vide and conquer strategy in selecting the active set of transmitters and receivers, it is possible to minimize ...... the receiver in order to perfectly decode the transmitted signal, in all these scenarios. ...... statistics of Ëd(p) to determine its p.d.f..
Efficient Signal Processing Techniques for Future Wireless Communications Systems
A Thesis Submitted to the Faculty of Drexel University by Sarod Yatawatta in partial fulfillment of the requirements for the degree of Doctor of Philosophy December 2004
c Copyright 2004 Sarod Yatawatta.
All Rights Reserved.
ii
Dedications To my Parents and my Teachers
iii
Acknowledgements I would like to thank my advisor Professor Athina Petropulu for her guidance, encouragement and support. I would also like to thank my committee members Prof. Kapil Dandekar, Prof. Ruifeng Zhang from ECE Department, Prof. Robert Boyer from the Mathematics Department and Prof. Xia Gen Xia from University of Delaware, ECE Department, for their careful review and comments. I would also like to thank my friends and colleagues including Ivan Bradaric, Xueshi Yang, Riddhi Dattani, Rui Lin, Jie Yu, Yuanning Yu, Surujlal Dasrath, Pierluigi Salvo Rossi, Vasileios Nasis, Lun Dong, Hailong Yang, and Jeff Englinger. Finally, I would like to thank my mother, father, brother, sister, grandmother and everyone else in my family for their guidance and support.
iv
Table of Contents
List of Tables
ix
List of Figures
x
Abstract
xii
1 Introduction
1
1.1
Emerging Trends in Wireless Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Challenges Facing Future Wireless Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2.1
The Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.2.2
Multiuser Interference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.3
Exploiting Spatial Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
1.2.4
Hardware Cost and Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
The Contributions Made by This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.4
Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2 Exploiting Cyclostationarity for Blind Channel Estimation
7
2.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
2.2
Theoretical Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2.1
Cyclostationarity of Communications Signals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.2.2
Estimation of Cyclostationary Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3
Frequency Domain Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.4
Proposed Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.5
2.4.1
FIR Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4.2
FIR Channels with Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 2.5.1
FIR Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
v
2.5.2
FIR Channels with Pulse Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.6
Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3 Orthogonal Frequency Division Multiplexing in Multiuser Communications
36
3.1
Orthogonal Frequency Division Multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2
Multiuser OFDM Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.3
Channel Estimation in OFDM Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.4
3.3.1
Single-user Single Antenna OFDM Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3.2
Single-user OFDM Systems with Multiple Transmit/Receive Antennae. . . . . . . 41
3.3.3
Multi-user OFDM Systems without Multi-user interference . . . . . . . . . . . . . . . . . . . 42
3.3.4
Multi-user OFDM Systems with Multi-user Interference . . . . . . . . . . . . . . . . . . . . . . . 42
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 Blind Channel Estimation in Multiuser OFDM Systems Using Precoding
44
4.1
Linear Precoding in OFDM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.2
Linear Precoding for Blind Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3
Blind Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4
Resolving the Diagonal Ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 4.4.1
Resolving Ambiguity Based on Pilots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.4.2
Resolving Ambiguity in a Blind Fashion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5
Implementation Issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.6
Selection of Precoding Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.7
Multiuser OFDM Systems with Space Time Coding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.8
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 4.8.1
2 by 2 Multiuser System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.8.2
2 Users with STBC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.8.3
Comparison with Existing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
vi
4.9
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 Precoder Design for Channel Estimation
63
5.1
Performance Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2
Channel Estimation Error. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.3
Analytical Bit Error Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.4
Signal to Interference Plus Noise Ratio (SINR) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.5
Finding Optimal Precoders Using Genetic Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.6
Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
5.7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6 Multiuser cooperation in OFDM systems
76
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.2
Space Time Cooperation in a Multiuser OFDM System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 6.2.1
Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.2.2
Proposed Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
6.2.3
Multiuser OFDM without Cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2.4
Maintaining Constant Transmission Energy Between Cooperative and Noncooperative Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2.5 6.3
Time Division Duplexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
Channel Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 6.3.1
Comparison of Cooperative and Non-cooperative Schemes . . . . . . . . . . . . . . . . . . . . 91
6.4
Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
6.5
Determining the Level of Cooperation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
6.6
Extension to more than Two Users . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
6.7
Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
6.8
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
vii
7 Energy Efficient Channel Estimation in MIMO Systems
110
7.1
Energy Consumption in MIMO Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
7.2
General Methodology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.3
Minimizing Energy at the Transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
7.4
Minimizing Energy at the Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
7.5
Minimizing Energy Both at the Transmitter and Receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.6
Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
7.7
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
8 Conclusions
120
Bibliography
122
A Appendices of Chapter 4
132
A.1 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 A.2 Proof of (4.18) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 A.3 Proof of (4.38) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 B Appendices of Chapter 5
135
B.1 Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 B.2 Construction of T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 C Appendices of Chapter 6
140
C.1 Capacity after interblock precoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 C.2 Proof of Proposition 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 C.3 Proof of Proposition 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 C.4 Proof of Proposition 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 D Appendices of Chapter 7
149
D.1 Channel estimation error and energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
viii
D.2 Proof of Proposition 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 D.3 Proof of Proposition 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 Vita
153
ix
List of Tables 4.1
Assignment of precoding matrices with block number and source . . . . . . . . . . . . . . . . . . . . . . 53
4.2
STBC for a single user system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.3
Space-time coding for a single and 2 user system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
7.1
MSE and energy for different channel estimation schemes for 50, random 8 by 8 channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
x
List of Figures 2.1
A digital communications system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2
The magnitude of the cyclic correlation for a random FIR channel of length L = 5. . . 15
2.3
The magnitude of the cyclic spectrum for a random FIR channel of length L = 5. . . . . 16
2.4
The phase of the cyclic spectrum for a random FIR channel of length L = 5. . . . . . . . . . 17
2.5
Calculation of phase due to channel zeros corresponding to the stopband. As ω increases, the phase contribution from these zeros change as if the location of the zeros rotate clockwise. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.6
(a) ONMSE and (b) BER for cepstrum based estimates and proposed estimates given in section 2.4.1. We generated 100 random FIR channels of length 8 and for each channel, obtained the estimates for 25 Monte Carlo runs. 1000 symbols were used in channel estimation and perfect channel length knowledge was assumed. In (a), we have given the ONMSE results where both a rectangular window of length 2L was used, and a window was not used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.7
(a) ONMSE and (b) BER for the method proposed in section 2.4.2 and for the subspace method. We generated 100 channels with pulse shaping and for each channel, 25 Monte Carlo runs were taken at each SNR. The modulation scheme used was 4 QAM and perfect channel length knowledge was used for equalization. . . 31
2.8
(a) ONMSE and (b) BER for the method proposed in section 2.4.2 and for the subspace method. We generated 100 channels with pulse shaping and for each channel, 25 Monte Carlo runs were taken at each SNR. The modulation scheme used was 4 QAM and the channel length was overestimated by one symbol interval. . 32
2.9
Variation of the ONMSE with rolloff for the channels given in equation (2.59). . . . . . . . 33
2.10 Channel obtained using experimental data. Lines with circles gives the channel obtained by training. Lines with stars gives the mean of the estimated channel using 400 symbols. Solid lines give the real part while broken lines give the imaginary part of the channel.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.11 Equalized data: (a) Received constellation (b) Equalized constellation using channel obtained via training and (c) Equalized constellation using the estimated channel. 400 symbols were used in the estimation and 2000 symbols were used in the equalization. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 2.12 Signal generator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.13 Vector signal analyser. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3.1
An OFDM transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2
An OFDM receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.3
An MT user OFDM system with MR receiving antennae. The data is precoded before transmission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
xi
4.1
Block diagram of a 2 user STBC system. Each user will implement STBC independently.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
4.2
BER for (a) fast ( fd = 10−5 ) varying and (b) slow ( fd = 10−6 ) varying channels. . . . . 61
4.3
BER comparison with and without STBC. Random channels using the Jakes model with fd = 10−6 were used in this simulation. 400 OFDM blocks were used in channel estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.4
BER comparison with method of Bolcskei et al. 400 OFDM blocks were used in channel estimation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.1
Comparison of analytical and simulated BER and SINR for 40 different channels. . . . 74
5.2
Comparison of analytical and simulated NMSE for 40 different channels. . . . . . . . . . . . . 74
5.3
BER for different combinations of α, β. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
5.4
BER comparison of ad-hoc precoding scheme and the solution obtained using genetic algorithm for different SNR. BER below 10−6 was taken to be 10−6 . . . . . . . . . . . . 75
6.1
A 2 user OFDM system with one receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.2
An OFDMA transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
6.3
Cooperative STBC OFDM scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
6.4
Cooperative STBC OFDM scheme with Time Division Duplexing . . . . . . . . . . . . . . . . . . . . . 104
6.5
Excess capacity C1 + C2 − 2C˜1 − 2C˜2 (a),(c),(e) Simulated (b),(d),(f) Theoretical upper bounds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.6
Minimum of C12 , C21 , (C1 +C2 )/2 for various α and β values . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.7
Capacity region for cooperation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
6.8
Voronoi diagram for users u1 , u2 , u3 , u4 , u5 . In each Voronoi cell, the SNR of each user at the receiver is given. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.9
Performance of a 2 user OFDM system with and without cooperation for 4 QAM . . . . 107
6.10 Performance of a 2 user OFDM system with and without cooperation for 4 QAM . . . . 108 6.11 Performance of a 2 user OFDM system with and without cooperation for 4 QAM . . . . 108 6.12 Performance of cooperation and rate reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.13 Performance of cooperation with interuser SNR variation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 7.1
MIMO channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
7.2
MSE variation with N, M and J . We see that the MSE is independent of N, has a linear variation with M and is inversely proportional to J.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
xii
Abstract Efficient Signal Processing Techniques for Future Wireless Communications Systems Sarod Yatawatta Athina Petropulu, Ph. D. Wireless communications systems are evolving to be more diverse in use and more ubiquitous in nature. It is of fundamental importance that we consume the resources available in such systems, i.e., bandwidth and energy, to preserve room for more users and to preserve longevity. Signal processing can greatly help us achieve this. In this thesis we consider improving the utility of resources available in wireless communications systems. The basic obstacle for most wireless communications systems is the multipath channel that causes intersymbol interference. Channel estimation is a crucial step for recovering the transmitted symbols. Moreover, as more devices are equipped with wireless capabilities, the bandwidth becomes scarce and it is important to allow more than one device or more than one user to use the same frequency range or the same channel. However, this introduces multiuser interference, which is again eliminated only if the channel is known. Furthermore, most wireless systems are battery powered, at least at the transmitter end. Hence it is crucial that energy consumption is minimized to preserve the longevity of the system. The contribution of this thesis is three fold: (i) We propose novel bandwidth efficient blind channel estimation algorithms for single input multiple output systems, and for multiuser OFDM systems. The former exploits cyclostationarity inherent in communications signals. The latter exploits the structure introduced to the transmitted signal via precoding. We consider design of such precoders by optimizing performance metrics such as the bit error rate and signal to interference plus noise ratio. (ii) In the multiuser systems case, we propose a novel cooperative OFDM system and show that, when users face significantly different channel conditions, cooperation can improve the performance of all the cooperating users. (iii) We consider energy efficient training based system estimation in large MIMO systems. The goal there is to minimize energy consumption both in transmission of training symbols and in performing computations. We show that by using a divide and conquer strategy in selecting the active set of transmitters and receivers, it is possible to minimize energy consumption without degrading the accuracy of the channel estimate.
