Post on 28-Feb-2022
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
Jean-Francois Frigon
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 ...
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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
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3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 67
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 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 81
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 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 116
5.3.1 Comparison with MIMO Equalization . . . . . . . . . . . . 117
5.3.2 Interpolation Effects . . . . . . . . . . . . . . . . . . . . . 119
6 Spread Spectrum MIMO . . . . . . . . . . . . . . . . . . . . . . . . 129
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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 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 144
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 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 166
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
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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
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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
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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
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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
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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
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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
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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
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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
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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
Jean-Francois Frigon
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
Number of Receive Antennas
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
Number of Transmit/Receive Antennas
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
Number of Receive Antennas
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
Number of Transmit Antennas
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
Number of Transmit Antennas
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
Number of Transmit Antennas
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
Number of Receive Antennas
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
Number of Transmit Antennas
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
Number of Transmit/Receive Antennas
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
Number of Transmit/Receive Antennas
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
Number of Transmit/Receive Antennas
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.
3.4 Simulation Results
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
4.3 Simulation Results
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)
MMSE withoutCancellation
MMSE with Cancellation &MMSE VBLAST
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 withoutCancellation
MMSE with Cancellation &MMSE VBLAST
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
−j 2πncpP + vf (p)
=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
−j 2πncpP + vf (p)
=1√P
Nc∑nc=0
P−1∑i=0
dt(i)e−j2πipP Cnce
−j 2πncpP + vf (p)
=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.
5.3 Simulation Results
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)
6.4 Simulation Results
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.
7.3 Simulation Results
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
42 iterations 475 iterations
RAKE Receiver with Ls = 32 28 iterations 300 iterationsGeneralized RAKE Receiverwith Ls = 32
37 iterations 400 iterations
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