

σ2vI). In this section, the performance of the MIMO DFE receiver with can-

cellation in the presence of colored noise is investigated for frequency selective

fading channels. For simulation purposes, a model similar to the one presented

in Section 2.4.2 is used. It emulates a combination of a colored interference from

94

another MIMO device and white noise. The noise vector v is given by:

v =√γint dintCint +

√1− γint n. (4.25)

dint is a 1 × Mint(Nc,int + Nf + 1) vector of zero-mean independent Gaussian

random variable with variance σ2v/Mint. n is a 1×N(Nf + 1) vector of zero-mean

independent Gaussian random variable with variance σ2v . The elements of dint

and n are jointly uncorrelated and independent of the transmitted data sequence

d. Cint = [C0,int · · ·CNf ,int] is the interference channel matrix where

Cj,int =

0Mintj×N

Ctint

0Mint(Nf−j)×N

(4.26)

and

Ctint =

c1,1,int(0) · · · c1,N,int(0)...

. . ....

cMint,1,int(0) · · · cMint,N,int(0)...

. . ....

c1,1,int(Nc,int) · · · c1,N,int(Nc,int)...

. . ....

cMint,1,int(Nc,int) · · · cMint,N,int(Nc,int)

.

The noise vector v autocorrelation matrix is:

Rv = σ2v

(γintC

∗intCint + (1− γint)I

). (4.27)

The properties of the colored noise are controlled by the following parameters:

Mint and Nc,int. The balance between colored and white noise in the system

interference v depends on γint. For γint = 0 the interference is white and for

γint = 1 it is entirely colored. For the simulation results presented in this section,

the SNR is set to 10 dB, Nc,int = Nc = 2 and γint,rms = γrms = 1.56.

95

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

White with Cancellation

White withoutCancellation

Colored with Cancellation

Colored with Cancellation − White Noise Estimation

Colored withoutCancellation

Figure 4.12: CDF in the Presence of Colored Noise for M = N = 4.

Figure 4.12 compares the slicer-SNR CDF of the MIMO DFE with receiver

cancellation for different noise properties and receiver configurations. The num-

ber of transmit and receive antennas is 4. The colored noise parameters are

Mint = M = 4 and γint = 1. This emulates a communication system with identi-

cal multiple MIMO users and limited by interference noise. The performance of

the system with white noise (γint = 0) is also included for comparison purpose.

The performance in colored noise of a MIMO receiver which does not estimate

the complete noise correlation matrix is also simulated. This receiver assumes

white noise and only estimates the variance of each noise component. That is,

Rv for the white noise estimation receiver is diagonal, and its diagonal elements

are the diagonal elements of the auto-correlation matrix of the noise vector v.

The MIMO DFE receiver with and without cancellation performs better in

the presence of colored noise as predicted by the theoretical channel capacity in

96

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

γint

=0γint

=0.25

γint

=0.5

γint

=0.75

γint

=1

Figure 4.13: Effect of γint on the Slicer-SNR CDF for M = N = 4.

Section 2.4.2. The MIMO receiver with cancellation takes better advantage of the

colored properties of the noise to improve the slicer-SNR than the receiver without

cancellation. For a 5% outage, the slicer-SNR of the receiver with cancellation

improves by 1.3 dB. A receiver that only estimates the noise variance is not able

to improve the slicer-SNR and has a 1.6 dB SNR degradation versus the optimal

receiver at a 5% outage.

Figure 4.13 shows the effect of γint on the slicer-SNR of the receiver with

cancellation. The number of transmit and receive antennas is 4 and Mint =

M = 4. The slicer-SNR improves as the environment changes from noise limited

(γint = 0) to interference limited (γint = 1). However, the improvement is only

significant when at least half the noise consists of colored interference (γint ≥ 0.5).

The previous simulation results analyzed the performance of the system in

a multi-users MIMO environment where the interference is coming from other

97

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

White Noise

Mint

=1

Mint

=2

Mint

=4

Mint

=8

Mint

=1 −

White Noise Estimation

Figure 4.14: Effect of Mint on the Slicer-SNR CDF for M = N = 4.

MIMO users. This type of interference does not have a distinct spatial signature

that can be efficiently cancelled when M = N = 4. Sources of interference could

have a different number of transmit antennas than the number of antennas for the

MIMO transmission of interest. Figure 4.14 illustrates the effect of the number of

transmit antennas at the interference source (Mint). The number of transmit and

receive antennas is 4 and γint = 1. For Mint = 1 the noise has a distinct spatial

signature and the MIMO DFE receiver with cancellation is able to mitigate the

interferer to provide a significant slicer-SNR improvement. For a large number

of transmit antennas at the interference source, the system behavior converge

to a system in the presence of white noise. For a 5% outage, an improvement

of 0.6 dB, 1.2 dB, 2.7 dB, and 4.9 dB in slicer-SNR is observed when the noise

source is colored with 8, 4, 2, and 1 transmit antennas at the interference source,

respectively. The white noise estimation receiver performs poorly when Mint = 1

98

0 5 10 1510

−3

10−2

10−1

100

SNR (dB)

BE

R

White Noise

γint

=1 − Mint

=M

γint

=1 − Mint

=1

Figure 4.15: BER Performance in the Presence of Colored Noise for M = N = 4.

and is not able to take advantage of the spatial signature of the interference.

There is a 5.5 dB degradation with this receiver at a 5% outage.

Figure 4.15 shows the bit error rate of the MIMO DFE receiver with cancel-

lation in the presence of colored noise for different SNR. The number of transmit

and receive antennas is 4 and γint = 1. The BER for Mint = 1 and Mint = M = 4

are presented. The optimal receiver with error-free cancellation and feedback was

simulated. The results clearly show the improvement that can be obtained with

the proposed receiver when the noise source has a spatial signature, even if no

degree of freedom is available at the receiver. For a BER of 10−3, the MIMO

DFE receiver with cancellation provides an improvement of 1.9 dB and 9.2 dB in

the presence of colored noise with Mint = 4 and Mint = 1, respectively.

99

4.4 Flat Fading Channels

In order to compare the performance of the MIMO DFE receiver with cancella-

tion with other receiver architectures, the system was simulated for flat fading

channel conditions (Nc = 0). The narrowband equivalent of the MIMO DFE

with cancellation is obtained by setting Nf = Q = ∆ = 0. For the flat fading

channel, the MIMO channel matrix C is given by:

C =

c1,1(0) · · · c1,N(0)

.... . .

...

cM,1(0) · · · cM,N(0)

and each element cm,n(0) of the channel matrix C is Rayleigh distributed with unit

average power. The notation presented in Section 4.1 withNc = Nf = Q = ∆ = 0

is used to analyze the different receiver architectures.

4.4.1 Zero-Forcing Channel Inversion

The simplest narrowband receiver is obtained by using a matrix inversion of the

MIMO channel. This is a zero-forcing algorithm that cancels the co-channel

interference while ignoring the noise. The estimate of the transmitted data d(i)

is given by:

d(i) = uC+. (4.28)

where C+ is the pseudo-inverse of the MIMO channel matrix C. The pseudo-

inverse of a matrix A with i rows and j columns is given by:

A+ =

A∗(AA∗)−1 i ≤ j

(A∗A)−1A i > j,

100

if A has min(i, j) strictly positive singular values. Otherwise, the pseudo-inverse

can also be defined in terms of the SVD decomposition of A = USV∗ =∑ri=1 uiσiv

∗i (ui is the ith column of U and vi the ith column of V). r is the number

of strictly positive singular values and defines the rank of A. The pseudo-inverse

of A is then given by:

A+ =r∑i=1

viσ−1i u∗i

= V

S−11:r×1:r 0r×i−r

0r×j−r 0j−r×i−r

U∗.

The MMSE vector J for the zero-forcing receiver is given by:

J = diag(Rd −C+∗Rdu −RduC+ + C+∗RuC+

)(4.29)

where

Rd = σ2dIM

Rdu = C∗Rd

Ru = C∗RdC + Rv.

4.4.2 VBLAST

The Vertical Bell Labs Layered Space-Time (VBLAST) receiver [FGV99, GFV99,

WFG98] is a well known receiver architecture for MIMO channels. Like the

MIMO DFE with cancellation, it uses available decisions to suppress their in-

terference from the received data vector. The VBLAST receiver architecture is

depicted in Figure 4.16. When detecting the data stream m, the VBLAST al-

gorithm first cancels the interference due to the data streams m + 1 to M from

the received vector u to obtain the vector y. The data estimate is then com-

puted from a linear combination of y. Data streams are detected from transmit

101

Nwm

dm

N

dm

u

d1(i)

dM(i)

v1(i)

vN(i)

u1(i)

uN(i)

c11

Receiver

Cm

f

y

M-m

Figure 4.16: VBLAST Receiver.

antenna M to antenna 1. Note that any detection order can be used through the

appropriate reordering of the rows of the channel matrix C. The data estimate

for stream m, 1 ≤ m ≤M , for a VBLAST receiver is given by:

dm(i) = (u− {d}m+1:MCfm)wm (4.30)

where {d}m+1:M = [dm+1(i) · · ·dM(i)], Cfm is the interference cancellation matrix

for data stream m and consists of rows m+ 1 to M of C. wm is the mth column

of Cwm

+. The matrix Cwm consists of rows 1 to m of C. The MMSE of data stream

m for the VBLAST receiver is given by:

Jm = σ2d −w∗mRdy

m −Rdym∗wm + w∗mRy

mw∗m (4.31)

where Rdym is the mth column of Cw

m∗Rd

m and

Rym = Cw

m∗Rd

mCwm + Rv.

Rdm = σ2

dI is the autocorrelation matrix of the data vector [d1(i) · · ·dm(i)].

102

4.4.3 MMSE VBLAST

VBLAST, as originally proposed, uses interference cancellation followed by zero-

forcing channel inversion. However, channel inversion suffers from poor perfor-

mance for low SNR and colored noise, as will be seen in Section 4.4.4. A better

approach is to compute wm using an MMSE criterion. The only added com-

plexity of the MMSE VBLAST versus the original VBLAST is to estimate the

properties of the noise in addition to the channel estimation. For the MMSE

VBLAST wm is given by:

wm = (Rym)−1 Rdy

m. (4.32)

Rym and Rdy

m are the same as for the VBLAST receiver described in the previous

section. For the MMSE VBLAST receiver, the data estimate and MMSE for

stream m are given by equations 4.30 and 4.31, respectively.

4.4.4 Simulation Results

Figure 4.17 compares the slicer-SNR CDF for the different MIMO receivers for

flat fading channels with four transmit and receive antennas. Although the

average SNR is high (20 dB), zero-forcing MIMO channel inversion performs

poorly. MMSE without cancellation clearly outperforms direct channel inversion

and should be used instead. However, algorithms using interference cancellation

provides a significant performance improvement. The VBLAST algorithm, for

a 20 dB average SNR, outperforms MMSE without cancellation by 5.4 dB at a

5% outage. An additional improvement of 1.8 dB is obtained by using MMSE

weighting in MMSE VBLAST instead of the zero-forcing channel inversion of

VBLAST. By estimating the noise properties it is therefore possible to obtain

a performance improvement. Note that MMSE VBLAST and the MIMO re-

103

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

MMSE withoutCancellation

MMSE with Cancellation &MMSE VBLAST

Channel Inversion

VBLAST

Figure 4.17: CDF for Flat Fading Channel for M = N = 4.

ceiver with cancellation introduced in this chapter have the same performance in

white noise. However, the novel receiver introduced in this dissertation has the

following advantages:

• Generalized form for frequency selective fading channels;

• Single channel inversion instead of M pseudo-inverse calculations;

• Suitable for adaptive algorithms that avoid computationally intensive ma-

trix inversion (see Chapter 7).

Figure 4.18 illustrates the performance of the narrowband receivers in a col-

ored noise environment. The number of transmit and receive antennas is 4, the

average SNR is set to 10 dB, Mint = 1, Nc,int = 0 and γint = 1. As was the case for

the white noise environment, zero-forcing channel inversion performs poorly and

104

0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)



Channel Inversion

VBLAST

Figure 4.18: CDF for Flat Fading Channel in the Presence of Colored Noise for

M = N = 4.

should be avoided. However, VBLAST now performs worst than MMSE without

interference cancellation. This is due to the lower average SNR and the spatial

signature of the noise. MMSE solutions reduce the interference strength while

the zero-forcing channel inversion used in VBLAST does not take advantage of

the noise structure. A significant slicer-SNR increase is again obtained by using

interference cancellation in addition to MMSE combining. The improvement is

even larger in the presence of colored noise (10.7 dB improvement with cancella-

tion at 5% outage versus 7.2 dB in white noise). Note that MMSE VBLAST and

the MMSE DFE receiver with cancellation are also equivalent in colored noise.

Figure 4.19 shows the BER performance of the MIMO receivers for flat fad-

ing channels with white noise. The MIMO communication systems have four

transmit and receive antennas. Channel inversion has the worst performance of

105

0 5 10 1510

−3

10−2

10−1

100

SNR (dB)

BE

R



MMSE with Decision Feedback Cancellation &MMSE VBLAST with Data Feedback

Channel Inversion

VBLAST

VBLAST with Decision Feedback

Figure 4.19: BER Performance for MIMO Flat Fading Channel for M = N = 4.

all receivers for all SNR’s. MMSE without cancellation performs slightly better

than the receivers with interference cancellation and decision feedback at low

SNR. However, MMSE receivers with cancellation and decision feedback quickly

performs better than MMSE without cancellation while the VBLAST receiver

with decision feedback is better than the MMSE without cancellation only for

SNRs larger than 12.5 dB. Decision feedback instead of error free interference

cancellation degrades the performance of VBLAST more than for MMSE based

interference cancellation (2.4 dB versus 1.5 dB). The performance improvement

at a BER of 10−2 obtained by using error free MMSE interference cancellation

instead of zero-forcing channel inversion VBLAST is a significant 2.9 dB. It is also

interesting to note that MMSE interference cancellation with decision feedback

always outperforms optimal VBLAST with error free cancellation.

106

CHAPTER 5

Multi-Carrier MIMO

Equalization is an effective technique to mitigate the effect of multipath prop-

agation but suffers from a large complexity increase as the excess delay of the

channel becomes longer. Multi-carrier modulation is an alternative communi-

cation technique that has been developed for frequency selective fading chan-

nels. Orthogonal Frequency Division Multiplexing (OFDM) is the most common

multi-carrier modulation scheme. In comparison with single carrier modulation

systems, such as equalization, OFDM modulates the data on orthogonal carriers

which are added and transmitted simultaneously [Cha66, WE71, PR80, Jr85,

Kal89, Bin90, DJC97, NP00]. This effectively divides the wideband channel

into a number of narrowband transmission sub-channels. The use of Discrete

Fourier Transform (DFT) for baseband modulation and demodulation of the

OFDM signal [WE71] allows efficient implementations by making use of the Fast

Fourier Transform (FFT). Transmission of a cyclic prefix during the guard inter-

val between OFDM symbols [PR80] preserves the sub-carrier orthogonality and

eliminates inter-symbol interference. A receiver can therefore be designed using

narrowband techniques on each sub-channel.

OFDM has enjoyed a lot of popularity recently and is being used for many

wired and wireless applications. In wired environments, OFDM is also known as

Discrete Multitone (DMT) modulation and is used, for example, in Asymmet-

ric Digital Subscriber Line (ADSL) communication systems. In wireless applica-

107

tions, OFDM has been adopted in standards for wireless local area networks such

as HiperLAN and IEEE 802.11. Extension of OFDM to the MIMO frequency

selective fading channel is an attractive solution to increase the throughput of

current wireless local area networks. This problem has been recently investi-

gated [RC98, ATN98]. In this chapter, the solution to the MIMO DFE receiver

with cancellation developed in Chapter 4 is extended to MIMO OFDM. This

new MIMO OFDM solution provides a better performance due to the interfer-

ence cancellation algorithm and allows the use of the efficient MIMO adaptive

algorithms developed in Chapter 7.

5.1 MIMO OFDM Model

In this section, the MIMO OFDM communication system is modeled. The guard

interval prevents OFDM symbols consisting of P superposed orthogonal sub-

carriers to interfere with each other, therefore eliminating ISI. The interval be-

tween OFDM symbols must be larger than the maximum excess delay expected

for the channel. A cyclic prefix is transmitted during the guard interval to pre-

serve the sub-channel orthogonality, as will be seen later. Since there is no OFDM

symbol ISI, each OFDM symbol transmission is independent and the analysis

presented in this section is based on a single OFDM symbol.