1 Chapter 1. Introduction
1.1
Emerging Trends in Wireless Systems The process of communication is undergoing a major change in both application and scope.
Traditional communications systems are fixed using copper wire or optical fiber. Moreover, the purpose of these systems are mainly for voice or data transmission. However, this is changing. For instance, the number of trivial electronic devices with communication capabilities is increasing day by day. From sensor networks (smart dust[72]) to household appliances, the need for communication becomes essential as they become more intelligent. The fundamental reason for the appearance of smarter and smaller devices is the reduction of hardware cost used for computing. This enables inexpensive deployment of wireless devices. At the opposite end, the Internet is reaching beyond the boundaries laid out by copper wire and optical fibers. There is a visible merging of the major networks in the world, i.e., the telephony network, cable television network, the Internet and even the power grid. The last mile obstacle is being overcome by using wireless systems such as the worldwide interoperability for microwave access (WiMax) [69]. The combined effect of these phenomena will be the transformation of traditional communication systems to ubiquitous communications systems. Unlike their predecessors, future communications systems will not be used for only one purpose such as voice or data transmission. For instance, control [54] and monitoring [45] applications that employ wireless systems are emerging. Hence, the future communications systems needs to accommodate diverse traffic with diverse QoS (Quality of Service) requirements. Consequently, it is fundamentally important to apply our best efforts and knowledge for the improvement of such systems.
2 1.2
Challenges Facing Future Wireless Systems
1.2.1
The Channel
In a wireless system, the transmitter emits an electromagnetic waveform which propagates through the atmosphere and is received by a receiving antenna some distance away from the transmitter. Inevitably, there will be obstacles in the propagation medium that are both man made such as buildings, and natural such as mountains. Due to these obstacles the propagating waveform is both attenuated and reflected, resulting in multiple propagating wavefronts. Moreover, it is possible that two of these wavefronts cancel each other out at the receiver, which results in the phenomenon called multipath fading. Another form of fading is caused by the random fluctuations of the conditions in the atmosphere termed as scintillation. Furthermore, thermal noise as well as interference from other signals degrade the signal reception at the receiver. The aforementioned effects due to the propagation medium are represented as an abstract entity called the channel. The effect of multiple wavefronts is represented as multiple paths in a channel. It should be noted that if either the transmitter or the receiver is mobile, the channel is time-varying. In order to recover a perfect replica of the transmitted signal at the receiver, it is essential to know some information about the channel. The cancellation of channel effects is referred to as equalization. In coherent detection of the transmitted signal at the receiver, complete knowledge of the channel is required. It is possible to construct the equalizer directly without explicitly estimating the channel, or indirectly, by first estimating the channel. In either case, the transmitter should send a signal known a priori by the receiver which is called training. Most wireless devices will be battery powered. Hence the transmission of training signals will seriously affect the longevity of such devices. Moreover, training increases the overhead of the transmitted signal, thus reducing the net data transmission rate. However, by using blind channel estimation methods, it is possible to reduce the amount of training required (and energy spent) significantly. Typically, some special property of the transmitted signal is exploited for blind channel estimation. As an illustration of the rich variety of methods available, we could give methods that
3 exploit cyclostationarity, methods that exploit higher order statistics (HOS), methods that exploit algebraic structure of the received signal, and even machine learning methods using neural networks. In most communications systems, the transmitted signal is originating from an information source, which emits information symbols that belong to a finite alphabet. Normally, for complete recovery of the transmitted symbols, it is sufficient to sample the received signal at a rate equal to the symbol rate. However in fractional sampling, the sampling is done at a rate higher than the symbol rate. One major class of blind channel estimation methods exploit the statistical properties of the information source and require fractional sampling at the receiver. By linear precoding of the input, where the input is passed through a known linear transform, it is also possible to artificially introduce certain statistical properties to the source signal that can be exploited for blind channel estimation.
1.2.2
Multiuser Interference
As wireless devices increase in number, it is possible that two devices transmit simultaneously using the same frequency band. This would result in interference of each transmitter’s signal by the other transmitter. Due to this, neither signal would be recoverable at the receivers and this effect is called multiuser interference. Normally this is resolved by each user retransmitting its own signal until it is received without interference by the receiver. However, in some situations, it might be economically beneficial to allow multiple users to use the same channel, yielding a multiuser system. This results in a multiuser detection problem at the receiver. However, by using signal processing techniques it is possible to recover the transmitted signals of each user.
1.2.3
Exploiting Spatial Diversity
The ability to communicate without wires enables us the full usage of the spatial dimension. The spatial dimension can be exploited to increase the data rates as well as reduce the error rates, i.e., by using spatial multiplexing [19]. In order to achieve this, more than one antennae is employed at the transmitter and the receiver. This effectively results in a multiple input, multiple output (MIMO)
4 system. Due to practical limitations, it is not always possible to employ multiple transmitting and receiving antennae. However, when multiple users are present, it is possible for some users to act as relays for other users by transmitting the other users signal. This effectively produces a MIMO system without increasing the number of transmitting antennae for each user. If all users relay other user’s signals, it is called cooperation. In recent research work, it is shown to improve the overall performance of all users.
1.2.4
Hardware Cost and Complexity
The fundamental reason for the proliferation of wireless devices is the low hardware cost. However, in order to produce cheap hardware, the algorithms run using that hardware should be simple and should consume less energy. Hence it is also important to design simpler signal processing algorithms to reduce computational (energy) and hardware cost.
1.3
The Contributions Made by This Thesis There are several novel contributions made by this thesis to the rich field of wireless communi-
cations. They can be itemized as follows: • A novel algorithm for blind channel estimation in single input single output systems by fractional sampling. [80, 82, 52] • A novel algorithm for blind channel estimation in Multiuser/MIMO OFDM systems using linear non redundant precoding, also focusing on optimal precoder design. [79, 78, 77] • A novel multiuser cooperation scheme using OFDM under multipath channels, with comparisons to a non cooperating systems both analytically and numerically. [81, 83] • A novel divide and conquer method of minimizing energy in channel estimation of large MIMO systems. [84]
5 We have considered several vastly different problems encountered in wireless systems in this thesis as seen from the above. However, our motive is clearly to improve the efficiency of wireless systems. We should point out that our use of the term efficiency here does not refer to a numerical quantity. On the contrary, we denote an increase in efficiency to be the combined effect of reducing training overhead, bandwidth usage, energy consumption and hardware cost.
1.4
Outline of the Thesis In this thesis we focus on meeting the challenges mentioned in the beginning of this chapter.