The system consists of M transmit antennas and N receive antennas, and

the OFDM symbols have P orthogonal sub-channels. Figure 5.1 shows the block

diagram of a MIMO OFDM transmitter. Let define the data vector d(p) =

[d1(p) · · ·dM(p)] for p = 0, . . . , P − 1. The data vectors d(p), 0 ≤ p ≤ P −

1, are transmitted into a single MIMO OFDM symbol. dm(p) is mapped to

the pth orthogonal sub-channel and is transmitted from antenna m. The data

sequence is drawn from a QAM constellation with power σ2d and is assumed to

108

d(p)M Vector

IDFT

dt(i)M Add Cyclic

Prefix

dc(i)

dc,1(i)

dc,M

(i)

1 to P

Serial to

Parallel

P to 1

Parallel

to Serial

Figure 5.1: Architecture of MIMO OFDM Transmitter.

be independently identically distributed in the frequency and space domain:

E[d(p)∗d(q)] =

σ2dI p = q

0 p 6= q.

σ2d is the data signal variance and is such that the total transmit Mσ2

d is constant.

The sequence d(p) is mapped to the P parallel frequency sub-channels and its

vector Inverse DFT (IDFT) is taken. The parallel output of the vector IDFT is

then re-mapped to a serial sequence of P samples in the time domain. The time

domain sequence is denoted by dt(i) = [dt,1(i) · · ·dt,M(i)] for i = 0, . . . , P − 1.

dt(i) is given by:

dt(i) =1√P

P−1∑p=0

d(p)ej2πipP . (5.1)

Note that the IDFT transformation is such that the signal variance of dt,m(i) is

maintained equal to σ2d.

The maximum number of multipaths is assumed to beNc+1, therefore a guard

interval of length Nc is necessary to prevent OFDM symbols ISI. To maintain the

sub-carriers orthogonality, a cyclic prefix is pre-appended in the guard interval

to create the sequence dc(i) = [dc,1(i) · · ·dc,M(i)] for i = 0, . . . , P + Nc − 1. The

109

Z-1dc(i)

M

C0

C1

Z-1

CNc

N

vt(i)

uc(i)

Figure 5.2: MIMO OFDM Channel Model.

last Nc vector samples of the sequence dt(i) are pre-appended generating the

following dc(i) sequence:

dc(i) =

dt(i + P−Nc) i = 0, . . . , Nc − 1

dt(i−Nc) i = Nc, . . . , P +Nc − 1.

(5.2)

dc,m(i) is the ith sample of the OFDM symbol transmitted from antenna m.

Figure 5.2 illustrates the channel model used for the OFDM analysis. The

MIMO channel has M inputs and N outputs. Let define cm,n(nc), 1 ≤ m ≤ M ,

1 ≤ n ≤ N , 0 ≤ nc ≤ Nc, as the coefficient of the (nc + 1)th multipath for the

channel connecting transmit antenna m to receive antenna n. The MIMO matrix

multipath Cnc, 0 ≤ nc ≤ Nc, is defined as:

Cnc =

c1,1(nc) · · · c1,N (nc)

.... . . · · ·

cM,1(nc) · · · cM,N(nc)

.The MIMO channel impulse response is then as follows:

C(z) =Nc∑nc=0

Cncz−k (5.3)

and the output of the MIMO channel uc(i) for i = 0, . . . , P +Nc− 1 is given by:

uc(i) =Nc∑nc=0

dc(i− nc)Cnc + vt(i), (5.4)

110

uc(i)

vt,1(i)

vt,N(i)

uc,1(i)

uc,N(i)

N Remove

Cyclic

Prefic

ut(i) Vector

DFT

1 to P

Serial to

Parallel

MIMO

Receiver

u(p)

P to 1

Parallel

to Serial

M d(p)

Figure 5.3: Architecture of MIMO OFDM Receiver.

where vt(i) = [vt,1(i) · · ·vt,M (i)] is the IID stationary noise vector.

The samples uc(i) at the receiver are corrupted by inter samples multipath

interference and MIMO co-channel interference. However, the received samples

from i = 0 to i = Nc − 1 are corrupted by samples from the previous OFDM

symbol (i.e., they are linearly dependent on dc(i)’s for which i < 0) and are

therefore affected by inter OFDM symbol interference. The other samples from

the received OFDM symbol are only linearly depending on other samples from

the same OFDM symbol and are not corrupted by OFDM symbol ISI. To avoid

OFDM ISI, the first Nc vector samples from the received sequence uc(i) are

removed to generate the sequence ut(i), i = 0, . . . , P −1, as shown in Figure 5.3.

The sequence ut(i), i = 0, . . . , P − 1, after the cyclic prefix removal is as follows:

ut(i) = uc(i + Nc)

=Nc∑nc=0

dc(i + Nc − nc)Cnc + vt(i + Nc).(5.5)

The vector DFT of the block of data ut(i), i = 0, . . . , P − 1, is then taken.

111

The output of the vector DFT u(p) for p = 0, . . . , P − 1, is given by:

u(p) =1√P

P−1∑i=0

ut(i)e−j 2πip

P

=1√P

P−1∑i=0

Nc∑nc=0

dc(i + Nc − nc)Cnce−j 2πip

P +1√P

P−1∑i=0

vt(i + Nc)e−j2πipP

=1√P

Nc∑nc=0

P−nc−1∑i=−nc

dc(i + Nc)Cnce−j 2π(i+nc)p

P + vf (p)

=1√P

Nc∑nc=0

[ −1∑i=−nc

dc(i + Nc)e−j2πipP +

P−nc−1∑i=0

dc(i + Nc)e−j2πipP

]Cnce

−j 2πncpP + vf (p)

=1√P

Nc∑nc=0

[ −1∑i=−nc

dt(i + P)e−j2πipP +

P−nc−1∑i=0

dt(i)e−j 2πip

P

]Cnce


=1√P

Nc∑nc=0

[P−1∑

i=P−nc

dt(i)e−j 2π(i−P )p

P +

P−nc−1∑i=0

dt(i + Nc)e−j2πipP

]Cnce


=1√P

Nc∑nc=0

P−1∑i=0

dt(i)e−j2πipP Cnce


=1√P

P−1∑i=0

dt(i)e−j 2πip

P

Nc∑nc=0

Cnce−j 2πncp

P + vf (p)

= d(p)Cf (p) + vf (p)

(5.6)

112

since

1√P

P−1∑i=0

dt(i)e−j 2πip

P =1√P

P−1∑i=0

1√P

P−1∑k=0

d(k)ej2πkiP e−j

2πipP

=1

P

P−1∑k=0

d(k)P−1∑i=0

ej2π(k−p)i

P

=1

PPd(p)

= d(p)

and

P−1∑i=0

ej2π(k−p)i

P =

1−ej

2π(k−p)PP

1−ej2π(k−p)

P

= 0 k 6= p

P k = p.

Cf (p) is the matrix Fourier transform of the MIMO channel:

Cf (p) =Nc∑nc=0

Cnce−j 2πncp

P (5.7)

and is therefore the MIMO frequency response of the wideband channel for sub-

channel p, p = 0, . . . , P − 1. vf (p) = [vf,1(p) · · ·vf,N(p)] is the frequency noise

vector given by:

vf (p) =1√P

P−1∑i=0

vt(i + Nc)e−j2πipP (5.8)

for sub-carrier p, p = 0, . . . , P − 1. The auto-correlation matrix Rvf(p) of the

frequency noise vector is given by:

Rvf (p) = E[vf (p)∗vf (p)]

=1

P

P−1∑i=0

P−1∑k=0

E[vt(i + Nc)∗vt(k + Nc)]ej2π(i−k)p

P

=1

P

P−1∑i=0

P−1∑k=0

Rvt(i−k)ej 2π(i−k)p

P

(5.9)

113

where Rvt(i−k) = E[vt(i)∗vt(k)] is the auto-correlation matrix of the time domain

stationary vector process vt(i). Note that if the time domain noise process is

white (i.e., Rvt(i−k) = σ2vIδ(i− k)) then the frequency domain noise is also white

with auto-correlation Rvf (p) = σ2vI.

The MIMO receiver process, for an OFDM symbol, the input vectors u(p)

received on each sub-channels to obtain the best estimate d(p) = [d1(p) · · · dM(p)]

of the data d(p) transmitted on the sub-carriers. In this chapter, the narrowband

version of the MIMO DFE receiver with cancellation is proposed to compute the

estimate vector for each sub-channel. The data estimate d(p) for each sub-carrier

p, 0 ≤ p ≤ P − 1, is given by:

d(p) = −df (p)B(p) + u(p)F(p)

= [df (p) u(p)]

−B(p)

F(p)

= y(p)W(p),

(5.10)

where df (p) = d(p), B(p) is an M ×M stricly lower triangle matrix, and F(p)

is an N ×M matrix. The output of the MIMO OFDM receiver is then converted

to a serial stream d(p), p = 0, . . . , P − 1, and is the estimate of the transmitted

sequence d(p), p = 0, . . . , P − 1, in the current OFDM symbol. The MIMO

OFDM receiver error e(p) = [e1(p) · · · eM(p)] for p = 0, . . . , P − 1, is given by

e(p) = d(p)− d(p)

= d(p)− y(p)W(p).(5.11)

5.2 MIMO OFDM MMSE Solution

The optimal MMSE solution for the proposed MIMO OFDM receiver with cancel-

lation is easily obtained by using the results derived in Section 4.2. The optimal

114

coefficient matrix W(p)o for frequency sub-channel p, p = 0, . . . , P − 1, is given

by:

W(p)o = L(p)−1{

L(p)−∗

Rd(p)y(p)

}strictly-lower

(5.12)

where L(p) is the Cholesky decomposition of Ry(p). The auto-correlation matrix

of y(p) is given by:

Ry(p) = E[y(p)∗y(p)] =

Rdf (p) Rdf (p)u(p)∗

Rdf (p)u(p) Ru(p)

(5.13)

where:

Rdf (p) = Rd(p) (5.14)

Rdf (p)u(p) = Cf (p)∗Rd(p) (5.15)

Ru(p) = Cf (p)∗Rd(p)Cf(p) + Rvf (p) (5.16)

Rd(p) = σ2dIM . (5.17)

The cross-correlation matrix Rd(p)y(p) is:

Rd(p)y(p) = E[y(p)∗d(p)]

=

Rd(p)df (p)

Rd(p)u(p)

=

Rd(p)

Cf (p)∗Rd(p)

.(5.18)

The MMSE vector of the MIMO OFDM receiver with cancellation for each

sub-carrier p, p = 0, . . . , P − 1, is defined as J(p) = [J1(p) · · ·JM(p)] where

Jm(p) = E[em(p)∗em(p)]. The MMSE for the MIMO OFDM receiver is therefore

115

given by:

J(p) = diag(Re(p))

= diag(E[e(p)∗e(p)

])= diag

(Rd(p) −W(p)o

∗Rd(p)y(p) −Rd(p)y(p)

∗W(p)o +

W(p)o∗Ry(p)W(p)o

).

(5.19)

The MMSE optimal solution for the MIMO OFDM receiver without can-

cellation is obtained using the same technique and the optimal matrix for each

sub-carrier p is given by:

W(p)o = L(p)−1{

L(p)−∗

Rd(p)y(p)

}lower(M)

. (5.20)

The MMSE for the MIMO OFDM receiver without cancellation is also given by

Equation 5.19.


The performance of the optimal MIMO OFDM receiver with cancellation is eval-

uated in this section through simulations. First, the performance of the MIMO

OFDM receiver is compared to the MIMO DFE receiver. Then, the impact of

various interpolation algorithm on the MIMO OFDM receiver performance is

analyzed. Unless specified otherwise, the simulation channel conditions are the

following. The SNR was fixed to 20 dB and the noise is assumed to be white

(Rvt(i−k) = σ2vIδ(i − k)). The channel impulse response was generated using

the Rayleigh fading exponential power delay profile specified in Section 2.4.1 for

τ rms = 0.5Ts and the number of OFDM sub-carriers was set to P = 64. The

MIMO communication system consists of 4 transmit antennas and 4 receive an-

tennas (M = N = 4).

116

6 8 10 12 14 16 18 20 22 240

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

DFE − τrms

=0.5Ts

DFE − τrms

=Ts

OFDM − τrms

=0.5Ts

and τrms

=Ts

Figure 5.4: Comparison of CDF for MIMO OFDM and MIMO DFE Receivers.

5.3.1 Comparison with MIMO Equalization

The performance of the MIMO OFDM receiver with cancellation is first com-

pared with the MIMO DFE receiver with cancellation. The slicer-SNR CDF for

these communication systems is presented in Figure 5.4. The systems have been

simulated for frequency selective fading channels with τ rms = 0.5Ts and τ rms = Ts.

The parameters of the MIMO DFE receiver for both environments are Nf = 4,

Q = 3 and ∆ = 4. For each rms delay spread, the CDF is generated using 1000

independent MIMO channel realizations. For each time domain channel real-

ization, the optimal MMSE for the MIMO DFE receiver with cancellation was

computed and the slicer-SNR for each data stream m, 1 ≤ m ≤ M , calculated.

Then, the Fourier Transform of the time domain channel realization is computed

and the optimal MIMO OFDM MMSE solution is found. For both the OFDM

117

and DFE receiver, the sub-optimal detection ordering algorithm presented in Sec-

tion 4.3 is used. The slicer-SNR for each data stream m, 1 ≤ m ≤M , and every

sub-channel p, 1 ≤ p ≤ P , is calculated. Thus, for each channel realization,

MP slicer-SNR are obtained for the MIMO OFDM receiver with cancellation.

The slicer-SNR CDF for the MIMO DFE receiver is generated using the recorded

slicer-SNR for each data stream and each channel realization, while the slicer-

SNR CDF for the MIMO OFDM receiver is plotted using the recorded slicer-SNR

for all data streams and frequency sub-channels for all channel realizations.

The slicer-SNR CDF for the MIMO OFDM receiver are identical for τ rms =

0.5Ts and τ rms = Ts, as is the case for SISO OFDM. This is due to the fact

that after the Fourier Transform, the frequency channel matrix Cf (p) is Rayleigh

distributed with unit power, independently of the rms delay spread of the channel.

Therefore, for any τ rms the slicer-SNR CDF is the same as for a flat fading

channel, as can be observed by comparing the results with the MIMO DFE

CDF for τ rms = 0Ts in Figure 4.6. The MIMO DFE receiver is able to take

directly advantage of the frequency diversity and improves the worst-case slicer-

SNR for larger τ rms. The MIMO OFDM receiver can not directly exploit it

and therefore has larger variation of the slicer-SNR at the output. However,

unlike a communication system in a flat fading environment, the MIMO OFDM

receiver also offers frequency diversity that can be used to improve the system

performance. That is, for certain frequency sub-channels the MIMO slicer-SNR

is lower, while for others it is higher. A MIMO system which uses coding across

the frequency sub-carriers is able to then take advantage of this MIMO frequency

diversity.

118

5.3.2 Interpolation Effects

Computing the optimal MMSE solution for each of the P sub-carriers of a MIMO

OFDM system is computationally intensive. Fortunately, the MIMO frequency

channel matrix Cf(p) of two sub-channels p and q can be considered as corre-

lated if their frequency separation p − q is less than the coherence bandwidth.

Interpolation of the optimal solutions can then be used to reduce the receiver

complexity and the pilot overhead requirements. In this section, different inter-

polation algorithms are evaluated.

For a MIMO system with cancellation, in addition to interpolation for the

weight matrix between sub-carriers, detection ordering needs to be considered

since B(p) depends on the detection order. That is, two optimal MMSE solu-

tions computed with a different data detection order can not be used for interpo-

lation. For the first interpolation algorithm, a single detection order is used for

all sub-channels. The optimal MMSE OFDM solution for the receiver without

cancellation is first computed for the sub-carriers p, where p ∈ {ak} is a frequency

channel used for interpolation. The set {ak} is defined as

{ak} = {a0, a1, a2, . . . }

=

{0, psep, 2psep, . . . ,

⌊P − 1

psep

⌋psep, P − 1

}.

The global ordering MMSE vector Jord = [J1,ord · · ·JM,ord] given by

Jord =∑p∈{ak}

J(p) (5.21)

is then used to determine the ordering order. The data for all sub-channels p,

0 ≤ p ≤ P − 1, is detected in the ascending order of their global MMSE Jm,ord.

The MIMO OFDM optimal matrix W(p)o for the receiver with cancellation is

then calculated for the interpolation sub-channels p, p ∈ {ak}, using the global

119

detection order. The weight matrix for the frequency sub-channels p, p 6∈ {ak},

is then computed using linear interpolation between the adjacent optimal MMSE

weight matrix. The interpolated coefficient matrix W(p)int for sub-carrier p,

p 6∈ {ak}, is given by:

W(p)int =p− aj

aj+1 − aj(W(aj+1)o −W(aj)

o), (5.22)

where aj is the largest element of the set {ak} smaller than p and aj+1 is the

element following aj in the set {ak}. The MMSE Jint(p) = [J1,int(p) · · ·JM,int(p)]

for the interpolated sub-carrier p is given by:

Jint(p) = diag(Rd(p) −W(p)int∗Rd(p)y(p) −Rd(p)y(p)

∗W(p)int +

W(p)int∗Ry(p)W(p)int).

(5.23)

Figures 5.5 to 5.7 show the performance of interpolated OFDM receivers for

different environements and configurations. Figure 5.5 and Figure 5.6 present the

slicer-SNR CDF for rms delay spread of τ rms = 0.5Ts and τ rms = Ts, respectively,

for interpolated MIMO OFDM systems with four transmit and receive antennas.

For comparison purpose, the slicer-SNR CDF of an interpolated SISO (M = N =

1) OFDM system is shown in Figure 5.7. Note that each discrete frequency index

represents a 1PTs

frequency separation. The relative frequency separation between

interpolation sub-channels can be defined as:

f relsep =

psep/PTs1/τ rms

=psepτ rms

PTs(5.24)

The interpolated MIMO OFDM system behaves similarly as a function of f relsep

for both frequency selective fading environments. The slicer-SNR degradation is

acceptable up to f relsep = 6.25%. For larger relative frequency separation between

interpolation sub-channels, the performance degrades quickly since the support-

ing points for the interpolation are no longer enough correlated to provide an

120

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

Optimal

Psep

=4

Psep

=8

Psep

=12

Psep

=16

Figure 5.5: CDF for Interpolated MIMO OFDM Receiver for τ rms = 0.5Ts.

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

Optimal

Psep

=2

Psep

=4

Psep

=6

Psep

=8

Figure 5.6: CDF for Interpolated MIMO OFDM Receiver for τ rms = Ts.

121

−5 0 5 10 15 20 25 300

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

Optimal

Psep

=4

Psep

=8

Psep

=12

Psep

=16

Figure 5.7: CDF for Interpolated SISO OFDM Receiver for τ rms = 0.5Ts.

acceptable linearly interpolated coefficient matrix. It is interesting to note that

despite the fact that a large coefficient matrix with feedback is interpolated for

MIMO OFDM with cancellation instead of a single coefficient for SISO OFDM,

the behavior of the MIMO system as a function of f relsep is similar to the SISO

system.

The performance of the MIMO OFDM interpolated system is also affected by

the use of single ordering instead of optimal ordering for each sub-carrier. This

explains why for f relsep = 3.125% the slicer-SNR CDF for the MIMO system is

different from the optimal solution while for the SISO system they are almost

identical. For some sub-carriers for the interpolated MIMO OFDM system the

detection order is different from the optimal detection order. Therefore, although

the worst slicer-SNR for these sub-carriers is lower than for the optimal order,

the best slicer-SNR might be higher (remember that optimal ordering maximizes

122

Table 5.1: 5% Outage Slicer-SNR Decrease for Interpolated OFDM

f relsep (%)

MIMOτ rms = 0.5Ts

MIMOτ rms = Ts

SISO

3.125% 1.1 dB 1.9 dB 0.3 dB6.25% 2.7 dB 3.6 dB 2.2 dB9.375% 5.5 dB 6.5 dB 5.7 dB12.5% 8.4 dB 10.0 dB 10.1 dB

the minimum SNR). Thus, while the low outage slicer-SNR is worst, the high

outage slicer-SNRs are better for the interpolated system, even after considering

interpolation errors.

Table 5.1 compiles the 5% outage slicer-SNR degradation for the different

systems. The MIMO OFDM interpolated receiver is slightly more affected by the

sub-optimal ordering for a large rms delay spread. The results confirms the similar

behavior for the different configurations as a function of the relative frequency

separation. It is interesting to note that the slicer-SNR degrades more quickly

for the SISO system than for the MIMO. This might be due to the fact that the

interpolation errors in the MIMO coefficient matrix compensate each other and

provide some “interpolation diversity” versus the single coefficient interpolation

in SISO.

The performance of the interpolated MIMO OFDM system can be improved

by using a different detection order for each interpolation interval instead of a

single detection order for all sub-carriers. This interpolation algorithm works

as follows. For each interpolation interval, the optimal MMSE OFDM solution

for the receiver without cancellation is computed for sub-carriers ak and ak+1.

The detection order for sub-carriers ak to ak+1 − 1 is in ascending order of the

MMSE vector Jord = J(ak)+J(ak+1). The MIMO OFDM optimal matrix W(p)o

for the receiver with cancellation is then calculated for the interpolation sub-

123

5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

Optimal

Single Ordering

InterpolatedOrdering

Figure 5.8: Effect of Ordering Algorithm.

channels ak and ak+1 using this order. The interpolated weight matrix W(p)int

for sub-carriers p, ak ≤ p ≤ ak+1, in the interpolation interval is computed using

equation 5.22 and their MMSE is given by equation 5.23. Note that for sub-carrier

ak+1, except when ak+1 is the last element of the set {ak}, the weight matrix that

has been computed might not be the correct one for this algorithm since it uses

the detection order for the interpolation interval ak+1 to ak+2. Therefore, in the

worst case, the number of optimal solutions that needs to be computed is doubled.

Figure 5.8 shows the isolated impact of the ordering algorithm on the MIMO

OFDM system performance for τ rms = 0.5Ts and an effective frequency separa-

tion of f relsep = 12.5%. For the results presented in Figure 5.8 only the ordering

for each sub-carrier was computed using the single ordering algorithm or the in-

terpolated ordering algorithm. The optimal coefficient matrix and MMSE for

each sub-carrier was then calculated using the computed detection order. The

124

interpolated ordering algorithm performs better than the single ordering algo-

rithm. It was observed that the performance of the single ordering algorithm

is almost independent of f relsep for f rel

sep ≤ 12.5% while the interpolated ordering

algorithm performance improves as the relative frequency separation decreases.

For example, for f relsep = 3.125%, the slicer-SNR CDF for the interpolated ordering

algorithm is almost identical to the optimal ordering system. For an outage of

5%, the slicer-SNR decreases by 0.5 dB and 1 dB for the interpolated ordering al-

gorithm and single ordering algorithm, respectively. Comparing with the results

presented in Table 5.1, this confirms that the ordering algorithm contributes to

most of the degradation for f relsep = 3.125% and its impact is thereafter constant

and becomes negligible.

The performance of the complete single ordering and interpolated ordering

algorithms is compared in Figure 5.9 for an effective frequency separation of

f relsep = 6.5%. The slicer-SNR CDF of a third interpolation algorithm is also

included. This algorithm uses channel interpolation instead of coefficient matrix

interpolation. The channel is first estimated for the sub-carriers p, p ∈ {ak}. The

optimal weight matrix and MMSE for these sub-carriers is computed using the

MIMO OFDM with cancellation solution. Each sub-carrier uses its own optimal

detection order since the channel matrix is interpolated in this algorithm. The

channel matrix for the frequency sub-channels p, p 6∈ {ak}, is then computed

using linear interpolation between the nearest adjacent channel matrices. The

interpolated coefficient matrix Cf(p)int for sub-carrier p, p 6∈ {ak}, is given by:

Cf (p)int =p− aj

aj+1 − aj(Cf (aj+1)−Cf (aj)

), (5.25)

where aj is the largest element of the set {ak} smaller than p and aj+1 is the

element following aj in the set {ak}. The coefficient matrix W(p)int for each

125

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

Optimal

Weigth Interpolation −Single Ordering

Channel Interpolation

Weigth Interpolation −Interpolated Ordering

Figure 5.9: Performance of Different MIMO OFDM Interpolation Algorithms.

interpolated sub-carrier p, p 6∈ {ak}, is then computed as follows:

W(p)int = L(p)−1int{

L(p)−∗int

Rd(p)y(p)int}

strictly-lower(5.26)

where the detection order is locally determined optimally using the interpolated

channel matrix. L(p)int

is the Cholesky decompostion of Ry(p)int. The correlation

matrices Ry(p)int and Rd(p)y(p)

int are computed using the interpolated channel

matrix Cf (p)int. The MMSE for the interpolated sub-carrier p is then given

by equation 5.23 (for the MMSE calculation the actual channel matrix Cf (p)

is used to compute Ry(p) and Rd(p)y(p)). Although this algorithm requires the

same amount of computation as the optimal solution, the number of sub-carriers

dedicated to pilots is decreased.

The results show that using interpolated ordering algorithm can slightly im-

prove the slicer-SNR at low outage. For example at 5% outage, the interpolated

126

ordering algorithm improves the slicer-SNR by 0.7 dB. Meanwhile, the channel

estimation algorithm provides, at an additional complexity cost, a much larger

improvement at low outage. At a 5% outage the slicer-SNR degradation due to

channel estimation interpolation is 0.7 dB. The better performance of channel

estimation interpolation is due to the following factors. First, optimal ordering

is used for each sub-carrier. Also, after the Fourier Transform the channel ma-

trix for each sub-channels are correlated. Therefore, interpolation of the channel

matrix should be performed. Interpolation of the weight matrix is only an ap-

proximation to interpolating the channel and then computing the corresponding

coefficient matrix.

Although, the ordering algorithms for the MIMO OFDM receiver with cancel-

lation using interpolation degrade the system performance compared to the opti-

mal solution, attempt to determine a detection order should be done as demon-

strated by the results presented in Figure 5.10. The CDFs have been computed

for an effective frequency separation of f relsep = 6.5%. The following systems are

compared: single interpolated ordering MIMO OFDM with cancellation, non-

ordered MIMO OFDM with cancellation (i.e. channel as randomly generated

with detection from antenna M to 1), and MIMO OFDM without cancellation.

At a 5% outage, single ordering provides a slicer-SNR improvement of 2.4 dB

over the non-ordered algorithm and 6.3 dB over the MIMO OFDM receiver with-

out cancellation. The improvement due to the ordering, albeit sub-optimal, is

significant and worth the additional complexity.

127

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

Optimal

Single Ordering Cancellation

Without Cancellation

Non−Ordered Cancellation

Figure 5.10: CDF for Different Cancellation Algorithms for MIMO OFDM Re-

ceivers Using Interpolation.

128

CHAPTER 6

Spread Spectrum MIMO

Equalization and OFDM are appropriate techniques to mitigate and exploit the

effects of multipath propagation in indoor frequency selective fading channels.

However, in outdoor environments, the maximum excess delay of the channel

is several order of magnitude larger, which has undesirable effects on these re-

ceiver techniques (increase in equalizer computational complexity and longer

cyclic prefix and wasted bandwidth for OFDM). Spread spectrum modulation

techniques are an alternative approach for communications over wideband chan-

nels [Rap96, St96]. In spread spectrum, a low data rate signal is transmitted over

a bandwidth that is several orders of magnitude larger than required. Spread

spectrum communication is not bandwidth efficient for a single user case, but

performs well in multiple users environment and in the presence of interferers.

Spread spectrum techniques are therefore used in many military and commercial

cellular communication systems.

Direct Sequence Spread Spectrum (DSSS) Code Division Multiple Access

(CDMA) communication is the most popular spread spectrum technique and has

emerged as the technology of choice for cellular systems. DSSS CDMA has first

been introduced for second generation digital networks in the US and Wideband

CDMA (WCDMA) has been chosen as the physical layer technology for world-

wide Third Generation Universal Mobile Telecommunications System (UMTS)

networks [HT02]. In DSSS CDMA, the narrowband data is multiplied by a

129

pseudo-noise spreading signal that has a chip rate several order of magnitude

higher than the symbol rate. The ratio of the chip rate to the symbol rate is

defined as the spreading gain of the system (sometimes it is also referred as the

spreading factor) and is related to its ability to support multiple users and reject

interference. The advantages of CDMA are numerous. Among others, the fol-

lowing can be underlined: ease of deployment of new base stations, soft capacity

and quality of service, soft handover between cells, and resistance to narrowband

and wideband interference. Also, for outdoor channels with a large excess de-

lay spread, a low complexity RAKE receiver can be used to mitigate multipath

propagation.

MIMO is an attractive solution to improve the data throughput of current

CDMA technology for next generation cellular networks. A RAKE receiver for

MIMO CDMA has been introduced in [HVF99] and [Nag97]. In this chapter, a

new optimal generalized MIMO RAKE Receiver with cancellation is derived and

analyzed.

6.1 MIMO CDMA Model

In this section the model for a MIMO DSSS CDMA communication system is

introduced. The MIMO system consists of M transmit and N receive anten-

nas. Figure 6.1 depicts the architecture of the MIMO CDMA transmitter that is

considered in this dissertation. The CDMA system consists of U + 1 users simul-

taneously transmitting using the same carrier frequency. At each time instant a

data vector du(t) = [du,1(i) · · ·du,M(i)] is generated for each user u, 0 ≤ u ≤ U .

To simplify the notation and the analysis, it is assumed that all users generates

data synchronously, however the solution can be extended to asynchronous users.

The data sequence for each user is drawn from a QAM constellation with power

130

d0(t)

MdS

1(i)

dSM(i)

s0(i)

dU(t)

M

sU(i)

dS(i)

0γ

Uγ

Figure 6.1: Architecture of MIMO CDMA Transmitter.

σ2d and is assumed to be independently identically distributed in the user, space

and time domain:

E[du(t)∗dk(j)] =

σ2dI u = k and t = j

0 otherwise.

σ2d is the data signal variance and is such that the total transmit Mσ2

d is constant.

The data vectors for each user u are then multiplied by their own spreading

sequence su(i). Note that i is the chip time index and t is the symbol time index.

The spreading signal has a chip rate several orders of magnitude larger than the

symbol rate. The ratio between the chip rate and the symbol rate is defined as the

spreading factor F . Without loss of generality, it is assumed that all users have

the same spreading factor. Each spreading sequence is approximately orthogonal

to all other codewords to minimize the inter-user interference. In systems where

users synchronization can be achieved at the chip level (e.g., WCDMA down-

link) the spreading sequence is a multiplication of Orthogonal Variable Spreading

Factor (OVSF) spreading codes and a scrambling sequence [3rd02]. The users

131

are then perfectly orthogonals. For the model used in this chapter, the spread-

ing sequences are complex with unit power and periodic with period Ls (Ls is a

multiple of F ). The amount of power allocated to each user is controlled by the

parameter γu. The spreaded sequences from all users are added together to form

the spreaded data vector dS(i) = [dS1 (i) · · ·dSM(i)] given by:

dS(i) =U∑u=0

√γusu(i)du

(⌊i

F

⌋). (6.1)

dSm(i) is transmitted at chip time index i from antenna m.

The MIMO channel model for MIMO OFDM presented in Section 5.1 is used

for the MIMO spread spectrum model. The MIMO channel has M inputs, N out-

puts and Nc + 1 multipaths. The output of the channel uS(i) = [uS1 (i) · · ·uSM(i)]

is given by:

uS(i) =Nc∑nc=0

dS(i−Nc)Cnc + v(i), (6.2)

where Cnc is the M × N MIMO channel matrix for multipath nc and v(i) =

[v1(i) · · ·vM(i)] is the IID stationary noise vector with auto-correlation Rv(i−k) =

E[v(i)∗v(k)]. uSn(i) is the sample received at chip time index i from antenna n.

To simplify the notation it is assumed that Nc ≤ F . The analysis can also be

generalized for any value of Nc.

The architecture of the novel MIMO generalized RAKE receiver with cancel-

lation introduced in this dissertation is shown in Figure 6.2. The RAKE receiver

consists of K fingers. Traditional RAKE receivers use RAKE fingers at offsets

corresponding to channel multipaths. In the generalized version, this restriction

is removed and fingers can be positioned at any offset τk. The objective of the re-

ceiver is to process the sequence of data vector uS(i) to compute the best estimate

of d0(t), the data transmitted by user u = 0. The receiver therefore performs

132

uS(i)

v1(i)

vN(i)

uS1(i)

uSN(i)

N

s0

*(i-T)

Z-1

N

i=T+(t+1)F-1

F1

s0

*(i-T)

Z-1

NFK

M d0(t) d

0(t)

B

u1(t)

uK(t)

i=T+(t+1)F-1

Z-(T-τk)

Z-(T-τ1)

Figure 6.2: MIMO Generalized RAKE Receiver with Cancellation.

on each finger a time correlation operation using the sequence s0(i) in order to

detect only the desired user.