First we focus on blind channel estimation with simpler algorithms for communications systems. By estimating the channel blindly it is possible to reduce the energy consumption by eliminating the need for training. Chapter 2 gives some theoretical background and a novel algorithm for blind channel estimation of a SISO system. We also focus on accommodating more than one user in the same channel and using the available bandwidth efficiently. We concentrate on OFDM systems that overcome the multipath channel by dividing it into a set of sub channels in the frequency domain. However, when multiple users share the same channel, multiuser interference is still present. Hence we propose a novel algorithm for blind channel estimation (and thus eliminating multiuser interference). In chapter 3, we give the theoretical background on OFDM and in chapter 4, we present the proposed channel estimation method. In chapter 5, we study the performance of the proposed method and find ways of optimizing it. Multiple users in communications systems do not always have detrimental effects on performance. It is possible for the users to cooperate and by doing so exploit the spatial dimension to improve performance. In chapter 6, we consider a multiuser OFDM system and find limits of performance that can be achieved by cooperation. Cooperation has so far being considered in scalar channels. However, in chapter 6, we consider cooperation under multipath channels where both multipath diversity and cooperation diversity coexist. The energy required for transmission and computation grows as the number of interfering users
6 or the number of transmitters and receivers increase. In chapter 7, we focus exclusively on minimizing energy spent in channel estimation in multiuser or MIMO systems. We propose a divide and conquer strategy select active set of transmitters and receivers in order to minimize energy consumption without deteriorating the system performance. Finally, in chapter 8, we give possible extensions to the work done in this thesis and present concluding remarks.
7 Chapter 2. Exploiting Cyclostationarity for Blind Channel Estimation
2.1
Introduction In general, for the estimation of an LTI system excited by an unknown stationary non-Gaussian
input, based on the system output, one has to use higher-order statistics (HOS) of the output sampled at baud rate. When the input is cyclostationary, which is the case in communications signals, system estimation can be carried out using second-order cyclic statistics of the oversampled output [20, 67]. Based on this premise, several methods have been proposed for blind channel estimation (for a review of existing results the reader is referred to [41]). They can be categorized as time-domain and frequency-domain, with reference to the transform domain in which the problem is formulated. A serious deficiency in time domain-methods ([67, 48]) is their sensitivity to channel order (or length) estimation errors. Most often in practice the channel (the combined effect of transmit-receive filters and multipaths) is a continuous function of time and its magnitude will not suddenly drop to zero. The notion of channel length depends on the application. Some methods have been proposed to overcome the need for channel length information [37, 22, 51], yet this could still pose a problem when the channel is time varying. On the other hand, frequency-domain methods do not require channel length, but cannot be applied to band-limited channels. In that case, the phase of the full spectrum cannot be recovered because the cyclic spectrum has limited support [14]. In practice, most channels are bandlimited, both by the physical nature of the channel and by the pulse shaping employed. In this chapter we propose a novel frequency-domain blind channel estimation approach. The channel phase frequency response is obtained in closed form in terms of the appropriately discretized cyclic phase spectrum. For the case of bandlimited channels with a small rolloff, we propose an approximate expression for the channel phase response, which has low computational complexity. The resulting channel estimate will always contain an error. However the error will be negligible if the channel has negligible energy within the stopband. Despite this bias, the proposed
8 approach can still be of great practical use since it involves low cost. As a matter of practice, choosing the correct algorithm for a particular purpose does not depend on the accuracy of the result only, but also on economy in hardware implementation. With the introduction of wireless communication capabilities to most electronic devices in the offing, the proposed approach could provide a viable alternative in reducing cost.
2.2
Theoretical Background Let us consider the baseband representation of a digital communication system as given in Fig.
2.1. The input s(t) is derived from a communication source, which is generally cyclostationary. The noise component is given as w(t). The combined effect of transmitter pulse shaping, multipath interference and fading and receiver pulse shaping is given as h(t) which is called the channel impulse response.
w(t)
x(t)
s(t)
channel h(t)
Figure 2.1: A digital communications system
Assuming the system h(t) is linear and time invariant (LTI) and is stable in the sense that
9 bounded input produces bounded output (BIBO), we can express the received signal as
x(t) = s(t) ⊗ h(t) + w(t) =
Z ∞
−∞
s(τ)h(t − τ)dτ + w(t)
(2.1)
In practice, the information source is discrete, i.e. it emits symbols in discrete steps of time, which we assume to be constant and call the symbol interval T . Moreover, the emitted symbols will not take arbitrary values but a finite set of values that belong to a predefined constellation or a finite alphabet. In this case, we can refine (2.1) as ∞
s(t) =
∑
ν=−∞
sν δ(t − νT ), x(t) =
∞
∑
ν=−∞
sν h(t − νT ) + w(t), ν ∈ Z, sν ∈ S
(2.2)
where the integration has been replaced by a summation and the input takes a finite set of values given by sν ∈ S and the discrete time step is T . We should note that the output signal x(t) is still a continuous signal even though the input sν is discrete. However, in practice, the output is sampled and converted into a discrete value by an A/D converter. Normally the output is sampled at a sampling interval equal to the symbol interval T , or at baud rate. However, let us consider fractional sampling with a sampling interval ∆ being a fraction of the symbol interval ∆ = 4
x(n) = x(n∆) =
2.2.1
T p
where p is the oversampling ratio. The sampled output is ∞
∑
ν=−∞
sν h(n − νp) + w(n), n ∈ Z, p ∈ Z+ .
(2.3)
Cyclostationarity of Communications Signals
Let us consider x(n) to be a signal generated by some stochastic process. The autocorrelation of x(n) is given by 4
Rx (t1 ,t2 ) = E{x(t1 )x? (t2 )}
(2.4)
where t1 and t2 are arbitrary time instances. For a stationary signal, we have,
Rx (t1 ,t2 ) = Rx (t1 − t2 )
(2.5)
10 and for a cyclostationary signal, we have
Rx (t1 ,t2 ) = Rx (t1 + T f ,t2 + T f )
(2.6)
where T f is the fundamental period. Moreover, the mean of a cyclostationary signal should be periodic with the same period. Due to this periodicity, it is possible to express the autocorrelation of a cyclostationary signal by its Fourier Series expansion. Thus, for a cyclostationary signal, with fundamental period T f , we define the cyclic correlation as 4
Rkα x (τ) =
1 Tf
τ 2π τ Rx (t + ,t − )e− jkαt dt, α = , k ∈ Z 2 2 Tf −T f /2
Z Tf /2
(2.7)
The cyclic spectrum (also termed as spectral correlation density) of a cyclostationary signal is given by 4 Sxkα (Ω) =
Z ∞
−∞
− jΩτ Rkα dτ x (τ)e
(2.8)
which is the Fourier Transform of the cyclic correlation (2.7). We next apply these definitions to the signal given by (2.2). Before proceeding, we make the following assumptions. 0
• The source signal sν is white with variance σ2 , i.e. E{sν sν0 } = σ2 δ(ν − ν ) • The noise w(t) is stationary white with variance σ2n and is uncorrelated with the source. Then we have Rx (t1 ,t2 ) = σ2
∞
∑
ν=−∞
h(t1 − νT )h? (t2 − νT ) + σ2n δ(t1 − t2 )
(2.9)
Proof: First note that from (2.2) we have Rs (t1 ,t2 ) = E{s(t1 )s? (t2 )} = σ2
∞
∑
ν=−∞
δ(t1 − νT )δ(t2 − νT ) = σ2 δ(t1 − t2 )
∞
∑
ν=−∞
δ(t1 − νT ) (2.10)
11 By substituting (2.2) in (2.4) we have
Rx (t1 ,t2 ) = = =
(2.11)
Z ∞Z ∞
Rs (ρ, λ)h(t1 − ρ)h? (t2 − λ)dρdλ + σ2n δ(t1 − t2 ) −∞ −∞ Z ∞ Z ∞ ∞ 2 σ δ(ρ − λ) δ(ρ − νT ) h(t1 − ρ)h? (t2 − λ)dρdλ + σ2n δ(t1 − t2 ) −∞ −∞ ν=−∞ ∞ h(t1 − νT )h? (t2 − νT ) + σ2n δ(t1 − t2 ) σ2 ν=−∞
∑
∑
yielding us (2.9).
We see that from (2.9) that (2.6) is satisfied with the fundamental period T f = T . Thus we conclude that the output signal given in Fig. 2.1 is cyclostationary. Next we find its cyclic correlation. Using (2.9) in (2.7) we get Rkα x (τ)
σ2 = T
Z ∞
τ τ h(t + )h? (t − )e− jkαt dt + σ2n δ(τ)δ(k) 2 2 −∞
(2.12)
Proof: substitution of (2.9) in (2.7) results in
Rkα x (τ) =
(2.13) T
σ2
∞
τ τ h(t + − νT )h? (t − − νT )e− jkαt dt + n 2 2 T −T /2 ν=−∞
Z σ2 T /2
Z σ2 T /2
∑
Z ∞
τ 2 sin(kαT /2) τ h(t + )h? (t − )e− jkαt dt + δ(τ) T −∞ 2 2 T kα Z σ2 ∞ τ ? τ − jkαt = h(t + )h (t − )e dt + σ2n δ(τ)δ(k) T −∞ 2 2
(a)
=
σ2n
−T /2
δ(τ)e− jkαt dt
where (a) is due to the summation over ν of the period [−T /2, T /2] being equal to the real axis. If H(Ω) is the Fourier Transform of h(t), the cyclic spectrum of x(t) is given by
Sxkα (Ω) =
σ2 kα kα H(Ω + )H ? (Ω − ) + σ2n δ(k) T 2 2
(2.14)
12 Proof: Using (2.8) and (2.12) we have
Sxkα (Ω)
σ2 = T
Z ∞Z ∞ −∞
τ τ h(t + )h? (t − )e− jkαt e− jΩτ dtdτ + σ2n δ(k) 2 2 −∞
Z ∞
−∞
δ(τ)e− jΩτ dτ
(2.15)
In order to evaluate the first integral, we use the transform X = t + τ/2, Y = t − τ/2 with dtdτ = dX dY . Then we have
Sxkα (Ω) =
σ2 T
Z ∞
−∞
h(X )e− j(Ω+kα/2)X dX
Z ∞
−∞
h? (Y )e j(Ω−kα/2)Y dY + σ2n δ(k)
(2.16)
Noting that H(Ω) =
Z ∞
−∞
h(t)e− jΩt dt
(2.17)
we get (2.14).