For each RAKE finger k, 1 ≤ k ≤ K, uS(i) is delayed by T − τk. T is chosen

such that T ≥ Nc. The delayed sequence is then multiplied at the chip rate

by s∗0(i − T ) and accumulated. The correlator is reset to zero at i = T + tF

and dumped at the symbol rate to uk(t) = [uk,1(t) · · ·uk,N(t)] at chip index

i = T + (t + 1)F − 1. t is the symbol time index at the receiver and is delayed

133

by T chips with respect to the transmitter symbol time index. uk(t) is given by:

uk(t) =

T+(t+1)F−1∑i=T+tF

uS(i− (T− τk))s∗0(i− T )

=

(t+1)F−1∑i=tF

[Nc∑nc=0

dS(i + τk − nc)Cnc + v(i + τk)

]s∗0(i)

=

(t+1)F−1∑i=tF

[Nc∑nc=0

U∑u=0

√γusu(i− (nc − τk))du

(⌊i− (nc − τk)

F

⌋)Cnc +

v(i + τk)

]s∗0(i)

(6.3)

Let define the following quantities:

δ−τ,u,t =

∑tF+τ−1

i=tF su(i− τ)s∗0(t) τ > 0

0 otherwise

(6.4)

δτ,u,t =

∑(t+1)F−1

i=tF+τ su(i− τ)s∗0(t) τ ≥ 0∑(t+1)F+τ−1i=tF su(i− τ)s∗0(t) otherwise

(6.5)

δ+τ,u,t =

∑(t+1)F−1

i=(t+1)F+τ su(i− τ)s∗0(t) τ < 0

0 otherwise.

(6.6)

To simplify the notation it is assumed that a summation where the upper index

is smaller than the lower index is equal to zero. uk(t), the output of RAKE finger

134

k can then be expressed as:

uk(t) =U∑u=0

√γu

Nc∑nc=0

[tF+nc−τk−1∑

i=tF

su(i− (nc − τk))s∗0(i)du(t− 1) +

min((t+1)F+nc−τk−1,(t+1)F−1)∑i=max(tF+nc−τk ,tF )

su(i− (nc − τk))s∗0(i)du(t) +

(t+1)F−1∑i=(t+1)F+nc−τk

su(i− (nc − τk))s∗0(i)du(t + 1)

]Cnc +

(t+1)F−1∑i=tF

v(i + τk)s∗0(i)

=U∑u=0

√γu

Nc∑nc=0

[δ−nc−τk ,u,tdu(t− 1) + δnc−τk ,u,tdu(t) +

+ δ+nc−τk,u,tdu(t + 1)

]Cnc +

(t+1)F−1∑i=tF

v(i + τk)s∗0(i)

(6.7)

The generalized MIMO RAKE receiver with cancellation estimate of d0(t)

is obtained by combining the output of the K RAKE fingers and canceling the

interference caused by the data streams for which decisions are available. Let

introduce the RAKE vector u(t) = [u1(t) · · ·uK(t)] and the cancellation feedback

vector df (t) = d0(t). The data estimate d0(t) is given by:

d0(t) = −df (t)B +K∑k=0

uk(t)Fk

= −df (t)B + u(t)F

= [df (t) u(t)]

−BF

= y(t)W.

(6.8)

Fk is the N×M coefficient matrix for RAKE finger k, F is the generalized RAKE

135

MIMO feedforward coefficient matrix given by:

F =

F1

· · ·

FK

, (6.9)

and B is the M×M strictly lower triangular cancellation coefficient matrix. The

generalized MIMO RAKE receiver with cancellation error vector is given by:

e(t) = d0(t)− d0(t)

= d0(t)− y(t)W.(6.10)

6.2 MIMO Generalized RAKE Receiver MMSE Solution

The optimal MMSE solution for the novel MIMO generalized RAKE receiver

with cancellation is obtained by using the results derived in Section 4.2. The

optimal coefficient matrix W(t)o at symbol time index t is given by:

W(t)o = L(t)−1{

L(t)−∗

Rd0(t)y(t)

}strictly-lower

(6.11)

where L(t) is the Cholesky decomposition of Ry(t). W(t)o is different for each

received symbol since, as will be demonstrated, the correlation matrices Ry(t)

and Rd0(t)y(t) depends on t. The auto-correlation matrix of y(t) is given by:

Ry(t) = E[y(t)∗y(t)] =

Rdf (t) Rdf (t)u(t)∗

Rdf (t)u(t) Ru(t)

(6.12)

where:

Rdf (t) = Rd0 (6.13)

Rdf (t)u(t) = Rd0(t)u(t) (6.14)

Rd0 = σ2dIM , (6.15)

136

and the cross-correlation matrix Rd0(t)y(t) is the following:

Rd0(t)y(t) = E[y(t)∗d0(t)] =

Rd0(t)df (t)

Rd0(t)u(t)

=

Rd0

Rd0(t)u(t)

. (6.16)

These matrices depend on the time varying correlation matrices Rd0(t)u(t) and

Ru(t) given by:

Rd0(t)u(t) = E[u(t)∗d0(t)] =

Rd0(t)u1(t)

...

Rd0(t)uK(t)

(6.17)

where

Rd0(t)uk(t) = E[uk(t)∗d0(t)] =√γ0

Nc∑nc=0

Cnc

∗Rd0δ∗nc−τk,0,t, (6.18)

and

Ru(t) = E[u(t)∗u(t)] =

Ru1(t)u1(t) · · · R∗u1(t)uK(t)

.... . .

...

Ru1(t)uK(t) · · · RuK(t)uK(t)

(6.19)

where

Ruj(t)ui(t) = E[ui(t)∗uj(t)]

=U∑u=0

γu

Nc∑nl=0

Nc∑nm=0

Cnl

∗Rd0Cnm

(δ−∗nl−τi,u,tδ

−nm−τj ,u,t +

δ∗nl−τi,u,tδnm−τj ,u,t + δ+∗nl−τi,u,tδ

+nm−τj ,u,t

)+ Rv(t),j,i.

(6.20)

Rv(t),j,i is the noise correlation at the output of the RAKE receiver and is given

by:

Rv(t),j,i = E

(t+1)F−1∑l=tF

s0(l)v(l + τi)∗

(t+1)F−1∑m=tF

s∗0(m)v(m + τj)

=

(t+1)F−1∑l=tF

(t+1)F−1∑m=tF

s0(l)s∗0(m)Rv(l+τi−m−τj).

(6.21)

137

In the case where the noise is uncorrelated in the time dimension Rv(k−l) =

Rvδ(0) and

Rv(t),j,i =

min((t+1)F+τj−τi−1,(t+1)F−1)∑i=max(tF+τj−τi,tF )

s0(l)s∗0(l − (τj − τi))Rv

= δ∗τj−τiRv.

(6.22)

The MMSE vector of the MIMO generalized RAKE receiver with cancellation for

the symbol received at time index t is defined as J(t) = [J1(t) · · ·JM(t)] where

Jm(t) = E[em(t)∗em(t)] and is given by:

J(t) = diag(Re(t))

= diag(E[e(t)∗e(t)

])= diag

(Rd0 −W(t)o∗Rd0(t)y(t) −Rd0(t)y(t)

∗W(t)o + W(t)o∗Ry(t)W(t)o).

(6.23)

However, having a different optimal solution W(t)o for each received symbol

is not practical. It is more desirable to have a single solution for all received

symbol. The time dependency comes from the fact that the spreading sequences

are different for each symbol. Fortunately, the spreading sequences are periodic

with period Ls and therefore the correlation matrices Rd0(t)y(t) and Ry(t) are also

periodic with period LsF

. That is, it can be shown that given su(i) = su(i+ kLs),

k an integer, it follows that for any t and any integer j:

Ry(t) = Ry(t+jLsF ) (6.24)

and

Rd0(t)y(t) = Rd0(t+jLsF )y(t+jLs

F ). (6.25)

The global (i.e., time invariant) optimal MMSE solution Wo for the MIMO

generalized RAKE receiver with cancellation is selected to minimize the time

138

averaged MMSE J = [J1(t) · · ·JM(t)] given by:

J(t) =F

Ls

t=Ls/F−1∑t=0

J(t)

= diag

(Rd0 −Wo∗ F

Ls

t=Ls/F−1∑t=0

Rd0(t)y(t) −F

Ls

t=Ls/F−1∑t=0

Rd0(t)y(t)∗Wo +

Wo∗ F

Ls

t=Ls/F−1∑t=0

Ry(t)Wo

)= diag

(Rd0 −Wo∗Rd0y −Rd0y

∗Wo + Wo∗RyWo)

(6.26)

where

Ry =F

Ls

t=Ls/F−1∑t=0

Ry(t) (6.27)

and

Rd0y =F

Ls

t=Ls/F−1∑t=0

Rd0(t)y(t). (6.28)

The global time invariant MMSE solution is then the following:

Wo = L−1{

L−∗

Rd0y

}strictly-lower

(6.29)

where L is the Cholesky decomposition of Ry.

6.3 Combiner Spread Spectrum MIMO Receivers

The optimal MIMO generalized RAKE receiver presented previously globally

optimizes the coefficient matrices of all RAKE fingers and the cancellation matrix.

To decrease the receiver computational complexity, it is proposed in this section,

as shown in Figure 6.3, to perform the MIMO reception locally on a per RAKE

139

uS(i)

v1(i)

vN(i)

uS1(i)

uSN(i)

N

s0

*(i-T)

Z-1

NF1

s0

*(i-T)

Z-1

NF

K

d0,1(t)

d0(t)

B1

Combiner

d0,K(t)

d0(t)

BK

d0(t)

u1(t)

uK(t)

i=T+(t+1)F-1

i=T+(t+1)F-1

Z-(T-τk)

Z-(T-τ1)

Figure 6.3: Combining MIMO Generalized RAKE Receiver with Cancellation.

finger basis, and then combine the local data estimate from each finger to compute

the global data estimate.

For each RAKE finger k, 1 ≤ k ≤ K, the data estimate of d0(t) is given by:

d0,k(t) = −df (t)Bk + uk(t)Fk

= [df (t) uk(t)]

−Bk

Fk

= yk(t)Wk,

(6.30)

and the error vector for this finger is:

e(t) = d0(t)− d0,k(t) = d0(t)− yk(t)Wk. (6.31)

The time invariant optimal coefficient matrix for RAKE finger k for the combining

MIMO generalized RAKE receiver with cancellation is then given by:

Wko = L

−1{

L−∗

Rd0yk

}strictly-lower

(6.32)

140

where L is the Cholesky decomposition of Ryk. Ryk

and Rd0ykare the time av-

eraged correlation matrices. The time averaged auto-correlation matrix of yk(t)

is given by:

Ryk=LsF

Ls/F−1∑t=0

E[y(t)∗y(t)]

=LsF

Ls/F−1∑t=0

Rdf (t) Rdf (t)uk(t)∗

Rdf (t)uk(t) Ruk(t)

=

Rd0 Rd0uk

∗

Rd0ukRuk

,(6.33)

and the cross-correlation matrix by:

Rd0yk=LsF

Ls/F−1∑t=0

E[y(t)∗d0(t)]

=LsF

Ls/F−1∑t=0

Rd0(t)df (t)

Rd0(t)uk(t)

=

Rd0

Rd0uk

.(6.34)

Rd0ukand Ruk

are the time averaged correlation matrices, respectively, of

Rd0(t)uk(t) (equation 6.18) and Ruk(t) (equation 6.20) given in the previous sec-

tion. The average MMSE vector Jk = [Jk,1 · · ·Jk,M ] of RAKE finger k is then:

Jk = diag(Rd0 −Wk

o∗Rd0yk−Rd0yk

∗Wko + Wk

o∗RykWk

o). (6.35)

The combiner weights for each data streamm, 1 ≤ m ≤M , the data estimates

from each MIMO RAKE finger to compute the global data estimate. Let define

for m = 1, . . . ,M the finger data estimate vector zm(t) = [z1,m(t) · · · zK,M(t)] =

141

[d0,1,m(t) · · · d0,K,m(t)] and the coefficient matrix Wm:

Wm =

{B1}m · · · {BK}m{F1}m · · · 0N×1

.... . .

...

0N×1 · · · {FK}m

, (6.36)

where {A}m is the mth column of A. It can then be verified that:

zm(t) = y(t)Wm. (6.37)

The global data estimate d0,m(t) for the combiner MIMO generalized RAKE

receiver with cancellation is given by:

d0,m(t) = zm(t)

WCm,1

...

WCm,K

= zm(t)WCm. (6.38)

WCm is the RAKE combiner weighting vector for data stream m.

The time averaged MSE Jm for data stream m for the combiner MIMO gen-

eralized RAKE receiver with cancellation is given by:

Jm =LsF

Ls/F−1∑t=0

E[(d0,m(t)− d0,m(t))∗(d0,m(t)− d0,m(t))]

=LsF

Ls/F−1∑t=0

(σ2d −WC

m

∗Rd0,m(t)zm(t) −Rd0,m(t)zm(t)

∗WCm + WC

m

∗Rzm(t)W

Cm

)= σ2

d −WCm

∗Rd0,mzm −Rd0,mzm

∗WCm + WC

m

∗RzmWC

m,

(6.39)

where

Rzm =LsF

Ls/F−1∑t=0

Rzm(t) = Wm∗RyWm (6.40)

142

and

Rd0,mzm =LsF

Ls/F−1∑t=0

Rd0,m(t)zm(t) = Wm∗{Rd0,y}m. (6.41)

Two approaches to compute the RAKE combiner weighting vector are pro-

posed. The first technique selects the weight vector WCm to minimize Jm. This

is the optimal combining vector for this receiver and is given by:

WCm = Rzm

−1Rd0,mzm. (6.42)

The second technique uses a Maximum Ratio Combining (MRC) approach to

compute the weight vector WCm. MRC scales the output from each RAKE finger

proportionally to the signal coefficient and inversely proportional to the noise

power. The output zk,m from RAKE finger k used for combining can be expressed

as:

zk,m = d0,m(t) = d0,m(t)− ek,m(t). (6.43)

The signal coefficient is 1 and the time averaged noise (error) power is Jk,m. The

combining vector for the MRC receiver is therefore given by:

WCm = αm

1/J1,m

...

1/JK,m

= αmWMRCm . (6.44)

MRC does not scale the output of the receiver. In order to compute the final

SNR and allow a fair comparison with the other receivers, the scaling factor αm

is introduced to minimize the MSE for the selected MRC coefficients. That is αm

minimizes the following:

Jm = σ2d − α∗mWMRC

m

∗Rd0,mzm −Rd0,mzm

∗WMRCm αm + α∗mWMRC

m

∗RzmWMRC

m αm

(6.45)

143

and is given by

αm =(WMRC

m

∗RzmWMRC

m

)−1WMRC

m

∗Rd0,mzm. (6.46)


The performance of MIMO spread spectrum is evaluated in this section through

simulations. The performance of the optimal generalized RAKE receiver with

cancellation is first analyzed and compared to the MIMO equalization solution.

Then, the performance of the combiner MIMO RAKE architectures is evaluated.

For all the simulations the spreading factor is set to F = 32 and the number

of users is U = 4. The MIMO communication channel consists of 4 transmit

antennas and 4 receive antennas (M = N = 4). The MIMO receivers with

cancellation use the sub-optimal detection ordering algorithm.

The spreading signal are generated according to the 3GPP WCDMA stan-

dard [3rd02]. The spreading sequence su(i) is defined as:

su(i) =1√2cF,nu(rem(i/F )) s(rem(i/Ls))

cF,nu(i) is the channelization spreading code and is an OVSF code. The OVSF

codes have a length F and provide orthogonality between the different users. The

generation method for the spreading code cF,nu = [cF,nu(0) · · · cF,nu(F − 1)] for

0 ≤ nu ≤ F − 1, is defined as follows:

cF,nu =

[cF/2,bnu/2c cF/2,bnu/2c

]for nu even[

cF/2,bnu/2c − cF/2,bnu/2c]

for nu odd

and c1,0 = 1. For the simulations F = 32 and n0 = 1, n1 = 12, n2 = 15, n3 = 21,

and n4 = 29. s(i) is a complex scrambling code and is common to all users.

For the simulation, the WCDMA downlink scrambling code n = 0 is used and

144

s(i) is generated as follows. Let define the two m-sequences x(i) and y(i) for

i = 0, . . . , 218 − 20:

x(i+ 18) = x(i+ 7) + x(i) modulo 2

y(i+ 18) = y(i+ 10) + y(i+ 7) + y(i+ 5) + y(i) modulo 2

The initial conditions are x(0) = . . . = x(17) = 0 and y(0) = . . . = y(17) = 1.

The Gold sequences zre(i) and zim(i) are defined for i = 0, . . . , 218 − 16 as:

zre(i) = x(i) + y(i) modulo 2

zim(i) = y(i+ 5) + y(i+ 6) + y(i+ 8) + y(i+ 9) + y(i+ 10) + y(i+ 11) +

y(i+ 12) + y(i+ 13) + y(i+ 14) + y(i+ 15) modulo 2.

The scrambling sequence s(i) is then given by:

s(i) = (−1)zre(i) + j(−1)zim(i) for i = 0, . . . , Ls − 1.

Unless mentioned otherwise, the sequence length is set to Ls = 38400.

The power dedicated to the demodulated user u = 0 is set to γ0 = 1/32 = −15

dB and γu = 31/128 = −6.2 dB for u = 1, . . . , 4. Note that∑U

u=0 γu = 1. The

system was simulated for white noise (i.e., Rv(k−l) = σ2vIδ(k− l)) and the average

SNR was set to 20 dB. The SNR is defined as the ratio between the total received

power from all users, all antennas and all multipaths, and the noise variance

at a received antenna. Note that the parameter settings are selected to allow

a fair comparison between the different systems. For example, in a flat fading

environment MIMO CDMA and MIMO equalization have the same performance.

6.4.1 Comparison with MIMO Equalization

The performance of the MIMO generalized RAKE receiver with cancellation

is illustrated in Figure 6.4 for a frequency selective fading environment with

145

0 5 10 15 20 250

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

DFE

RAKE Receiver

Linear Equalizer andGeneralized RAKE Receiver

Figure 6.4: CDF for MIMO Generalized RAKE Receiver with Cancellation.

τ rms = 0.5Ts. The multipath channel is generated using the channel model given

in Section 2.4.1 and the channel parameters are Nc = 2 and γrms = 1.56. For

comparison purpose, the performance of the MIMO DFE receiver with cancel-

lation was computed. The parameters of the MIMO DFE receiver are Nf = 3,

Q = 2 and ∆ = 3. The MIMO linear equalizer receiver with cancellation is

obtained by letting Q = 0. Two MIMO RAKE receivers are simulated. The

first is a traditional RAKE receiver where fingers are positioned at the multipath

positions. That is, the MIMO RAKE receiver with cancellation consists of three

fingers (K = 3) with τk = k−1 for k = 1, 2, 3. For the generalized MIMO RAKE

receiver with cancellation, an additional RAKE finger is added at τ4 = 3. The

CDF for each system is generated using 1000 independent MIMO channel real-

izations. For each channel, the optimal solution for the four different receivers is

calculated and the slicer-SNR for each data stream m, 1 ≤ m ≤M , is computed.

146

The slicer-SNR CDF for the simulated receivers is generated using the recorded

slicer-SNR for each data stream and each channel realization.

Compared to the DFE receiver, the RAKE receiver with K = 3 has a sig-

nificant performance degradation. For example, at a 5% outage, the slicer-SNR

decreases by 8.3 dB. However, when compared with the linear equalizer perfor-

mance, the degradation is only 0.4 dB at a 5% outage. The performance degra-

dation when going from a DFE to a linear equalizer illustrates the importance

of the feedback section to mitigate the ISI introduced by multipath propagation.

Therefore, the performance degradation of the RAKE receiver is mainly due to

the absence of the feedback section. The generalized RAKE receiver provides a

performance improvement by adding an additional RAKE finger, even if there is

no multipath at the finger position. Indeed, the performance of the generalized

MIMO RAKE receiver with cancellation is similar to the MIMO linear equalizer

with cancellation.

The previous channel illustrates the performance of the MIMO receiver for a

typical indoor environment. Outdoor channels typically have larger excess delay

with strong multipaths arriving with long delays due to reflections from distant

objects such as mountains and buildings. Figure 6.5 compares the performance

of the MIMO RAKE receiver and MIMO DFE receiver for an outdoor channel.

The channel consists of four equal power multipaths at delay of 0Ts, Ts, 5Ts

and 10Ts. τ rms = 3.94Ts for this channel and each multipath is an independent

Rayleigh fading random variable with variance 0.25. The MIMO DFE receiver

with cancellation has the same parameters as previously and the RAKE receiver

with cancellation consists of four fingers (K = 4) positioned at τ1 = 0, τ2 = 1,

τ3 = 5, and τ4 = 10. Although the MIMO RAKE receiver complexity (four

MIMO taps and 1 cancellation tap) is less that the MIMO DFE (six MIMO taps

147

−6 −4 −2 0 2 4 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

DFE

RAKE Receiver

Figure 6.5: Slicer-SNR CDF for Outdoor Channel.

and 1 cancellation tap), its performance is significantly better for this typical

outdoor channel. For example, at a 5% outage the MIMO RAKE receiver with

cancellation outperforms the DFE receiver by 3.6 dB. Also, although the multi-

path propagation conditions are more severe for the outdoor channel compared

to the indoor environment, the MIMO RAKE receiver performance only degrades

by 3.6 dB at a 5% outage versus a 15.6 dB slicer-SNR decrease for the MIMO

DFE receiver.

Figure 6.6 shows the performance of the generalized MIMO RAKE receiver for

a sequence length Ls = 32. The channel consists of three equal power multipaths

at delay of 0Ts, Ts, 2Ts. τ rms = 0.82Ts for this channel and each multipath is

an independent Rayleigh fading random variable with variance 1/3. The RAKE

receiver with cancellation consists of three fingers (K = 3) located at τ1 = 0,

τ2 = 1, τ3 = 2. The generalized MIMO RAKE receiver with cancellation has

148

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

Rake Receiver −L

s=38400

Generalized Rake Receiver − L

s=38400

Rake Receiver −L

s=32

Generalized Rake Receiver − L

s=32

Figure 6.6: Slicer-SNR CDF for Ls = 32.

an additional finger (K = 4) at τ4 = 3. This channel and the RAKE receiver

configurations are also used in the next section to simulate the performance of

the combiner RAKE receiver architectures. First, it can be noted that for the

sequence length Ls = 38400, the performance improvement provided by the addi-

tional finger for the generalized RAKE receiver is larger for this channel than for

the indoor channel with τ rms = 0.5Ts. In this environment, at a 5% outage, the

generalized RAKE receiver outperforms the RAKE receiver by 1.3 dB versus 0.4

dB for the indoor environment. For Ls = 32, the sequence length is equal to the

spreading factor F = 32. Therefore, the optimal solution is the same for every

received symbol and is equal to the time averaged MMSE solution. The RAKE

receiver is therefore able to take better advantage of the symbol cross-correlations

to improve the performance as seen in the simulation results. For example, at a

5% outage, the slicer-SNR improves by 4.7 dB for the sequence length Ls = 32.

149

−5 0 5 10 150

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

SNR (dB)

P(S

NR

< A

bsci

ssa)

Optimal RAKEReceiver

RAKE Receiver withMMSE Combining

Generalized OptimalRAKE Receiver

Generalized RAKE Receiverand RAKE Reiceverwith MRC Combining

Generalized RAKE Receiverwith MMSE Combining

Figure 6.7: Slicer-SNR CDF for the Combiner MIMO RAKE Receivers with

Cancellation.

6.4.2 Performance of Combiner Architectures

The performance of the proposed combiner architectures is illustrated in Fig-

ure 6.7. Both combiner MIMO RAKE receivers perform similarly and the per-

formance degradation is reasonable versus the significant complexity reduction

of these receivers. At a 5% outage, the slicer-SNR decreases by 2.3 dB for the

RAKE receiver with MRC combining and 2 dB for the RAKE receiver with

MMSE combining. However, the combining generalized RAKE receivers are not

able to take advantage of the additional fingers to significantly improve the re-

ceiver performance. For the MRC combining receiver, the additional finger does

not improve the slicer-SNR. This is due to the fact that the additional RAKE

finger has a much lower SNR than the other fingers and therefore has no signifi-

cant contribution to the combined output. The MMSE combining receiver is able

150

to extract some information from the additional finger to improve the slicer-SNR

by approximately 0.1 dB versus 1.3 dB for the optimal MIMO RAKE receiver.

The performance degradation of the combiner MIMO RAKE receivers is due

to the loss of the spatial information when combining the multipaths. That is, the

optimal RAKE receiver performs a joint combining in the space and time domain.

On the other hand, the combiner receivers first perform a weighting in the space

domain. Each finger computes its own MIMO coefficient matrix independently of

the other finger and therefore ignores the additional information available from

the other fingers. Thereafter, the space information for each data stream is lost

and cannot be used to correctly combine the output from each finger.

151

CHAPTER 7

Adaptive MIMO Algorithms

In Chapter 4 a new framework for the study of MIMO DFE with cancellation

was introduced. Although the problem of MIMO equalization has been previously

studied [YR94, TAS95, AS99, AS00, MHC99, CC99], only [MHC99] has explored

a recursive least square adaptive method for the simple case of a linear MIMO

equalizer. In this chapter, the framework developed in Chapter 4 is used to derive

novel adaptive algorithms for the MIMO DFE with cancellation. Specifically,

stochastic gradient algorithms and recursive least-square adaptive algorithms are

studied.

7.1 Stochastic Gradient Algorithms

7.1.1 LMS Algorithm

In order to obtain the LMS adaptive algorithm for the MIMO DFE with cancel-

lation, the steepest descent algorithm is first derived. This algorithm finds the

optimal coefficient matrix Wo using an iterative procedure that starts from an

initial guess and improves upon it, until it converges to the optimal solution. The

152

MSE cost function of the MIMO DFE receiver with cancellation is given by:

J(W) = E[|d(i−∆)− yW|2

]=

M∑m=1

E[|dm(i−∆)− yWm|2

]=

M∑m=1

Jm(Wm),(7.1)

where Wm is the mth column of W, and |x|2 = xx∗. This equation shows that

Jm(Wm) does not depend on the other columns Wi, i 6= m. Therefore, if each

column Wm independently converges to its optimal value, the associated cost

function Jm(Wm) also converges to its minimum, and the global cost function

then attains the global minimum. The steepest descent algorithm can thus be

applied to each of the vectors Wm. Recall that the first m elements of Wm are

equal to zero. It can then be shown that

∇WmJ(Wm) ={−Rm

d(i−∆)y∗ + Wm∗Ry

}l(m)

, (7.2)

where

[{A}l(m)

]j

=

[A]j

if m < j ≤ N(NF + 1) +M(Q+ 1),

0 if 1 ≤ j ≤ m,

(7.3)

and Rmd(i−∆)y is the mth column of Rd(i−∆)y. The steepest descent algorithm for

each Wm, 1 ≤ m ≤M , is given by:

Wmi = Wm

i−1 − µ∇WmJ(Wm)∗

= Wmi−1 − µ

{−Rm

d(i−∆)y∗ + Wm∗Ry

}∗l(m)

.(7.4)

Grouping the steepest descent algorithms for each column Wm of W Together,

the following MIMO DFE with cancellation steepest descent algorithm is ob-

tained:

Wi = Wi−1 + µ{

Rd(i−∆)y −RyWi−1

}stricty-lower

, (7.5)

153

or −Bi

Fi

=

−Bi−1

Fi−1

+ µ

{Rd(i−∆)y −Ry

−Bi−1

Fi−1

}stricty-lower

. (7.6)

By replacing the correlation matrices in the MIMO DFE with cancellation

steepest descent algorithm by their instantaneous values, the following MIMO

DFE with cancellation LMS algorithm is derived:

Wi = Wi−1 + µ {y∗i (d(i−∆)− yiWi−1)}stricty-lower (7.7)

or −Bi

Fi

=

−Bi−1

Fi−1

+ µ

{[df i ui

]∗(d(i−∆)−

[df i ui

]−Bi−1

Fi−1

)}stricty-lower

.

(7.8)

Other stochastic gradient algorithms, such as the Sign-LMS and the Normalized-

LMS algorithms, can be derived in a similar manner. The LMS algorithm has a

complexity on the order of O(M(N(Nf + 1) +M(Q+ 1))).

7.2 Recursive Least-Squares Adaptive Filters

7.2.1 RLS Algorithm

Similarly to the steepest descent algorithm, the solution Wm that minimizes

the least-squares error for each data stream in the MIMO DFE receiver with

cancellation can be found. This solution minimizes the least-squares error of the

overall receiver since each data stream error is independent of the weight vector

used to compute the estimate for the other data streams.

154

The exponentially-weighted regularized least-squares problem that needs to

be solved is to find the column vector Wm, m = 1, . . . ,M , that minimizes at

each time instant i the following cost function [Hay96, Say03]:

J(Wm) = Wm∗(λ−(i+1)Π0

)−1Wm +

i∑j=0

λi−j|dm(j −∆)−{yj}l(m)

Wm|2,

(7.9)

where Π0 is a N(Nf +1)+M(Q+1)×N(Nf +1)+M(Q+1) regularizing positive

definite matrix, and λ ≤ 1 is a forgetting factor.

The RLS iterative algorithm that solves this problem for each of the Wm for

the MIMO DFE receiver with cancellation can be summarized as follows [Say03].

Start with Wm−1 = 0N(Nf+1)+M(Q+1)×1 and Pm

−1 = Π0, then:

Wmi = Wm

i−1 + gmi

[dm(i−∆)− yiW

mi−1

], (7.10)

where

gmi =

λ−1Pmi−1

{yi}∗l(m)

1 + λ−1{yi}l(m)

Pmi−1

{yi}∗l(m)

(7.11)

Pmi = λ−1

(Pmi−1 − gm

i

{yi}l(m)

Pmi−1

). (7.12)

The RLS algorithm for the MIMO DFE receiver with cancellation has a com-

plexity on the order of O(M(N(Nf + 1) +M(Q+ 1))2).

For the case where the current decisions are not used to obtain the data

estimate, the vector gmi and the matrix Pm

i are given by:

gmi =

λ−1Pmi−1

{yi}∗l(M)

1 + λ−1{yi}l(M)

Pmi−1

{yi}∗l(M)

(7.13)

Pmi = λ−1

(Pmi−1 − gm

i

{yi}l(M)

Pmi−1

). (7.14)

The vector gmi and the matrix Pm

i are the same for the recursions performed on

each weight vectors Wm, 1 ≤ m ≤ M . The update equations for each Wm can

155

be grouped to obtain the following RLS algorithm for the MIMO DFE receiver

without cancellation matrix coefficient update:

Wi = Wi−1 + gi[d(i−∆)− yiWmi−1] (7.15)

where

gi =λ−1Pi−1

{yi}∗l(M)

1 + λ−1{yi}l(M)

Pi−1

{yi}∗l(M)

(7.16)

Pi = λ−1(Pi−1 − gi

{yi}l(M)

Pi−1

), (7.17)

P−1 is defined as before and W−1 = 0N(Nf+1)+M(Q+1)×M .

7.2.2 Inverse QR Algorithm

The conventional RLS algorithm suffers from stability problems and might di-

verge under fixed precision computation conditions. Furthermore, for the MIMO

DFE with cancellation a different RLS algorithm entity must be implemented for

each of the M data streams, which is computationally intensive. In this section,

the possibilities offered by an array form of the RLS algorithm known as the

inverse QR algorithm [SK98, Say03] are explored.

First, the inverse QR algorithm is substituted for the RLS algorithm for each

of the M data streams. Let start with Wm−1 = 0N(Nf+1)+M(Q+1)×1 and Pm1/2

−1 =

Π1/20 . Then, for each data stream m, m = 1, . . . ,M , and each time instant i ≥ 0,

the following array transformation is performed:1 1√λ

{yi}l(m)

Pm1/2

i−1

0 1√λPm1/2

i−1

Θmi =

γm−1/2

i 0

gmi γ

m−1/2

i Pm1/2

i

(7.18)

where Θmi is any orthogonal rotation that produces the zero vector in the first

row of the post-array. The vector estimate Wmi is updated as follows:

Wmi = Wm

i−1 +[gm

i γm−1/2

i

][γm−1/2

i

]−1[dm(i−∆)− yiW

mi−1

](7.19)

156

where the quantities{gm

i γm−1/2

(i)}

and{γm−1/2

i

}are read directly from the

entries of the post-array.

Now an example is used to show how the inverse QR algorithm can be simpli-

fied for the MIMO DFE with cancellation. Let assume that N = M = 2, Nf = 0

and Q = 0. Furthermore, suppose that at step i a single matrix P1/2i−1 of the form

P1/2i−1 =

pi−1,1,1 0 0

pi−1,2,1 pi−1,2,2 0

pi−1,3,1 pi−1,3,2 pi−1,3,3

(7.20)

is used instead of the distinct matrices Pm1/2

i−1 . Note that one column from P

was removed since only the elements 2 to N(Nf + 1) + M(Q + 1) of y need to

be considered to compute the vector W (i.e., the first row of W is always null).

Thereafter, P is a square matrix of dimension N(Nf + 1) + M(Q + 1) − 1, and

W a N(Nf + 1) +M(Q+ 1)− 1×M matrix. With this consideration in mind,

it is assumed that P11/2

i−1 = P1/2i−1 and P21/2

i−1 is a sub-matrix of P1/2i−1 consisting of

the last two rows and columns of P1/2i−1, that is:

P21/2i−1 =

pi−1,2,2 0

pi−1,3,2 pi−1,3,3

. (7.21)

This assumption is valid for i = 0 and by induction this hypothesis will be shown

to be valid for any i > 0.