It is straightforward to extend the definitions (2.4), (2.7) and (2.8) for a discrete signal x(n) as given by (2.3). We find that the discrete, oversampled signal x(n) is also cyclostationary with cyclic period being equal to the oversampling ratio p. Hence we have [20], p−1
Rkα x (ν) =
1
∑ Rx (n + ν, n)e− j2πkαn , α = p
(2.18)
n=0
as the cyclic correlation and
Sxkα (e jω ) =
∞
∑
ν=−∞
− jνω Rkα , n, ν, k ∈ Z x (ν)e
(2.19)
as the cyclic spectrum. For the signal given in (2.3) we have 2 Rkα x (ν) = σ
p−1
∞
∑ ∑
n=0 r=−∞
h(n + ν − rp)h? (n − rp)e− j2πkαn + pσ2n δ(k)δ(ν)
(2.20)
and Sxkα (e jω ) = σ2 H(ω)H ? (ω − 2πkα) + pσ2n δ(k)
(2.21)
13 Proof: From (2.3) we have ∞
∑
Rx (n + ν, n) = E{x(n + ν)x? (n)} = σ2
r=−∞
h(n + ν − rp)h? (n − rp) + σ2n δ(ν)
(2.22)
Using (2.18) we have
Rkα x (ν)
=σ
p−1 2
∞
∑ ∑
n=0 r=−∞
h(n + ν − rp)h? (n − rp)e− j2πkαn + pσ2n δ(k)δ(ν)
(2.23)
and using (2.19) we get
Sxkα (e jω ) = σ2
(2.24) ∞
p−1
ν=−∞ n=0 r=−∞ p−1
= σ2 H(ω) ∑
(a)
∞
∑ ∑ ∑
h(h + ν − rp)h? (n − rp)e− jνω e− j2πkαn + pσ2n δ(k)
∞
∑
n=0 r=−∞
∞
∑
ν=−∞
e− jνω δ(ν)
h? (n − rp)e− j(n−r p)(ω−2πkα) e− j2πkαr p + pσ2n δ(k)
(b)
= σ2 H(ω)H ? (ω − 2πkα) + pσ2n δ(k)
where (a) is from the fact that
∞
H(ω) =
∑
h(n)e− jnω
(2.25)
n=−∞ ∞ and (b) is from the fact that ∑n=0 ∑∞ r=−∞ ≡ ∑(n−r p)=−∞ . p−1
2.2.2
Estimation of Cyclostationary Statistics
The fundamental quantity that needs to be estimated in a cyclostationary signal is the cyclic correlation. It is true that for the signal to be properly cyclostationary, the mean has to be periodic as well. However, in most communications systems, the signal have zero mean and hence this requirement is automatically satisfied. Let us assume that we have l samples of data from x(0) to x(l − 1). We also use an N point DFT
14 in our calculations. We first estimate the autocorrelation of the signal x(n) as 1 J−1 Rbx (n + ν, n) = ∑ x(n + ν + pl)x∗ (n + pl), 0 ≤ n ≤ p − 1, 0 ≤ ν ≤ N − 1. J l=0
(2.26)
The number of products in the summation above is given by
J=b
l−N c p
(2.27)
Next, we estimate the cyclic autocorrelation Rbkα x (ν) using equation (2.18) for 0 ≤ ν ≤ N − 1.
We should mention that in most communications systems, the channel h(n) has compact sup-
port. We generally call the support region as the length L of the channel. Due to this compact support, we see that the support region of (2.18) will be (2.20) −L ≤ ν ≤ L. If some estimate of the channel length, i.e., Lˆ is available, then we can window Rbkα x (ν) by a rectangular window of length
ˆ Finally, we estimate the cyclic spectrum by taking an N point FFT, i.e., 2L. 1 Sbxkα (l) = FFT(Rbkα x (ν)), α = , 0 ≤ l ≤ N − 1. p
(2.28)
We give a numerical example at this point. We have considered a communications systems with 4 QAM modulation. The channel h(n) was modeled as an FIR filter of length L = 5. We used 400 data samples (after oversampling by 4) for estimation. In Fig. 2.2 we have given the magnitudes of the true cyclic correlation (calculated using (2.20)) and the estimated cyclic correlation. We have selected the DFT length N to be equal to 37. Note in particular that for 10 ≤ ν ≤ 20 the magnitude is essentially zero. This is due to the channel having compact support in time domain. In Fig. 2.3 and 2.4 we have given the magnitude and phase of the cyclic spectrum for the same system. We note that in Fig. 2.4 there is a phase shift between the true and estimated phases. It is important to note here that the phases of the estimated cyclic spectrum will not be symmetric, as the theoretical one should be. In order to make it symmetric, we should shift the estimated
15 7 Estimated Cyclic Correlation True Cyclic Correlation 6
5
x
|Rk α(ν)|
4
3
2
1
0
5
10
15
ν
20
25
30
35
Figure 2.2: The magnitude of the cyclic correlation for a random FIR channel of length L = 5. phase by b−N/2pc. 2.3
Frequency Domain Channel Estimation The basic frequency domain channel estimation procedure involves two steps. In the first step,
the channel magnitude is obtained using the magnitude of the power spectrum. From (2.21) we see that when k = 0, |Sxkα (e jω )| = σ2 |H(e jω )|2 + pσ2n
(2.29)
and for high SNR values, σ2n L
N − 2Lc . N
(2.41)
where L is the channel length. Assumption (A4) implies that ωc < π/2 rad/cycle. For any system that transmits at or below Nyquist rate [56], the channel sampled at symbol rate (p = 1) has full bandwidth. Hence to satisfy this condition we need p ≥ 3. In general, we have no prior knowledge of Nz to check the validity of assumption (A5). However, by careful design of the shaping function, we can ensure that the minimum number of Nz is greater than the one required by (2.41). Multipaths will only add more zeros thus the condition will hold under multipaths if the shaping function itself satisfies it, unless the channel becomes unidentifiable according to [15]. Let us discretize ψ(ω) in (2.32) on a grid with spacing 2π/N and select N and p so that whenever φ(ω) falls in the passband, φ(ω + 2πkα) falls in the stopband. We take k = 1 again for maximum support. This can be achieved as long as,
Lc < m < N/2 − Lc
(2.42)
where m is the closest integer to αN/2 or N/2p. A value for m greater than 1 is guaranteed by N > 4Lc . Proposition 1 Let h(n) be an FIR impulse response of length L, which satisfies assumptions (A1)d be the channel impulse response constructed based on the correct magnitude re(A5). Let h(n)
d constructed using the estimated phase of the sponse of h(n), i.e., |H(l)|, and phase response φ(l),
21 \ cyclic spectrum, ψ(l + m), as: −ψ(l \ + m); 0 ≤ l ≤ Lc − 1, N − Lc ≤ l ≤ N − 1 4 d φ(l) = 0; Lc ≤ l ≤ N − Lc − 1
with N and p satisfying (2.42). Then, it holds:
d 4 d = h(n) IDFT {|H(l)|e jφ(l) }
− j 2π N ko n
−2Ne
= ah(n + b) ⊗
Λ(1 − e
j 2π N n
)
Λ
× ∑δ n− i=0
(2i + 1)N 2Λ
!
(2.43) + E(n)
where ! (N−1)/Λ−1 2π 2π a N−Lc −1 E(n) = ∑ H(l)e j N ln H(l)e j{ N (b+n)l+π ∑i=Nz u(l+ko −iΛ)} , N l=L c a = e− jNz (N/2−m) , b = Λ=b
(2.44)
Nz , ko = 2m − Lc, 2
N − 2Lc c, Nz − 1
and ⊗ denotes N point circular convolution. Proof of Proposition 1: The channel Frequency Response can be given as: Ntotal
H(ω) =
∏ (1 − ziz−1 )|z=e
jω
(2.45)
i=0
Here the channel has Ntotal zeros, out of which Nz lie on the unit circle. Let us refer to Fig. 2.5 to calculate the phase of the terms of (2.45) that contain the zeros on the unit circle. As ω increases, the phase contributed by this set of zeros changes as if the zeros themselves are rotated by ω clockwise. Let φi be the phase of (1 − zi e− jω ) where |zi | = 1. It holds: φi (ω) =
π ωc − ω π − ωc + +i , 0 ≤ i < Nz . 2 2 Nz − 1
(2.46)
22 Zi r
φi di r≈1
ωc − ω
Figure 2.5: Calculation of phase due to channel zeros corresponding to the stopband. As ω increases, the phase contribution from these zeros change as if the location of the zeros rotate clockwise.