For the data stream m = 2, let start with the pre-array1 yp2 yp3 yp4

0 pi−1,1,1 0 0

0 pi−1,2,1 pi−1,2,2 0

0 pi−1,3,1 pi−1,3,2 pi−1,3,3

(7.22)

157

which is transformed after two elementary rotations Θi,j (more details on rota-

tions are available in Section 7.2.3), where the rotations have the form:

Θi,1 =

fpp 0 0 fpq

0 1 0 0

0 0 1 0

fqp 0 0 fqq

,Θi,2 =

fpp 0 fpq 0

0 1 0 0

fqp 0 fqq 0

0 0 0 1

, (7.23)

into the post-array γ2−1/2

i yp2 0 0

0 pi−1,1,1 0 0

g2i,2γ

2−1/2

i x2,1,pa p2i,2,2 0

g2i,3γ

2−1/2

i x3,1,pa p2i,3,2 p2

i,3,3

. (7.24)

Note, that the elements p2i , γ

2i , and g2

i of the post-array are independent of the

content of the second column of the pre-array. Therefore, this column can contain

any values, as long as they satisfy the lower triangular properties of the matrix

P1/2i , and the values of the post-array after two rotations yield the expected values

for data stream m = 2. The vector W2i can be computed from the elements from

the first column of this post-array and the last two rows and columns contain

P21/2

i .

For data stream m = 1, the pre-array given in equation 7.22 is used. The pre-

array is transformed using three elementary rotations Θi,j, where the rotations

i = 1, 2 are the same as the one used for data stream m = 2, and the third one

has the form:

Θi,3 =

fpp fpq 0 0

fqp fqq 0 0

0 0 1 0

0 0 0 1

. (7.25)

158

The post-array is given byγ1−1/2

i 0 0 0

g1i,1γ

1−1/2

i p1i,1,1 0 0

g1i,2γ

1−1/2

i p1i,1,1 p1

i,2,2 0

g1i,3γ

1−1/2

i p1i,1,1 p1

i,3,2 p1i,3,3

=

γ1−1/2

i 0 0 0

g1i,1γ

1−1/2

i p1i,1,1 0 0

g1i,2γ

1−1/2

i p1i,1,1 p2

i,2,2 0

g1i,3γ

1−1/2

i p1i,1,1 p2

i,3,2 p2i,3,3

=

γ1−1/2

i 0 0 0

g1i,1γ

1−1/2

i pi,1,1 0 0

g1i,2γ

1−1/2

i pi,1,1 pi,2,2 0

g1i,3γ

1−1/2

i pi,1,1 pi,3,2 pi,3,3

.(7.26)

The vector W1i can be computed from the elements from the first column of

this post-array. Note, that the elements of the last two columns and rows of

the post-array are the same as the one obtained previously for the second data

stream m = 2, since the first two rotations are the same and the third one leaves

the last two columns unchanged. Therefore, the matrix P1/2i obtained from the

final post-array is equal to P11/2

i , and P21/2

i is a sub-matrix of P1/2i consisting of

the last two rows and columns of P1/2i , which confirms the initial hypothesis by

induction.

From this example, assuming that the same initial condition Π1/20 is used for

all data streams m, m = 1, 2, the following facts can be established:

• The matrix P21/2

i consists of the last two columns and rows of the matrix

P1/2i = P11/2

i ;

• Only a single matrix P1/2i which is read from the post-array for m = 1 needs

to be propagated;

• The elements g2i γ

2−1/2(i) and γ2−1/2

(i) are read directly from the entries of

the post-array, as computed for stream m = 1, after two rotations;

159

• The elements g1i γ

1−1/2(i) and γ1−1/2

(i) are read directly from the entries of

the post-array, as computed for stream m = 1, after three rotations.

These facts can be extended for any value of M ,N ,Nf , and Q.

The following inverse QR algorithm for the MIMO DFE receiver with cancella-

tion is thus proposed. Start with Wm−1 = 0N(Nf+1)+M(Q+1)−1×1 and P

1/2−1 = Π

1/20 ,

where Π1/20 is a N(Nf +1)+M(Q+1)−1 diagonal matrix. For each time instant

i ≥ 0, create the pre-array 1 1√λz∗iP

1/2i−1

0 1√λP

1/2i−1

, (7.27)

where zi consists of the elements 2 to N(Nf + 1) + M(Q + 1) of yi. Perform

N(Nf + 1) + MQ − 1 elementary orthogonal rotations on the pre-array, where

each rotation i, i = 1, . . . , N(Nf + 1) +MQ− 1, rotates around the first element

of the first row and nulls out the N(Nf + 1) +M(Q + 1) + 1− i element of the

first row in the post-array. Then, for j = 0, . . . ,M − 1, perform an orthogonal

rotation around the first element of the first row that nulls out the M − j + 1

element of the first row in the post-array. After each rotation j, the first column

of the post-array yields {γM−ji

}−1/2

gM−ji

{γM−ji

}−1/2

. (7.28)

The vector estimate Wmi, where m = M − j, is updated as follows:

Wmi = Wm

i−1 +[gm

i

{γmi}−1/2][{

γmi}−1/2]−1[

dm(i−∆)− yiWmi−1

](7.29)

where the quantities{gm

i γm−1/2

i

}and

{γm−1/2

i

}are read directly from the entries

of the post-array after the rotation j = M −m.

160

After the last rotation (i.e., after N(Nf + 1) + M(Q + 1) − 1 rotations that

produced a zero pattern in the last N(Nf + 1) + M(Q + 1) − 1 elements of the

first row of the post-array), the post-array is given by γ1−1/2

i 0

g1i γ

1−1/2

i P1/2i

. (7.30)

The matrix P1/2i read from this post-array is propagated to the next iteration

i+ 1. This algorithm reduces by a factor of approximately M the computational

complexity with respect with the conventional RLS algorithm. The inverse QR

RLS algorithm for the MIMO DFE receiver with cancellation has a complexity

on the order of O((N(Nf + 1) +M(Q + 1)− 1)2).

7.2.3 Elementary Circular Rotations

In the inverse QR algorithm, a series of elementary orthogonal rotations is used to

obtain a lower triangular post-array from the pre-array. Each orthogonal rotation

Θ rotates the pre-array around the first element of the first row and annihilates

an element q of the first row. The orthogonal rotation Θ, known as Givens

rotations [KSH00], has the form:

Θ =

fpp 0 · · · 0 fpq 0 · · · 0

0 1 · · · 0 0 0 · · · 0...

.... . .

......

.... . .

...

0 0 · · · 1 0 0 · · · 0

fqp 0 · · · 0 fqq 0 · · · 0

0 0 · · · 0 0 1 · · · 0...

.... . .

......

.... . .

...

0 0 · · · 0 0 0 · · · 1

,

161

where fpq and fqq are in the qth column, and fqp and fqq are in the qth row.

Let focus on the elementary 2 × 2 rotation Θ that takes a row vector X =

[xp xq] and rotates it to lie along the basis vector [1 0]. The Givens rotation

Θ is given by:

Θ =

fpp fpq

fqp fqq

=

c∗ −ss∗ s

=

x∗p√|xp|2+|xq|2

−xq√|xp|2+|xq|2

x∗q√|xp|2+|xq|2

xp√|xp|2+|xq|2

.(7.31)

It can be seen that ΘΘ∗ = I and

XΘ = [√|xp|2 + |xq|2 0].

The Givens rotation Θ involves the computation of a square-root, which is

computationally intensive and should be avoided. A square-root free rotation is

now described. Suppose that a rotation transforms X = YD1/2 into X = Y D1/2

.

162

The transformation can be shown to be as follows:

X = [xp xp]

= [xp xq]Θ

= [xp xq]

x∗p√|xp|2+|xq|2

−xq√|xp|2+|xq|2

x∗q√|xp|2+|xq|2

xp√|xp|2+|xq|2

= [yp yq]

d1/2p 0

0 d1/2q

y∗pd

1/2p√

|yp|2dp+|yq|2dq−yqd1/2

q√|yp|2dp+|yq|2dq

y∗qd1/2q√

|yp|2dp+|yq|2dqypd

1/2p√

|yp|2dp+|yq|2dq

= [yp yq]

y∗pdp|yp|2dp+|yq|2dq −yq

y∗qdq|yp|2dp+|yq|2dq yp

√|yp|2dp + |yq|2dq 0

0d

1/2p d

1/2q√

|yp|2dp+|yq|2dq

= [yp yq] Θ

dp1/20

0 dq1/2

= [yp yq]

dp1/20

0 dq1/2

= Y D

1/2

(7.32)

The new rotation

Θ =

fpp fpq

fqp fqq

=

y∗pdp|yp|2dp+|yq|2dq −yq

y∗qdq|yp|2dp+|yq|2dq yp

(7.33)

that transforms Y into Y is square-root free and depends on the weighting matrix

D. After the transformation the new weighting matrix is given by:

D =

|yp|2dp + |yq|2dq 0

0 dpdq|yp|2dp+|yq|2dq

(7.34)

The transformation Θ can be applied to any matrix Z = VD−1/2 to obtain

Z = ZΘ = VD−1/2

. The weighting factors that are associated with the columns

163

of the resulting matrix are given by the corresponding elements of the weighting

matrix D. Therefore, the desired matrix V can be obtained by applying the

normalization V = ZD1/2

. However, note that in the case of the inverse QR

algorithm, the updated vector Wmi depends on the ratio of elements of the

first column. Since they all use the same weighting factor, there is no need to

normalize the transformed values. Therefore, the inverse QR algorithm can be

computed square-root free. The modifications to the original algorithm are the

following:

• Begin with a N(Nf + 1) +M(Q+ 1) identity weighting matrix D;

• After each square-root free rotation Θ update the weigthing matrix D;

• Propagate the weigthing matrix D from iteration i− 1 to i;

• At the start of each iteration i, update the first element of the propagated

matrix D to one, since the first column is re-initialized at each iteration.

This modified square-root free rotation tends to suffer from possible over-

flow/underflow problems. For example, it can be noted that the weighting factor

corresponding to the first column is always greater than one. This is due to the

fact that the pivoting element (i.e., the first element in the first row of the pre-

array) is set to one at the beginning of each iteration and furthermore after each

rotation the pivoting element is also one. Therefore, it can also be observed that

the weighting factors of the other columns are always decreasing after each rota-

tion, which leads to an underflow problem with the weighting matrix. Inversely,

the entries of the matrix P1/2i suffers from overflow problems.

In [AP94], self-scaling Givens rotations are introduced. In this dissertation, a

two-way branch self-scaling algorithm adapted to complex value data is described.

164

The motivation of the algorithm is to select an appropriate rotation to achieve

a tighter clustering of the weighting values close to one. The algorithm is as

follows:

If dp ≥ dq and θ ≤ π/4 (i.e., |yp|2dp ≥ |yq|2dq), then let:

Θ =

1 0

y∗qdqy∗pdp

1

1−y∗pyqdp

|yp|2dp+|yq|2dq

0 1

D =

c2dp 0

0 c−2dq

=

|yp|2d2p

|yp|2dp+|yq|2dq 0

0dq

{|yp|2dp+|yq|2dq

}|yp|2dp

(7.35)

otherwise, if dp ≥ dq and θ > π/4 (i.e., |yp|2dp < |yq|2dq), then let:

Θ =

0 −1

1 ypyq

1 0

−y∗pyqdp|yp|2dp+|yq|2dq 1

D =

s−2dq 0

0 s2dp

=

|yp|2dp+|yq|2dq|yq|2 0

0 |yq|2dpdq|yp|2dp+|yq|2dq

(7.36)

otherwise, if dp < dq and θ ≤ π/4 (i.e., |yp|2dp ≥ |yq|2dq), then let:

Θ =

1 −yqyp

0 1

1 0

ypy∗qdq|yp|2dp+|yq|2dq 0

D =

c−2dp 0

0 c2dq

=

|yp|2dp+|yq|2dq|yp|2 0

0 |yp|2dpdq|yp|2dp+|yq|2dq

(7.37)

otherwise, if dp < dq and θ > π/4 (i.e., |yp|2dp < |yq|2dq), then let:

Θ =

y∗pdpy∗qyq1

1 0

1ypy∗qdq

|yp|2dp+|yq|2dq

0 −1

D =

s2dq 0

0 s−2dp

=

|yq|2d2q

|yp|2dp+|yq|2dq 0

0dp

{|yp|2dp+|yq|2dq

}|yq|2dq

(7.38)

165

The two-way branch self-scaling algorithm has the following properties:

• Square-root free rotations;

• Transformation by Θ only requires 2 complex multiplications;

• Transformation by Θ does not require the temporary copy of one of the

values being transformed;

• The larger weight is always decreased through the rotation, and likewise

the smaller weight is always increased through the rotation. This allows

small deviations of the weighting matrix around unity.


In this section, the convergence characteristics of the adaptive algorithms pre-

sented in this chapter are analyzed through simulations. The adaptive algorithms

are compared to the optimal MMSE solution and therefore provides a mean to

verify both the adaptive and optimal solutions. The stochastic gradient algorithm

and recursive least square algorithms for the MIMO DFE receiver with cancel-

lation are analyzed for various channel conditions. A different approach using

channel estimation to compute the optimal solution coefficient matrix is studied.

The adaptive algorithms are also extended to the MIMO OFDM and Generalized

MIMO RAKE receiver. For all the simulations the MIMO communication system

consists of 4 transmit antennas and 4 receive antennas (M = N = 4), and the

SNR was set to 20 dB.

166

0 250 500 750 1000 1250 1500 1750 20000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Time Index

MS

E

Optimal Solution

LMS Algorithm

RLS Algorithm

Figure 7.1: MSE Learning Curves for the MIMO DFE Receiver with Cancellation.

7.3.1 Adaptive MIMO Equalization

Figure 7.1 shows the MSE learning curve of the LMS and inverse QR RLS al-

gorithms for the MIMO DFE receiver with cancellation. The parameters of the

MIMO DFE receiver are Nf = 3, Q = 2 and ∆ = 3. The learning curves are

obtained by simulating 100 independent MIMO channel realizations. The LMS

update factor µ is set to 0.03. The inverse QR parameters are Π0 = 100I and

λ = 1. Each MIMO frequency selective fading channel is independently generated

using the exponential power delay profile channel model presented in Section 2.4.1

for τ rms = 0.5Ts. For each channel realization, the optimal MMSE is first com-

puted and averaged for the M data streams. The system is then simulated for

both adaptive algorithms, starting with the coefficient matrix W e0 = 0, to obtain

the MSE learning curve for the channel realization. Each time index i > 0 rep-

resents a new received symbol and an update of the adaptive algorithm resulting

167

0 250 500 750 1000 1250 1500 1750 20000

2

4

6

8

10

12

14

16

Time Index

SN

R (

dB)

Optimal Solution

LMS Algorithm

RLS Algorithm

Figure 7.2: SNR Learning Curves for the MIMO DFE Receiver with Cancellation.

in a new estimated coefficient matrix Wei . The mean squared error vector of the

MIMO DFE receiver with cancellation is then computed using equation 4.21 for

the weight matrix Wei . The averaged mean squared error for the M data streams

is the learning curve MSE for time index i for this channel realization. The final

learning curves shown in Figure 7.1 are obtained by averaging the MSE learning

curves for the 100 independent channel realizations. This procedure was used

to obtain all the learning curves presented in this section. The results show the

ability of the LMS and RLS algorithms to converge toward the optimal solution.

It also demonstrates that the solution found in Section 4.2 is optimal since the

adaptive algorithms don’t outperform the predicted MMSE.

Figure 7.2 displays the learning curves in term of SNR instead of mean square

error, which is more significant for communication systems. For this simulation,

the training period of the inverse QR RLS algorithm is limited to the first 500

168

received symbols while the LMS is continuously training. The RLS steady-state

error is caused by the fact that the RLS algorithm update is stopped, however if

the training was not terminated, RLS will asymptotically converge to the optimal

solution. The MIMO DFE receiver with cancellation consists of 102 complex

taps. For this system configuration and channel environment, the RLS algorithm

converges within 3 dB of the MMSE solution after 50 iterations and within 0.2 dB

after 500 iterations. The LMS algorithm has a slower convergence and exhibits a

steady-state misadjustment [Say03]. µ controls these properties. For large values

of µ, the convergence is faster but the steady state error is larger. For µ = 0.03,

the algorithm converges within 3 dB of the MMSE solution after 1100 iterations

and the steady-state error is below 2.2 dB. LMS has a much smaller complexity

but can only be used for a MIMO DFE receiver if the coherence time of the

channel is large enough and if the system designer can tolerate the steady-state

error.

Figure 7.3 compares the learning curves of the RLS and inverse QR algorithms

for one channel realization. It can be observed that for the first 2000 iterations,

both algorithms are virtually identical. However, after 2000 iterations, RLS ex-

hibits some instability and diverges from the optimal solution. It is well known

that the conventional RLS algorithm eventually diverges under fixed precision

computation conditions. However, these simulations were realized with double

floating point precisions. Therefore, the complexity of the MIMO DFE receiver

structure makes the RLS algorithm even more unstable. On the other hand, in-

stability and divergence for the inverse QR algorithm was never observed. For all

other simulations, the inverse QR implementation of the RLS algorithm is used.