The sum of the phases of all terms (1 − zi e− jω ) for which |zi | = 1, i.e. φz (ω) will then be equal
to Nz (π − ω2 ).
For ω ≈ θi , it holds: dφi (ω) r 1 = ((1 − r)2 − (1 + r)(θi − ω)2 ), ω ≈ θi 3 dω (1 − r) 2 We see that as r → 1,
dφi (ω) dω
(2.47)
becomes infinite, thus indicating discontinuity of φi at ω = θi .
We see from Fig. 2.5 that at this discontinuity, the overall change in φi (ω) is π. Hence, the overall phase response is linear with jumps of π at frequencies corresponding to each zero on the unit circle. Note that these jumps will not occur within the passband of the channel. Hence, if Nz is large enough, within the stopband, φz (ω) will dominate the overall phase response because all its derivatives become large whenever ω ≈ θi for any i ∈ [0, Nz − 1]. If the phase due to zeros away from the unit circle is φt (ω), the phase of the channel can be given as φ(ω) = Nz (π −
Nz −1 ω π − ωc ) + φt (ω) − π ∑ u(ω − ωc − i 2 ) 2 Nz − 1 i=0
(2.48)
where u(ω) is the step function. Using (2.48), we can rewrite φ(ω + 2πα) in discrete form (ω = φ(l + 2m) = Nz (N/2 − m) −
2π N l)
as
Nz −1 Nz π l + φt (l + 2m) − π ∑ u(l + 2m − Lc − iΛ) N i=0
(2.49)
23 where l is the discrete frequency index. Under the assumptions made earlier, φ(l) will lie on the passband while φ(l + 2m) will lie on the stopband. Hence the jumps of π will dominate (2.49) thus we can ignore the contribution from φt (l + 2m) to simplify our analysis. Hence we can write (2.35) as Nz −1 Nz π l − Nz(N/2 − m)π ∑ u(l + 2m − Lc − iΛ). −ψ(l + m) ≈ φ(l) + N i=0
(2.50)
Then we can re write (2.43) as: Nz −1 φ(l) + NNz π l − Nz (N/2 − m) + π ∑i=0 u(l + 2m − Lc − iΛ); d≈ φ(l) 0 ≤ l ≤ Lc − 1, N − Lc ≤ l ≤ N − 1 0; Lc ≤ l ≤ N − Lc − 1.
d d = IDFT {|H(l)|e jφ(l) Then h(n) } yields:
d= h(n) a N
(2.51)
Lc −1
∑ H(l)e j{
l=0
N−1
+
Nz −1 2π N (b+n)l+π ∑i=0 u(l+ko −iΛ)}
∑
H(l)e
∑
(N−1)/Λ−1
H(l)e j{ N (b+n)l+π ∑i=Nz 2π
u(l+ko −iΛ)}
l=Lc N −1
z j{ 2π N (b+n)l+π ∑ i=0 u(l−N+1+ko −iΛ)}
l=N−Lc
=
+
N−Lc −1
+ E(n)
2π a N−1 H(l)e j{ N (b+n)l} G(l) + E(n) ∑ N l=0
with M
G(l) = 2 ∑ rect( i=0
l + ko − i2Λ ) − 1, M ∈ Z Λ
where E(n) can be deduced from (2.51) and rect( Λτ ) is a rectangular pulse of width Λ.
(2.52)
24 We see that the Inverse Discrete Fourier Transform (IDFT) of G(l) is e− j N ko n (1 − e− j N Λn ) (1 − e j2πn ) 2π
g(n) =
where g(n) 6= 0 for n = be
N N 2Λ , 3 2Λ , . . ..
2π
(1 + e− j N Λn ) 2π
2π
(1 − e j N n )
By simplification of (2.53), (2.43) follows.
(2.53)
d will contain delayed copies of the original channel separated Equation (2.43) suggests that h(n)
N Λ.
If condition (2.41) holds, these copies will be separated well enough to allow one to extract
h(n). Note also that the error E(n) will be small (by assumption (A3)) if the energy within the stopband is negligible, i.e., |E(n)| ≤
2 N−Lc −1 ∑ |H(l)|. N l=L c
(2.54)
The magnitude of each replica appearing in (2.51) is determined by the magnitudes of the impulses in (2.43), which in turn is determined by
f (n) =
1 2π
|1 − e j N n |
.
(2.55)
This expression has its maxima at n = 0 and n = N, is monotonically decreasing as n goes from 0 to N/2 and, is monotonically increasing as n goes from N/2 to N. Hence the first and last replicas will have the highest magnitudes. As a consequence, to extract one copy of the channel estimate out d we could do the following. Let the anticipated channel length be Le . We find the segment of h(n)
with maximum energy out of the first Le samples and the last Le samples. After extracting the maximum energy sequence, we obtain a channel estimate with a scaling ambiguity and an unknown delay.
2.5
Numerical Results Simulations were done to verify the performance of the proposed methods and to compare with
other methods. In section 2.5.1, we give the results obtained for FIR channels with full bandwidth
25 support, and in section 2.5.2 we give results for bandlimited channels. The basic procedure is as follows. First, we generated a Quadrature Amplitude Modulation (QAM) signal x(n) = ∑∞ ν=−∞ sν h(n − νp) + w(n). The input symbols {sν } were from an i.i.d. QAM source. The noise w(n) was stationary, white and Gaussian. The cyclic spectrum was estimated as diescribed in section 1.11. b (l), the modulo 2π phase of Sbxα (l), according The phase of the channel was estimated using ψ
b (l) symmetric, it was shifted to methods proposed in sections 2.4.1 and 2.4.2. In order to make ψ (0) circularly by b−N/2pc. The power spectral density, Sbx (l) was used for amplitude estimation.
b = |Sbx(0) (l)|1/2 exp( jb H(l) φ(l))
(2.56)
Finally, the impulse response was computed as: b b h(n) = IFFT(H(l)).
(2.57)
The performance was measured using and the Overall Normalized Mean Square Error (ONMSE) and also Bit Error Rate (BER). We tested a total of Mh different channels. For each channel we performed Mc Monte Carlo runs corresponding to independent input realizations. The ONMSE was defined as 1 ONMSE = Mh
j 1 Mc ∑Ln=0 |b hi (n) − h j (n)|2 ∑ ∑ ∑L |h j (n)|2 n=0 j=1 Mc i=1 Mh
(2.58)
j where hˆ i is the estimate of the i-th Monte Carlo run of the j-th channel.
Based on each channel estimate we implemented a zero forcing equalizer and obtained the input estimate. The BER rate was computed for each channel and then averaged over the different channels tested.
26 2.5.1
FIR Channels
Each channel was taken to be FIR of length 8, and its coefficients were generated randomly. For each channel we performed Mc = 25 Monte Carlo runs, using the estimation method of Section 2.4.1. Finally the ONMSE and BER were obtained as the average over Mh = 100 different channels. For comparison purposes, we also implemented the nonparametric method (cepstrum based) of [36], where the phase of the channel is estimated using the Fourier Series expansion of the unwrapped phase of the SCD. The estimate of Rbαx (ν) was obtained with a rectangular windowing of length 2L as well as
without windowing.
The obtained estimate, usually contains more than one replicas of the channel response. Although the precise reason for such behavior needs to be further investigated, simulations suggest that this is a result of phase errors and the way they propagate to the final solution. For large N, i.e., N > 3L, there is enough room for the replicas to be separated. One could then extract the channel by identifying the segment with maximum energy, with length equal to the anticipated channel length, or length equal to the window size used in the equalizer. There will be at least one such disjoint segment in the estimate where the energy will be distinctly greater than the rest of the estimate. In the proposed method we chose N = 131 and p = 4. In the simulations we extracted the segment with maximum energy of length L, and we have given results in Fig. 2.6. ONMSE results are shown in Fig.2.6(a). From Fig. 2.6(a), we see that the windowing used to estimate Rbαx (ν), was not that critical in the final estimate. Since the window
requires some knowledge of channel length information, the above observation confirms that the methods tested here are not very sensitive to the channel length information. Corresponding BER results are shown in Fig. 2.6(b). Here we used channel length knowledge in order to compute the Zero Forcing equalizer. The results were obtained for 1000 symbols. The above results suggest that the proposed approach is only slightly better than that of [36] in terms of ONMSE and has same performance in BER. However, in terms of complexity, the proposed approach is less intensive than that of [36], since it requires no phase unwrapping. The matrix A
27 is independent of the data, hence we only need its inverse and moreover, the inverse only has ones and zeros. Hence solving equation (2.38) does not involve multiplications. In contrast, the cepstrum based method [36] requires phase unwrapping and two additional FFTs.
2.5.2
FIR Channels with Pulse Shaping
We generated channels of the form h(t) = a1 c(t + 1.5T, 0.11) + a2 c(t − T1 , 0.11) + a3 c(t − 1.5T, 0.11) W6T (t)
(2.59)
where a1 , a2 , a3 are normally distributed random numbers with mean zero and variance one and T1 is uniformly distributed in [−T, T ]. The shaping function c(t − τ, β) is a raised cosine [56] with delay τ and rolloff β, and W6T (t) is a rectangular window of width 6 symbol intervals T . We here considered Mc = 25, Mh = 100, N = 128 and p = 4. We compared the method proposed in section 2.4.2 with the high complexity subspace based method of [48]. In Fig. 2.7 we have given ONMSE and BER results obtained with perfect channel length knowledge, and in Fig. 2.8 we have given results where the channel length was overestimated by 1 symbol (4 samples). When perfect channel length knowledge is available, the method of [48] performs exceptionally well for high SNR, while the proposed method performs slightly better for low SNR. On the other hand, the proposed approximate method outperforms the former when there is a slight mismatch in channel length estimation. To check whether the channels given in (2.59) satisfy (2.41) we can do the following. If we oversample by 4 (p = 4, T = 1), we see that L = 24. Moreover, Lc ≈ 20, and for N = 128 (2.41) gives Nz > 16.5. By evaluating the zeros of the channel we find Nz = 18, indeed satisfying (2.41). Finally, we investigated the effect of rolloff to the error in approximation. We generated 30 channels of the form (2.59) with variable rolloff. We varied the rolloff from 0 to 0.4, and performed 10 Monte Carlo simulations for each value of the rolloff applying the proposed method. From Fig. 2.9 we see that the ONMSE is at acceptable levels for rolloffs below 0.3.
28 2.6
Experimental Results We also used experimentally obtained data to verify the performance of the proposed method.
Since the real channels are bandlimited we applied the method proposed in Section 2.4.2. The experimental set-up included an RF transmitter/receiver pair, in the indoor environment of the Communications and Signal Processing Laboratory of Drexel University. The equipment consisted of an Agilent ESG 4431B Vector Signal Generator (250 kHz to 6.0 GHz), an Agilent VSA 89640 Vector Signal Analyzer (dc to 2.7GHz), a VSA 89640 Analyzer software on a laptop, and two omni-directional antennae. The test data consisted of a known sequence of 1088 4-QAM modulated data symbols. The Vector Signal Generator uses a Fujitsu MB86060 D/A converter chip and implements square root raised cosine pulse shaping with rolloff 0.08. The data was transmitted at a frequency of 2.4 GHz and at a data-rate of 12 Msps. This sequence was transmitted repeatedly in a continuous mode. The receiver local oscillator was synchronized with the transmitter local oscillator by connecting the Reference clock-out of the transmitter to the reference clock-in of the receiver. To evaluate the obtained channel estimate we compared against the channel estimated based on a training approach. In the training approach we estimated the channel by cross correlating the known input sequence with the received signal. When implementing the proposed methods we used N = 196 and p = 4. The obtained results are give in Fig. 2.10. We also equalized the received signal using both channels, as seen in Fig. 2.11. We see that the results have close agreement.
2.7
Conclusions In this chapter, we presented a novel method for blindly identifying a linear time-invariant sys-
tem excited by cyclostationary input, within a scalar ambiguity and an unknown delay. The basic information used is the appropriately discretized modulo-2π phase of the cyclic spectrum of the system output. No system length information is needed, and no phase unwrapping is required.
29 We proposed a general method when the channel has full bandwidth and, considering the inherent inability of frequency domain methods to work when the channel is bandlimited, proposed an approximate method for bandlimited channels. Simulation results show that they offer competitive cost-performance tradeoffs compared to more complex alternatives.
30
−0.3
10
with windowing without windowing
−0.4
ONMSE
10
−0.5
10
Proposed Cepstrum based
−0.6
10
−0.7
10
0
5
10
15
20
25
30
35
SNR/(dB)
(a) 0
10
with windowing without windowing
BER
Proposed Cepstrum based
−1
10
−2
10
0
5
10
15
20
25
30
35
SNR/(dB)
(b) Figure 2.6: (a) ONMSE and (b) BER for cepstrum based estimates and proposed estimates given in section 2.4.1. We generated 100 random FIR channels of length 8 and for each channel, obtained the estimates for 25 Monte Carlo runs. 1000 symbols were used in channel estimation and perfect channel length knowledge was assumed. In (a), we have given the ONMSE results where both a rectangular window of length 2L was used, and a window was not used.
31
0
10
Proposed Subspace
−1
ONMSE
10
400 Symbols 1000 Symblos
−2
10
−3
10
0
5
10
15
20
25
30
35
SNR/(dB)
(a) 0
10
Proposed Subspace −1
BER
10
−2
10
400 Symbols 1000 Symblos −3
10
−4
10
0
5
10
15
20
25
30
35
SNR/(dB)
(b) Figure 2.7: (a) ONMSE and (b) BER for the method proposed in section 2.4.2 and for the subspace method. We generated 100 channels with pulse shaping and for each channel, 25 Monte Carlo runs were taken at each SNR. The modulation scheme used was 4 QAM and perfect channel length knowledge was used for equalization.
32
0
10
ONMSE
Proposed Subspace
−1
10
400 Symbols 1000 Symblos
−2
10
0
5
10
15
20
25
30
35
SNR/(dB)
(a) 0
10
BER
Proposed Subspace
−1
10
400 Symbols 1000 Symblos
−2
10
0
5
10
15
20
25
30
35
SNR/(dB)
(b) Figure 2.8: (a) ONMSE and (b) BER for the method proposed in section 2.4.2 and for the subspace method. We generated 100 channels with pulse shaping and for each channel, 25 Monte Carlo runs were taken at each SNR. The modulation scheme used was 4 QAM and the channel length was overestimated by one symbol interval.
33
0.25
0.2
ONMSE
0.15
0.1
0.05
0 0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
roloff
Figure 2.9: Variation of the ONMSE with rolloff for the channels given in equation (2.59).
channel obtained using training estimated channel 0.8
Real Imaginary
h(n)
0.6
0.4
0.2
0
−0.2 2
4
6
8
10
12
14
n
Figure 2.10: Channel obtained using experimental data. Lines with circles gives the channel obtained by training. Lines with stars gives the mean of the estimated channel using 400 symbols. Solid lines give the real part while broken lines give the imaginary part of the channel.
34
Scatter plot
1
Quadrature
0.5
0
−0.5
−1
−1
−0.5
0 In−Phase
0.5
1
(a) Scatter plot
Scatter plot
2 1
1.5
1 0.5
Quadrature
Quadrature
0.5
0
0
−0.5 −0.5 −1
−1.5
−1
−2 −2
−1.5
−1
−0.5
0 In−Phase
(b)
0.5
1
1.5
2
−1
−0.5
0 In−Phase
0.5
1
(c)
Figure 2.11: Equalized data: (a) Received constellation (b) Equalized constellation using channel obtained via training and (c) Equalized constellation using the estimated channel. 400 symbols were used in the estimation and 2000 symbols were used in the equalization.
35
Figure 2.12: Signal generator
Figure 2.13: Vector signal analyser.
36 Chapter 3. Orthogonal Frequency Division Multiplexing in Multiuser Communications
3.1
Orthogonal Frequency Division Multiplexing Let us consider the communication system with s(n) and y(n) the discretized input and output.
As we have seen in chapter 2 a muptipath channel results in the convolutive
x(ν) = h(ν) ⊗ s(ν) + w(ν)
(3.1)
output as in (3.1) causing inter symbol interference. The key behind Orthogonal Frequency Division Multiplexing OFDM is to eliminate this ISI by converting the convolutive channel into a set of scalar channels. Basically three steps are performed to accomplish this. First, instead of transmitting the data as a continuous stream, it is divided into blocks and transmitted block wise. Moreover, instead of transmitting the blocks as they are, a guard interval or an overhead is added to each block. As we see later, this eliminates the ISI. Furthermore, the data in each block is not the plain data but transformed data using the Fourier transform. We have illustrated a typical OFDM modulator in Fig. 3.1. Let N be the OFDM block length and let i be the block number. Let us consider the processing of the i th block. First, we take the IFFT of the block as given in (3.2). After taking the IFFT, we add a cyclic prefix of length P to each block. We should also note that OFDM systems exist where instead of adding a cyclic prefix, a set of P zeros are added as the guard interval. Let x be the N by 1 vector obtained by taking the IFFT of the symbol vector s.
s˜ i = F H si
(3.2)
where F is the N by N FFT matrix, s˜i = [s˜i (0), . . . , s˜i (N − 1)]T and si = [si (0), . . . , si (N − 1)]T . Let us assume the channel length to be L. At the receiver, if we consider the N + P symbols
37
F H si
si
s(n) S/P
IFFT
CP
P/S
S/P : Serial to Parallel P/S : Parallel to Serial CP : Cyclic Prefix
Figure 3.1: An OFDM transmitter
received which are xi (−P), xi (−P + 1), . . . , xi (N − 1), we have
xi (−P)
(N+P)×1
}
xi (−P + 1) .. . xi (−2) xi (−1) xi (0) xi (1) .. . xi (N − 1) | {z
h0 ... ... 0 hL−1 hL−2 . . . hL−1 hL−2 . . . h0 . . . 0 0 = 0 0 hL−1 hL−2 . . . h0 0 0 ... . . . hL−1 hL−2 . . . h0 | {z (N+P)×(N+P+L−1)
i−1 (N − L + 1) s ˜ s˜i−1 (N − L + 2) . .. s˜i−1 (N − 1) i s ˜ (N − P) s˜i (N − P + 1) (3.3) × . . . s˜i (N − 1) } s˜i (0) i s ˜ (1) .. . i s˜ (N − 1) | {z }
(N+P+L−1)×1
38
xi
F xi
yi (k) S/P
FFT
CP
P/S
S/P : Serial to Parallel P/S : Parallel to Serial CP : Remove Cyclic Prefix
Figure 3.2: An OFDM receiver
We have given a diagram of an OFDM demodulator in Fig. 3.2. Before any processing, the cyclic prefix of length P is discarded from each block. If P ≥ L by discarding first P symbols received we get
xi (0) xi (1) .. . xi (N − 1)
=
hL−1 hL−2 0
...
hL−1 hL−2
0
0
0
...
h0
...