The previous results illustrated the performance of the adaptive algorithms

for additive white noise. Figure 7.4 shows their performance in the presence

169

0 500 1000 1500 2000 25000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time Index

MS

E

RLS AlgorithmInverse QR Algorithm

Figure 7.3: Comparison of RLS and Inverse QR Algorithms.

of colored noise. The colored noise is generated using the model presented in

Section 4.3.3. The noise parameters areNc,int = Nc = 2, γint,rms = γrms = 1.56 and

γint = 1. The system was simulated for the case of multi-users MIMO interference

(Mint = M = 4) and for traditional SISO interference sources (Mint = 1). The

RLS algorithm converges quickly to the optimal solution for the different colored

noise conditions. In fact, the convergence properties of the RLS algorithm are not

affected by the nature of the noise. The RLS adaptive algorithm for the MIMO

DFE receiver with cancellation, in the presence of either AWGN noise, colored

noise with Mint = 4 or colored noise with Mint = 1, converges within 3 dB of their

respective MMSE optimal solution after 50 iterations, and within 0.2 dB after

500 iterations. For a given number of iterations, the resulting SNR for the LMS

adaptive algorithm improves in the presence of colored noise. However, the LMS

algorithm convergence significantly deteriorates in the presence of colored noise.

170

0 250 500 750 1000 1250 1500 1750 20000

2

4

6

8

10

12

14

16

18

20

Time Index

SN

R (

dB)

AWGN Optimal

AWGN LMSAWGN RLS

Mint

=4−Optimal

Mint

=4−LMSMint

=4−RLS

Mint

=1−Optimal

Mint

=1−LMSMint

=1−RLS

Figure 7.4: Learning Curves in the Presence of Colored Noise.

For example, LMS in the presence of white noise converges within 3 dB of the

optimal MMSE solution after 1100 iterations. For colored noise with Mint = 4,

the number of iterations required is now 1750 iterations, and forMint = 1 the LMS

algorithm does not converge within 3 dB after 2000 iterations (mis-adjustment of

4.6 dB after 2000 iterations). The steady-state error is also higher in the presence

of colored noise. Therefore, increasing the step size µ for colored noise would not

help the performance of the LMS algorithm.

7.3.2 Adaptive MIMO Channel Estimation

The adaptive algorithms derived for the MIMO DFE receiver with cancellation

directly estimate the coefficient matrix W. A different approach illustrated in

Figure 7.5 is to first estimate the channel matrix Ct, and then compute the

optimal coefficient matrix We for the estimated channel matrix Ce and noise

171

d

d

Ce

u

v1(i)

vN(i)

u1(i)

uN(i)

MIMO DFE

Receiver with

Cancellation

MIMO Channel

and Noise

Estimation

MIMO Weight

Computation

Rv

e

We

Figure 7.5: Architecture of the MIMO Channel Estimation Receiver.

auto-correlation matrix Rev using the results presented in Section 4.2. The MSE

at each iteration is then obtained by using We to compute the MSE using equa-

tion 4.21, where Ry and Rd(i−∆)y are generated using the actual channel matrix

Ct and noise auto-correlation matrix Rv.

The estimated channel matrix can be computed using either an LMS al-

gorithm or an inverse QR RLS algorithm. For both algorithms, let Ce−1 =

0M(Nc+1)×N , di = [d(i) · · ·d(i−Nc)] and ui = u(i). The LMS update equation

to estimate Cei is then:

Cei = Ce

i−1 + µd∗i (ui − diCei−1) . (7.39)

The inverse QR RLS algorithm is given as follows. For each iteration, the follow-

172

ing array transformation is performed:1 1√λd∗iP

1/2i−1

0 1√λP

1/2i−1

Θi =

γ−1/2i 0

giγ−1/2i P

1/2i

(7.40)

where Θi is any orthogonal rotation that produces the zero pattern in the first

row of the post-array, P1/2−1 = Π

1/20 , Π0 is an M(Nc + 1)×M(Nc + 1) regularizing

positive definite matrix and λ ≤ 1 is a forgetting factor. The channel estimate

Cei is then given by

Cei = Ce

i−1 +[giγ−1/2i

] [γ−1/2i

]−1

(ui − diCei−1) , (7.41)

where the quantities{

giγ−1/2i

}and

{γ−1/2i

}are read directly from the entries of

the post-array. For both algorithms, the estimated noise auto-correlation matrix

Revi is obtained by averaging the matrices e∗iei, where ei = [e(i) · · ·e(i−Nf)]

and e(i) = ui − diCei−1. For the simulated system, a sliding averaging window

over the last 50 e∗iei matrices was used.

Figure 7.6 compares the performance of the channel estimation algorithm with

the direct adaptive algorithms. The LMS update factor µ for the channel esti-

mation algorithm is set to 0.1 and the inverse QR parameters are Π0 = 100I and

λ = 1. The RLS channel estimation algorithm converges approximately twice as

fast as the RLS algorithm, converging within 3 dB of the optimal solution after

20 iterations, and within 0.2 dB after 225 updates. However, the price to pay for

the faster convergence is a costly matrix inversion to compute the weight matrix

estimate after the convergence of the channel estimate. The LMS channel esti-

mation algorithm also has good convergence properties and reaches within 3 dB

of the MMSE after 100 iterations. Note the LMS steady-state channel estima-

tion error which becomes apparent after 250 iterations. Therefore, increasing µ

to improve the convergence rate would also results in a larger steady state error.

173

0 50 100 150 200 250 300 350 400 450 5000

2

4

6

8

10

12

14

16

Time Index

SN

R (

dB)

Optimal

LMS

RLS

Channel Estimation−LMS

Channel Estimation−RLS

Figure 7.6: Performance of Channel Estimation Algorithms.

RLS channel estimation algorithm provides a good solution for fast time varying

channels if the computation power is available.

7.3.3 Adaptive MIMO OFDM

The LMS adaptive algorithm and recursive inverse QR RLS algorithm can be

easily adapted to the MIMO OFDM receiver with cancellation by executing a

separate instance of the algorithms independently for each interpolation sub-

carrier. The appropriate definition on a sub-carrier basis of the data vector,

input vector, and the coefficient matrix given in Section 5.1 should be used in

the adaptive algorithms described in Section 7.1 and 7.2 to obtain the estimated

coefficient matrix for a given sub-carrier. The following correspondence relations

174

for each sub-carrier p, 1 ≤ p ≤ P , are obtained:

yi ←→ y(p)i (7.42)

Wi ←→W(p)i (7.43)

d(i−∆)←→ d(p)i (7.44)

zi ←→ z(p)i (7.45)

gi ←→ gpi (7.46)

γ−1/2i ←→ γp

−1/2

i (7.47)

P1/2i ←→ Pp1/2

i (7.48)

W(p)−1 = 0M+N−1×M , Pp1/2

−1 = Π1/20 , Π0 is an M+N−1×M+N−1 regularizing

positive definite matrix, and z(p)i consists of the elements 2 to M +N of y(p)i.

i for the MIMO OFDM receiver adaptive algorithms corresponds to the OFDM

symbol time index.

Figure 7.7 shows the learning curves of the LMS and RLS adaptive algorithms

for the MIMO OFDM receiver with cancellation. The LMS update factor µ is

set to 0.15, and the inverse QR parameters are Π0 = 100I and λ = 1. The

number of sub-carriers is P = 64 and no interpolation is used (i.e., psep = 1). The

training period of the inverse QR RLS algorithm is limited to the first 200 received

OFDM symbols while the LMS is continuously training. The convergence for the

MIMO OFDM system is much faster than for the MIMO DFE since the number

of taps to train is now reduced to 22 per interpolation sub-carrier. The RLS

algorithm for the MIMO OFDM receiver with cancellation converges within 3 dB

of the MMSE solution after 9 iterations and within 0.2 dB after 100 iterations.

The LMS algorithm has a steady-state mis-adjustment smaller than 2.1 dB and

converges within 3 dB of the MMSE after 175 iterations. However, it should be

remembered that there is only one iteration of the adaptive algorithms per OFDM

175

0 50 100 150 200 250 300 350 400 450 5000

2

4

6

8

10

12

14

16

Time Index

SN

R (

dB)

Optimal Solution

LMS

RLS

Figure 7.7: Learning Curves for the MIMO OFDM Receiver with Cancellation.

symbol, including the cyclic prefix. Ignoring the cyclic prefix contribution, the

RLS algorithm converges within 3 dB after 576 channel samples. Also, for each

iteration, an update must be performed for each interpolation sub-carrier. For

τ rms = 0.5Ts, as shown in Section 5.3.2, Psep = 8 is appropriate. Therefore, there

is 9 interpolation sub-carriers and 81 RLS updates must be performed to converge

within 3 dB of the MMSE. On the other hand, each update is less intensive since

the number of taps and the size of Pp1/2

i are significantly smaller.

7.3.4 Adaptive MIMO CDMA

The LMS adaptive algorithm and inverse QR RLS algorithm can also be easily

adapted to the generalized MIMO RAKE receiver with cancellation by using the

appropriate definition of the data vector, input vector, and the coefficient matrix

176

given in Section 6.1 to the adaptive algorithms described in Section 7.1 and 7.2.

The following correspondence relations are obtained:

yi ←→ y(t) (7.49)

Wi ←→Wt (7.50)

d(i−∆)←→ d0(t) (7.51)

zi ←→ zt (7.52)

gi ←→ gt (7.53)

γ−1/2i ←→ γ

−1/2t (7.54)

P1/2i ←→ P

1/2t (7.55)

W−1 = 0M+NK−1×M , P1/2−1 = Π

1/20 , Π0 is anM+NK−1×M+NK−1 regularizing

positive definite matrix, and zt consists of the elements 2 to M +NK of y(t). t

corresponds to the MIMO CDMA receiver symbol time index.

Figure 7.8 shows the learning curves for the generalized MIMO RAKE re-

ceiver with cancellation. The MIMO CDMA communication system is set up as

described in Section 6.4. The system consists of 5 users (U = 4) with a spreading

factor F = 32. The spreading sequence length is Ls = 38400. The power ded-

icated to the demodulated user u = 0 is γ0 = 1/32 = −15 dB and the channel

SNR is set to 20 dB. The system was simulated for a full sequence length (i.e.,

1200 symbols). The channel consists of three equal power multipaths at delays

of 0Ts, Ts, 2Ts. Each multipath is an independent Rayleigh fading random vari-

able with variance 1/3. The adaptive algorithms were simulated for a traditional

RAKE receiver and the generalized version. The RAKE receiver with cancel-

lation consists of three fingers (K = 3) positioned at τ1 = 0, τ2 = 1, τ3 = 2.

The generalized MIMO RAKE receiver with cancellation has an additional finger

(K = 4) at τ4 = 3. The LMS update factor µ is set to 0.02 and the inverse

177

0 200 400 600 800 1000 1200−6

−4

−2

0

2

4

6

8

Time Index

SN

R (

dB)

Optimal Solution

LMS

RLS

GeneralizedOptimal Solution

GeneralizedLMS

GeneralizedRLS

Figure 7.8: Learning Curves for the MIMO Generalized RAKE Receiver with

Cancellation.

QR parameters are Π0 = 100I and λ = 1. The inverse QR RLS algorithm and

the LMS algorithm are continuously training. In order to have a vector y(t)

with similar variance for each element, the output of the correlators are scaled

by 1/(√γ0F ).

Although there is a time variation of the correlation properties of the received

signal, the adaptive algorithms are able to converge to the optimal MMSE so-

lutions. Note that the RLS algorithm has some difficulty converging initially

due to the high level of noise caused by multipath interference. Table 7.1 and

7.2 summarize the convergence properties for the RLS and LMS algorithm, re-

spectively. The convergence for the RLS generalized RAKE receiver is slightly

slower than for the conventional RAKE receiver since there is an additional tap

to adapt. However, the convergence rate is relatively good, given that there is no

178

Table 7.1: Convergence Properties for CDMA RLS

SystemConvergencewithin 3 dB

Convergencewithin 0.2 dB

RAKE Receiver withLs = 38400

35 iterations 375 iterations

Generalized RAKE Receiverwith Ls = 38400


RAKE Receiver with Ls = 32 28 iterations 300 iterationsGeneralized RAKE Receiverwith Ls = 32


Table 7.2: Convergence Properties for CDMA LMS

SystemConvergencewithin 3 dB

Mis-Adjustment after 1200iterations

RAKE Receiver withLs = 38400

125 iterations 1 dB

Generalized RAKE Receiverwith Ls = 38400

220 iterations 1.3 dB

RAKE Receiver with Ls = 32 325 iterations 1.3 dBGeneralized RAKE Receiverwith Ls = 32

475 iterations 1.7 dB

multipath at this position. The LMS algorithm, albeit slower than RLS, performs

relatively well for the CDMA receiver and its convergence rate is closer to the

RLS algorithm than for the other receivers. Also, it is interesting to note that

LMS is able to take advantage of the additional RAKE finger in the generalized

RAKE receiver, unlike the behavior observed in the presence of colored noise.

Figure 7.9 presents the performance of these algorithms when the sequence is

the length of a symbol (i.e. Ls = F = 32). Each received symbol as the same

correlation properties which leads to a higher MMSE. As shown in Table 7.1 RLS

is able to take advantage of the time invariant properties to converge faster than

for the case Ls = 38400. On the other hand, the convergence rate of LMS is

slower for Ls = 32. However, LMS is able to take advantage of the better cross-

179

0 200 400 600 800 1000 1200−2

0

2

4

6

8

10

12

Time Index

SN

R (

dB)

Optimal Solution

LMS

RLS

Generalized Optimal Solution

GeneralizedLMS

GeneralizedRLS

Figure 7.9: Learning Curves with Spreading Sequence Length Ls = 32.

correlation properties to improve the SNR. As was the case for Ls = 38400, the

convergence rate is slower for the generalized RAKE receiver for both RLS and

LMS.

The learning curves obtained for the MIMO DFE receiver with cancellation

and the generalized MIMO RAKE receiver with cancellation can not be compared

directly since the frequency selective fading channel are different. Generally, a

channel with more severe ISI requires a longer training period. Therefore, it can

be concluded that MIMO CDMA converges in less iterations than MIMO DFE.

However, each iteration for the MIMO CDMA adaptive algorithms requires F

channel samples (for the MIMO DFE there is one iteration per received sample).

On the other hand, for channels with a longer delay spread such as in outdoor

environments, the MIMO DFE length must be increased in order to maintain the

performance and would result in a slower convergence rate for the MIMO DFE.

180

CHAPTER 8

Conclusion

The objective of the research presented in this dissertation was to study commu-

nication systems featuring multiple transmit and receive antennas over frequency

selective fading channels. The theoretical capacity of wideband MIMO systems

under colored noise was first derived and used to demonstrate the multi-fold in-

crease in capacity offered by these systems. The frequency diversity of the chan-

nel provides a constant capacity improvement for every MIMO configurations

and becomes relatively non-significant when a large number of transmit/receive

antennas is used. The theoretical capacity was also used to illustrate the trade-

offs between linear capacity increase as a function of the minimum of the number

of transmit/receive antennas and the logarithmic transmit/receive diversity ca-

pacity improvement provided by the additional transmit/receive antennas. The

theoretical capacity for colored noise was used to demonstrate the capacity im-

provement provided by MIMO in a multi-user environment and its ability to

reject noise with spatial signature.

The impact of channel correlation and line-of-sight propagation was also an-

alyzed. It was shown that a correlation factor of 0.5 at both the transmitter and

receiver (i.e., antenna separation of 3.1 cm at 2.4 GHz) results in less that 10%

degradation in capacity. Similarly, a line-of-sight multipath with a power strength

4 times more than the Rayleigh random component (Ricean factor K = 6 dB)

decreases the MIMO capacity by 10%. The results also illustrated the fact that

181

frequency selective fading channel are less affected by line-of-sight propagation

than flat fading channels and both type of channel are equally impacted by chan-

nel correlation. Finally, channel knowledge at the transmitter can significantly

increase the MIMO channel capacity under poor channel conditions (low SNR,

greater number of transmit antennas than receive antennas, large channel corre-

lation, and strong line-of-sight propagation).

Smart antenna array systems are a special case of MIMO systems where the

number of transmit antenna is fixed to 1. This type of communication system

offers capacity improvement through receive diversity. A highly flexible proto-

type unit for high speed wireless communications using adaptive equalization and

smart antenna array was used to study the performance of a multiple antennas

system under realistic frequency selective fading channel conditions. The testbed

was used to conduct extensive field measurements that were compared to simu-

lation results. The field measurements demonstrated the dramatic signal quality

improvement provided by the smart antenna array in various environments (up

to 10 dB at 5% outage). They also showed that processing in both the space

domain and time domain (i.e., equalization) is required in indoor environments

to provide reliable high data rates links (30 Mbps in 5 MHz bandwidth). The

field measurements finally demonstrated the accuracy of the space-time channel

models.

Architectures for wideband MIMO receivers have been studied extensively in

this dissertation. Equalization, multi-carrier and spread spectrum form of MIMO

receivers have been considered. A novel problem formulation for the MIMO equal-

ization communication system allowed the derivation of an MMSE optimal finite

length MIMO DFE receiver with cancellation. This receiver has been studied

for different system configurations and transmission environments (AWGN, col-

182

ored noise, rms delay spread, channel correlation, and line-of-sight propagation).

The simulation results showed that the behavior of the MIMO DFE receiver

with cancellation was in accordance with the theoretical MIMO channel capac-

ity. A sub-optimal detection ordering algorithm was proposed and demonstrated

to perform closely to the optimal myopic ordering algorithm. Simulation results

also revealed that the MIMO DFE BER could be accurately predicted from the

MMSE. Capacity simulation of the uncoded MIMO DFE receiver with cancella-

tion finally showed that this system is able to provide a linear increase in capacity

as a function of the number of transmit/receive antennas, as predicted by theory.

The MSE of flat fading receivers such as zero-forcing, VBLAST and MMSE

VBLAST was derived and used to compare their performance with the MIMO

DFE receiver with cancellation. Simulations showed that the MMSE VBLAST

and the MIMO DFE performance are similar and they outperform the other two

types of receiver. However, MIMO DFE is superior to MMSE VBLAST since it

can be extended to frequency selective fading channel, has a lower computation

complexity, and is suitable to adaptive algorithms such as LMS and RLS.

The MIMO DFE problem formulation and optimal solution is general and can

be extended to other MIMO receivers with cancellation. A multi-carrier MIMO

OFDM receiver with cancellation was thus proposed based on the MIMO DFE

architecture. The performance of the MIMO OFDM receiver is similar to the

MIMO DFE receiver. Effects of interpolation were also studied. It was shown

that, as for traditional SISO OFDM system, MIMO OFDM receiver can used

interpolation between pilot sub-carriers. The frequency separation between the

pilot sub-carriers should be below 6.25% of the inverse of the rms delay spread of

the channel. Coefficient matrix interpolation is compared to channel interpola-

tion. The later is slightly better but is much more expensive in terms of compu-

183

tation complexity. Different detection ordering algorithms are also analyzed for

OFDM with interpolation and it is shown that the receiver performance degra-

dation due to sub-optimal interpolation ordering is non-significant compared to

the coefficient matrix interpolation error.

A generalized MIMO RAKE receiver with cancellation is proposed for out-

door environments and legacy CDMA cellular networks. The optimal MMSE

solution for this receiver is derived and used to perform the analysis. The solu-

tion is valid for various scrambling sequence length, spreading factor, number of

users and power balance. Simulations showed that the performance of the gener-

alized MIMO RAKE receiver with cancellation is worst than for a MIMO DFE,

but equivalent to the MIMO linear equalizer with cancellation for indoor chan-

nels. For outdoor channels with large excess delay, the MIMO RAKE receiver

outperforms the MIMO equalizer for an equivalent receiver complexity. Using

an additional MIMO RAKE finger at a time delay where no multipath exist can

provide an SNR improvement in excess of 1 dB, depending on the channel profile.

Low complexity combiner MIMO RAKE receivers are proposed and simulations

showed that the performance degradation of these receivers is approximately 2

dB. However, the generalized MIMO RAKE receivers for the combiner architec-

ture do not significantly improve the performance.

Novel LMS and RLS adaptive algorithms that do not require knowledge of

the channel have been proposed for the MIMO DFE receiver with cancellation. A

new MIMO inverse QR RLS algorithm with better stability properties and lower

complexity than the RLS algorithm is also introduced. Simulations demonstrated

the convergence properties of these algorithms and validated the optimal MMSE

solutions. RLS and LMS MIMO channel estimation is also compared to these

adaptive algorithms. The convergence rate of the MIMO channel estimation

184

algorithms is slightly faster, however the computation cost is significantly larger.

Extension of the adaptive algorithms to MIMO OFDM and the generalized RAKE

receiver are also introduced. The convergence for these systems is faster at the

symbol rate than for the MIMO DFE receiver. However, for a similar channel

transmission rate, the absolute convergence time is faster for MIMO DFE.

The appropriate MIMO receiver for frequency selective fading channels de-

pends on the computation power available and the operating environment. The

MIMO DFE receiver with cancellation using RLS channel estimation has the

best performance and the fastest convergence rate. This receiver is appropriate

for fast fading environments and a large amount of computation power is avail-

able at the receiver. The RLS adaptive algorithm offers a similar performance

with a slightly lower convergence speed and computation complexity. MIMO

DFE with LMS is a good low complexity alternative for low Doppler rate chan-

nels. The MIMO OFDM receiver with cancellation provides a lower complexity

receiver architecture. However, the performance degrades by approximately 2 dB

due to interpolation errors and sub-optimal ordering. MIMO OFDM also has a

lower convergence rate than MIMO DFE. Furthermore, OFDM requires highly

linear amplifier, which makes it an expensive solution for outdoor communication

systems with high transmit power. MIMO OFDM receivers are therefore appro-

priate for indoor channels with low Doppler rate. The generalized MIMO RAKE

receiver with cancellation is a low complexity architecture for outdoor multi-users

cellular communication systems. Its performance is significantly lower than the

MIMO DFE receiver for indoor channels (SNR degradation of 8 dB). On the

other hand, for a similar complexity, the MIMO CDMA system performance is

much better in large excess delay channels. The performance of the generalized

MIMO RAKE receiver could also be improved by using an hybrid of a DFE and a

RAKE receiver or through MIMO multi-user detection. The generalized MIMO

185

RAKE receiver also has a relatively slow convergence rate which might limit the

vehicular speed of MIMO terminals.

The research presented in this dissertation covered the theoretical aspect,

optimal implementation, and practical adaptive algorithms for wideband MIMO.

It thus provides a solid basis for the designer of a MIMO communication system

for a frequency selective fading environment. Starting from these results, new

avenues of research can emerge. For example, practical implementation issues

of MIMO systems such as I/Q imbalance, RF gain imbalance, fixed point and

dynamic range, initial acquisition, timing and frequency synchronization and

tracking, just to name a few, provide interesting topics of research. Tradeoffs

between the coding strategy and the receiver architecture as a function of the

channel parameters is also a fertile field of research. MIMO provides a new spatial

paradigm that can be exploited in new Multiple Access Control (MAC), Radio

Link Control (RLC), and networking protocols. Finally, the dynamic behavior

of these systems for different mobile speeds is also critical and requires further

research.

186

References

[3rd02] 3rd Generation Partnership Project. TS 25.213: Spreading and Mod-ulation (FDD) - Release 99 - Version 3.8.0, June 2002.

[AP94] A.A. Anda and H. Park. “Fast Plane Rotations with Dynamic Scal-ing.” SIAM Journal on Matrix Analysis and Applications, 15(1):162–174, January 1994.

[AS99] N. Al-Dhahir and A.H. Sayed. “A Computationally-Efficient FIRMMSE-DFE for Multi-User Communications.” In Proc. of the Thirty-Third Asilomar Conference on Signals, Systems, and Computers, pp.207–209, Pacific Grove, CA, Oct. 1999.

[AS00] N. Al-Dhahir and A.H. Sayed. “The Finite-Length Multi-Input Multi-Output MMSE-DFE.” IEEE Transactions on Signal Processing,48(10):2921–2936, October 2000.

[ATN98] D. Agrawal, V. Tarokh, A. Naguib, and N. Sashadri. “Space-TimeCoded OFDM for High Data-Rate Wireless Communication over Wide-band Channels.” In Proc. VTC’98, pp. 2232–2236, Ottawa, Canada,May 1998.

[Bin90] J.A.C. Bingham. “Multicarrier Modulation for Data Transmission:An Idea whose Time Has Come.” IEEE Communications Magazine,28(5):5–14, May 1990.

[CC99] W.J. Choi and J.M. Cioffi. “Multiple Input / Multiple Output (MIMO)Equalization for Space-Time Block Coding.” In Proc. of IEEE PacificRim Conference on Communications, Computers and Signal Process-ing, pp. 341–344, Victoria, Canada, Aug. 1999.

[Cha66] R.W. Chang. “Synthesis of Band-Limited Orthogonal Signals forMultichannel Data Transmission.” Bell System Technical Journal,45(10):1775–1796, December 1966.

[Cla68] R. Clarke. “A Statistical Theory of Mobile Radio Reception.” BellSystem Technical Journal, 47:957–1000, 1968.

[Cox94] F.T. Cox. “Generation of Channel Impusle Responses from theHIPERLAN Channel Model.” 1994 HIPERLAN Contribution, Feb.1994.

187

[CT91] Thomas M. Cover and Joy A. Thomas. Elements of Information The-ory. Wiley, New-York, 1991.

[DJC97] B. Daneshrad, L.J. Cimini Jr., M. Carloni, and N. Soleenberger.“Performance and Implementation of Clustered OFDM for WirelessCommunications.” ACM Journal on Mobile Networks Applications(MONET) (Special Issue on PCS), 2(4):305–314, December 1997.

[FD02] J.F. Frigon and B. Daneshrad. “Field Measurements of an IndoorHigh-Speed QAM Wireless System Using Decision Feedback Equal-ization and Smart Antenna Array.” IEEE Transactions on WirelessCommunications, 1(1):134–144, January 2002.

[FDP00] J.F. Frigon, B. Daneshrad, J. Putnam, E. Berg, R. Kim, T. Sun, andH. Samueli. “Field Trial Results for High-Speed Wireless Indoor DataCommunications.” IEEE Journal on Selected Areas in Communica-tions - Wireless Series, 18(3):297–309, March 2000.

[FG98] G.J. Foschini and M.J. Gans. “On Limits of Wireless Communicationsin a Fading Environment when Using Multiple Antennas.” WirelessPersonal Communications, 6(3):311–335, March 1998.

[FGV99] G.J. Foschini, G.D. Golden, R.A. Valenzuela, and P.W. Wolniansky.“Simplified Processing for High Spectral Efficiency Wireless Communi-cations Employing Multi-Element Arrays.” IEEE Journal on SelectedAreas in Communications, 17(11):1841–1852, Nov. 1999.

[Fos96] G.J. Foschini. “Layered Space-Time Architecture for Wireless Com-munications in a Fading Environment using Multi-Element Antennas.”Bell Labs Technical Journal, 1(2):41–59, Autumn 1996.

[GFV99] G.D. Golden, G.J. Foschini, R.A. Valenzuela, and P.W. Wolniansky.“Detection and Initial Laboratory Results Using V-BLAST Space-Time Communication Architecture.” Electronics Letters, 35(1):14–16,Jan. 1999.

[Hay96] Simon Haykin. Adaptive Filter Theory. Prentice Hall, Upper SaddleRiver, New Jersey, third edition, 1996.

[HT02] Harri Holma and Antti Toskala. WCDMA for UMTS. John Wiley andSons, 2nd edition, 2002.

[HVF99] H. Huang, H. Viswanathan, and G.J. Foschini. “Achieving HighData Rates in CDMA Systems Using BLAST Techniques.” In Proc.GLOBECOM’99, pp. 2316–2320, Rio de Janeiro,Brazil, Dec. 1999.

188

[Jr85] L.J. Cimini Jr. “Analysis and Simulation of a Digital Mobile ChannelUsing Orthogonal Frequency Division Multiplexing.” IEEE Transac-tions on Communications, 33(7):665–675, July 1985.

[JR97] J.M. Johnson and Y. Rahmat-Samii. “The Tab Monopole.” IEEETransactions on Antennas and Propagations, 45(1):187–188, Jan.1997.

[Kal89] I. Kalet. “The Multitone Channel.” IEEE Transcations on Commu-nications, 37(2):119–124, February 1989.

[KSH00] Thomas Kailath, Ali H. Sayed, and Babak Hassibi. Linear Estimation.Prentice Hall, New Jersey, 2000.

[MHC99] A. Maleki-Tehrani, B. Hassibi, and J.M. Cioffi. “Adaptive Equaliza-tion of Multiple-Input Multiple-Output (MIMO) Frequency SelectiveChannels.” In Proc. of the Thirty-Third Asilomar Conference on Sig-nals, Systems, and Computers, pp. 547–551, Pacific Grove, CA, Oct.1999.

[Nag97] A.F. Naguib. “Space-Time Receivers for CDMA Multipath Signals.”In Proc. of ICC’97, pp. 304–308, Montreal, Canada, June 1997.

[NP00] Richard Van Nee and Ramjee Prasad. OFDM for Wireless MultimediaCommunications. Artech House, 2000.

[NTS98] A. Naguib, V. Tarokh, N. Seshadri, and A.R. Calderbank. “A Space-Time Coding Modem for High-Data-Rate Wireless Communications.”IEEE Journal on Selected Areas in Communications, 16(8):1459–1477,October 1998.

[PL98] A.J. Paulraj and E. Lindskog. “Taxonomy of Space-Time Processingfor Wireless Networks.” IEE Proceedings - Radar, Sonar and Naviga-tion, 145(1):25–31, February 1998.

[PP97] A.J. Paulraj and C.B. Papadias. “Space-Time Processing for WirelessCommunications.” IEEE Signal Processing Magazine, 14(6):49–83,November 1997.

[PR80] A. Peled and A. Ruiz. “Frequency Domain Data Transmission UsingReduced Computational Complexity Algorithms.” In Proceeddings ofIEEE International Conference on Acoustics, Speech, Signal Processi-ing (ICASSP), pp. 964–967, Denver, CO, April 1980.

189

[Pro95] John G. Proakis. Digital Communications. McGraw-Hill, 3rd edition,1995.

[PS99] J. Putnam and H. Samueli. “A 4-Channel Diversity QAM Receiver forBroadband Wireless Communications.” In ISSCC Dig. Tech. Papers,Feb. 1999.

[Rap96] Theodore S. Rappaport. Wireless Communications - Principles andPractice. Prentice-Hall, 1996.

[RC98] G. Raleigh and J.M. Cioffi. “Spatio-Temporal Coding for Wire-less Communications.” IEEE Transactions on Communications,46(3):357–366, March 1998.

[RST91] T.S. Rappaport, S.Y. Seidel, and K. Takamizawa. “Statistical ChannelImpulse Response Models for Factory and Open Plan Building RadioCommunication System Design.” IEEE Transactions on Communica-tions, 39(5):794–807, May 1991.

[Say03] Ali H. Sayed. Fundamentals of Adaptive Filtering. John Wiley andSons, 2003.

[SK98] Ali H. Sayed and Thomas Kailath. DSP Handbook, chapter 21. CRCPress, 1998.

[St96] Gordon L. Stuber. Principles of Mobile Communications. Kluwer Aca-demic Publishers, 1996.

[SV87] A.A. Saleh and R.A. Valenzuela. “A Statistical Model for Indoor Mul-tipath Propagation.” IEEE Journal on Selected Areas in Communica-tions, SAC-5(2):128–137, Feb. 1987.

[TAS95] C. Tidestav, A. Ahlen, and M. Sternad. “Narrowband and Broad-band Multiuser Detection Using a Multivariable DFE.” In Proc. ofPIMRC’95, pp. 732–736, Toronto, Canada, Sept. 1995.

[TSC98] V. Tarokh, N. Seshadri, and A.R. Calderbank. “Space-Time Codesfor High Data Rate Wireless Communication: Performance Criteriaand Code Construction.” IEEE Transactions on Information Theory,44(2):744–765, March 1998.

[WE71] S.B. Weinstein and P.M. Erbert. “Data Transmission by Frequency-Division Multiplexing Using the Discrete Fourier Transform.” IEEETranscations on Communication Technology, 19(5):628–634, October1971.

190

[WFG98] P.W. Wolniansky, G.J. Foschini, G.D. Golden, and R.A. Valenzuela.“V-BLAST: An Architecture for Realizing Very High Data Rates Overthe Rich Scattering Wireless Channel.” In Proc. URSI InternationalSymposium on Signal, Systems, and Electronics, pp. 295–300, Pisa,Italy, Sept. 1998.

[YR94] J. Yang and S. Roy. “Joint Transmitter and Receiver Optimizationfor Multiple-Input-Multiple-Output Systems with Decision Feedback.”IEEE Transactions on Information Theory, 40(5):1334–1347, Septem-ber 1994.

191

System Architectures for Space-Time Communications - CiteSeer

Documents

Transcript of System Architectures for Space-Time Communications - CiteSeer