...
0
...
h0
...
0
...
h0
0
hL−1 hL−2 ...
hL−1 hL−2 . . . h0
×
s˜i (N − L − 1) s˜i (N − L) .. . s˜i (N − 1) s˜i (0) s˜i (1) .. . s˜i (N − 1)
(3.4)
39 which can be written as
h0
0
...
...
h1 h0 0 ... i . x (0) .. ... ... ... i x (1) = hL−1 hL−2 hL−3 . . . .. . 0 hL−1 hL−2 hL−3 i . x (N − 1) . ... ... ... . 0 0 ... ...
hL−1 hL−2 0
...
. . . h1
hL−1 hL−2 . . . h2
...
...
...
... ...
...
...
...
... ...
...
...
...
... ...
...
...
...
... ...
0
hL−1 hL−2 . . . h0
s˜i (0) s˜i (1) × .. . s˜i (N − 1)
(3.5)
˜ si xi = H˜
(3.6)
˜ is an N by N circulant matrix with the first column being [h0 , . . . , hL−1 , 0, . . . , 0]T . Next, where H we take the FFT of the received block xi and we have
˜ F H si yi = F xi = F H
(3.7)
˜ is circulant, we have where yi = [yi (0), . . . , yi (N − 1)]T . Since H ˜F FH
H
= diag[H(0), . . . , H(N − 1)]
(3.8)
where H(k) is the k the component of the N point FFT of the channel L−1
H(k) =
∑ hne− j2π
kn N
(3.9)
n=0
Due to the diagonal structure of H we have
yi (k) = H(k)si (k) + n(k), k ∈ [0, N − 1] where we have included the noise term n(k).
(3.10)
40 Thus we have effectively reduced the convolutive channel given in (3.1) into a set of scalar channels given in (3.10). This is the major advantage of OFDM because (3.10) is simpler to equalize.
3.2
Multiuser OFDM Systems The previous section covered OFDM system with one transmitter and one receiver. Next we
consider multiple transmitters and receivers as given in Fig. 3.3. We have MT transmitters and MR receivers and MT × MR channels as seen in Fig. 3.3.
d1i
precoder si1
h11
d2i
si2
h12
h21
d3i
si3
h13
h31
h1MT i dM T
siMT
yi1 yi2 yi3
hMR 1 hMR MT
yiMR
Figure 3.3: An MT user OFDM system with MR receiving antennae. The data is precoded before transmission.
Using superposition of (3.10), we can express the signal received at the m-th receive antenna during the i-th block as
yim (k) =
MT −1
∑
p=0
Hmp (k)sip (k) + nm (k), k = 0, ..., N − 1, m = 0, ..., MR − 1
(3.11)
where i is the block index; k is the carrier index; p is the user index; sip (k) is the transmitted symbol over the k-th carrier, which, as will be explained latter, is derived from the source symbols d ip (k) for k = 0, ..., N − 1; Hmp (k) is the gain of the k-th carrier between the m-th receive antenna and the p-th user; nm (k) denotes noise.
41 Similar to (3.9) we can express the channel gain between transmitter Let hmp (n) denotes the symbol rate cross-channel between the m-th receive antenna and the p-th user, then it holds: L−1
Hmp (k) =
∑ hmp (n)e− j
2π N kn
n=0
, k = 0, ..., N − 1
(3.12)
where L (L ≤ Ncp ) is the length of the longest cross-channel. 3.3
Channel Estimation in OFDM Systems We see that from (3.10) that in order to extract si (k) from the received signal yi (k) we need the
knowledge of the channel, H(k). Similarly, in (3.11) for MIMO OFDM systems, we need to find all channels Hmp (k). The various arrangements of OFDM systems and ways of blindly estimating the channels can be given as below.
3.3.1
Single-user Single Antenna OFDM Systems
Each symbol is sent over a different carrier, over which it encounters flat fading. Considering blocks of received and transmitted symbols, one can formulate a MIMO problem, where the channel matrix is diagonal. There is a cornucopia of available methods for blind channel estimation in this case [88][26],[49]. A weakness of single antenna OFDM systems is the lack of multipath diversity. Precoding can spread each symbol over multiple carriers thus increasing diversity. Unitary precoding has been shown to provide maximum spectral efficiency [11], however, it does not allow for blind channel estimation. A non-unitary linear precoding scheme was proposed in [53], which trades-off performance for blind channel estimation.
3.3.2
Single-user OFDM Systems with Multiple Transmit/Receive Antennae
Multiple transmit/receive antennas in combination with coding are used to improve diversity and rate. Some representative schemes include the space time block codes (STBC) [3], space time trellis codes, and layered space time [19]. In this case, the channel matrix is non diagonal. A
42 channel estimation scheme for such MIMO systems is the two-input one-output MIMO OFDM system studied in [42], where STBC was applied at the inputs and the system was estimated by exploiting the structure of the codes. This approach relies on transmission redundancy, i.e., each information symbol is transmitted twice in two consecutive time intervals through two different antennas. In [87], the same setup as in [42] was used, and the channel was estimated based on a subspace approach.
3.3.3
Multi-user OFDM Systems without Multi-user interference
Each user is allocated a disjoint set of subcarriers [73]. Such systems are also referred to as orthogonal frequency division multiple access (OFDMA) and have been adopted in IEEE 802.16 standard. In such systems there is no multi-user interference and the channel matrix is diagonal. Thus, any single-user frequency domain channel estimation method can be applied to this scenario.
3.3.4
Multi-user OFDM Systems with Multi-user Interference
All users use all available subcarriers independently. Each user can employ one or more antenna. Inevitably, in this case there is multi-user interference and the channel matrix is non diagonal. The majority of the single-user OFDM channel estimation methods do not apply to this case. There exists only a few channel estimation schemes for such systems. In [5], an OFDM system with cyclic prefix has been considered. Each block was multiplied with a scalar that varied periodically between blocks. Cyclic statistics of the received blocks, before the removal of the cyclic prefix, were used to yield the channel estimate. This kind of precoding results in variable power between blocks. In [85], a semi blind channel estimation method was proposed for multi-user OFDM systems that employ zero-padding instead of cyclic prefix. We should note here that the channel estimation methods developed for multi-user multi-antenna systems can be applied to most single user multi-antenna systems. At this point we should also mention that OFDM systems still obey the convolutional relationship given in (3.1). Hence, at this point one might wonder why we could not just apply to the
43 OFDM case already existing blind MIMO estimation methods that were proposed in different contexts. In fact, efforts along these lines have been reported. In the method proposed in [21], the input is first passed through an AR filter to add colour. At the receiver, Joint diagonalization of the output covariance [4] is done in order to estimate the channel. We should note that this method is computationally intensive because joint diagonalization [1] as well as deconvolution (to remove colour) has to be done. Higher-order statistics (HOS) based methods (see [7] and reference therein) could also be applied to solve the MIMO OFDM problem. However, such an approach would require high-complexity and would only work in the case of a small number of carriers. As it was also commented in [5], for large number of carriers, the Discrete Fourier Transform (DFT) performed at the OFDM modulation step would Gaussianize the data, thus the corresponding HOS estimates would be almost zero. In addition, HOS estimates have high computational complexity. Since the key idea behind OFDM systems is simplicity of equalization, one should seek simpler solutions for channel estimation that are consistent with the OFDM goals.
3.4
Conclusions In this chapter we have given the basic theory behind OFDM and its application in single user
and multi user as well as SISO and MIMO systems. It is essential that the channel is known at the receiver in order to perfectly decode the transmitted signal, in all these scenarios. In the next chapter, we present a novel blind channel estimation method, that can not only be applied to single user OFDM systems but also for multi user and MIMO OFDM systems. In fact, the proposed method in chapter 4 can be considered as a generalization of [53].
44 Chapter 4. Blind Channel Estimation in Multiuser OFDM Systems Using Precoding
4.1
Linear Precoding in OFDM Linear block precoding (LBP) has been used extensively for increasing throughput and diversity
gains in MIMO OFDM systems [43],[74],[40]. The design of codes to achieve these goals has recently become an area of huge interest and vast potential. For instance, algebraic number theory [17], coding theory [75], and iterative greedy optimization [55] have been used to design optimal codes. However, very little work has been done on developing LBP schemes that allow MIMO OFDM channel estimation. In this chapter we propose a blind estimation method of an MR × MT , (MR ≥ MT ) MIMO OFDM system with MT users and MR receivers, as given in Fig. 3.3. Non redundant linear block precoding is applied at the inputs before they enter the OFDM system, which increases multicarrier diversity and allows for blind channel estimation at the receiver. The proposed channel estimation approach employs computationally simple cross-correlation operations and yields the channel up to a diagonal ambiguity. It does not require channel length information, and is not sensitive to additive stationary noise. The precoding does not increase transmission power and unlike the method of [5] maintains even distribution of power between OFDM blocks. We provide the general description of precoding matrices. We also provide analytical expressions of symbol error probability and signal to interference ratio, which could be used to obtain optimum precoding schemes. The proposed method can be extended to multi-user STBC systems.
4.2
Linear Precoding for Blind Channel Estimation p=0,...,M −1
Our goal is to estimate the channel matrix H(k) = {Hmp (k)}m=0,...,MTR −1 , based on the received symbols. However, due to the fact that L N, in order to recover hmp (n), n = 0, ..., L − 1, we only need to estimate H(k) for L distinct values of k. On the other hand, in the presence of noise, our channel estimate would improve with increasing number of frequency domain samples estimates.
45 The best choice depends both on the available computational power and the accuracy of channel estimate required. In the proposed method, the trade off between computation and accuracy can be changed at the discretion of the user because we estimate channel matrices for each k individually and they all have the same ambiguity as shown later. Before we proceed, we make the following assumptions: (A0) We consider a synchronous communication system. (A1) Each user emits i.i.d. symbols from independent digitally modulated sources, with statistics: E{d ip (k)(dq (m))? } = δkm δi j δ pq σ2 , and E{|d ip (k)|4 } = η4 . j
(A2) The noise processes are white, circularly Gaussian, temporally and spatially uncorrelated and are independent of the data. It is assumed that E{n p (k)(nq (m))? } = δkm δ pq σ2n . (A3) The channel is considered to be slowly varying with the block index i. In particular, it is assumed to be invariant during the number of blocks that are necessary for good channel estimation. (A4) For channel identification, we assume there is a non-empty set of k’s, i.e. L , so that for k ∈ L , H(k) has at least one row full of non-zero elements.
For recovery of the transmitted symbols, we assume the channel matrices H(k) to be full rank for all k. The u-th row of H(k) represents the k-th carrier gains experienced between the u-th receive antenna and all sources. Assumption (A4) requires that for k ∈ L , there is some value of u so that all these gains are different than zero. 4
Let dip = [d ip (0) ... d ip (N − 1)]T denote the i-th block of symbols corresponding to the p-th user before any precoding. During the i-th block, the symbols of the p-th user to be transmitted over
46 carrier k are generated based on an N × 1 code vector, w p (k; i), as: sip (k) = wHp (k; i)dip
(4.1)
Equivalently, the i-th block of the p-th user is precoded according to: 4
sip = [sip (0) ... sip (N − 1)]T = [w p (0; i), w p (1; i), ...., w p (N − 1; i)]H dip = Wip dip
(4.2)
where Wip is a matrix whose k-th row equals wHp (k; i). The purpose of the precoding matrix is to introduce some correlation structure in the transmitted blocks. Also, by changing the coding matrix between subsequent blocks we will create diversity that will allow us to estimate the channel matrix. Since we will need to obtain autocorrelation estimates, the coding matrices have to stay the same for a number of blocks. Thus, in the following, we will take them to be periodic in i with period M, M ≥ MT . i.e., Wip = Wi+M p , ∀i, p
(4.3)
The selection of those values will be discussed in Section 4.6.
4.3
Blind Channel Estimation For the i-the received block, let us gather the symbols of all users that were received on carrier
k in vector yi (k). Then, based on (4.2) it holds: 4
yi (k) = [yi0 (k) ... yiMR −1 (k)]T = H(k)si (k) + n(k) 4
where si (k) = [si0 (k) .... siMT −1 (k)]T .
(4.4)
47 Based on (4.1), (4.4) can be written as:
yi (k) = H(k)Ψi (k)di + n(k)
where
4 i Ψ (k) =
(4.5)
wH 0 (k; i)
0
...
0
0 .. .
wH 1 (k; i)
... .. .
0 .. .
0
0
.. .
. . . wH MT −1 (k; i)
(4.6)
4
and di = [(di0 )T , (di1 )T , . . . , (diMT −1 )T ]T . For a fixed k, estimating H(k) in (4.5) is a typical MIMO channel estimation problem. We will next exploit the periodicity of the precoding matrices to estimate the channel in a computationally simple manner. Let us consider a sequence of received symbols that start at block i and are spaced apart by M blocks: y˜ i (k) = {yi (k), yi+M (k), yi+2M (k), . . .}
(4.7)
Due to (4.3), the precoding matrix corresponding to all blocks of y˜ i (k) will be the same. Consider the correlation matrix of y˜ i (k), i.e., 4
Rikl = E{˜yi (k)˜yi (l)H }, i = 0, ..., M − 1
(4.8)
We present the main result as the following proposition. Proposition 2 Let l ∈ L , and u such that the elements of HH (l)eu are all different than zero. Define
Ukl = 4
wH 0 (k; 0)w0 (l; 0) .. .
··· .. .
wH MT −1 (k; 0)wMT −1 (l; 0) · · ·
wH 0 (k; M − 1)w0 (l; M − 1) .. .
wH MT −1 (k; M − 1)wMT −1 (l; M − 1)
(MT × M) (4.9)
48 and 4
Cukl =
1 0 M−1 [R eu , R1kl eu , ..., Rkl eu ] σ2 kl
(MR × M).
(4.10)
If Ukl is a full row rank matrix, then it holds: 4 ˆ lu (k) = H Cukl U†kl
= H(k)Diag(HH (l)eu ) + δk,l σ2n eu 1TMT U†kl .
(4.11) (4.12)
The proof is given in Appendix A.1.
Remarks ˆ lu (k), l ∈ L equals the true channel matrix H(k) within a The above proposition indicates that H diagonal ambiguity matrix, with a bias term appearing for k = l. In the subsequent section we will propose ways to resolve the diagonal ambiguity. It is interesting to note that the noise contribution affects H(l) only. Assuming that N > L, there is enough redundancy in the frequency domain, to recover that particular sample based on the other recovered samples. We should note here that channel length information was not needed for obtaining Hlu (k). The full row rank requirement on Ukl guarantees that U†kl exist. A necessary condition for that is M ≥ MT . However, increasing M means increasing the number of correlations to estimate, thus requiring longer data samples. Hence the best possible choice for M would be MT . Let us also consider the complexity of channel estimation using Proposition 2. Calculating the inverse U†kl has complexity O(M 3 ) because M ≥ MT . In order to form Cukl , we need to estimate Rikl first. Since we only need one column of Rikl , the complexity of estimating correlations (for i = 0, . . . , M − 1) is O(MMR b MI c) = O(MR I) where I is the total data blocks used in estimation. Next, the matrix product Cukl U†kl requires O(M 2 MR MT ) operations. Hence the total complexity is O(M 3 + MR I + M 2 MR MT ) computations per carrier k. If we are estimating k for all carriers 0 to N − 1, the complexity increases by N times. However, we should note that the above analysis is for a generic case only. Depending on the precoding, we might have special structures for Ukl i.e.
49 diagonal, which lowers the complexity. Moreover, we need not estimate H(k) for all k because in time domain the channel length is L, and L N. The precoding and decoding operations require O(N 3 ) operations per information block. However, this is greatly reduced if the precoding matrices are sparse.
4.4
Resolving the Diagonal Ambiguity The diagonal ambiguity involves the l−th carrier gains across all channels between the u-th
receive antenna and all sources. There are two approaches that we can use to eliminate this diagonal ambiguity.
4.4.1
Resolving Ambiguity Based on Pilots
We can transmit pilots on the l-th carrier, based on which we can estimate H(l) directly. Note that in general, it is not possible to transmit pilots when precoding is done. Hence, we need to transmit pilots in a specific set of blocks corresponding to indices {p1 , p2 , p3 , . . . , pK }, that will not be precoded, neither will be used in the estimation of correlations. Based on these blocks we can estimate H(l) as:
ˆ H(l) = [y p1 (l), y p2 (l), . . . , y pK (l)][s p1 (l), s p2 (l), . . . , s pK (l)]† .
(4.13)
Finally, based on (4.12) the ambiguity can be resolved entirely. Moreover, the estimates corresponding to different values of u (see (4.10)) can be combined together to further improve the channel estimate as follows. For l ∈ L define 4 ˜ kl = C [C0kl , C1kl , . . . , C(MR −1)kl ] (4.14) = H(k) Diag[HH (l)e0 ]Ukl , Diag(HH (l)e1 )Ukl , . . . , Diag(HH (l)eMR −1 )Ukl (4.15)
˜ kl . = H(k)U
(4.16)
50 ˜ kl will have full rank. According to (A4), there will be at least one u with all nonzero HH (l)eu and U Thus, an estimate of H(k) can be obtained as: b ˜H ˜ ˜ H −1 H(k) = C˜kl U kl (Ukl Ukl ) . 4.4.2
(4.17)
Resolving Ambiguity in a Blind Fashion
We can also reduce the diagonal ambiguity into a unit modulus diagonal ambiguity without the need of pilots. Let 4
ˆ † (m)˜yi (m), m ∈ [0, N − 1] z˜ ilu (m) = H lu
(4.18)
and the correlation of z˜ ilu (m), 4
Qilu (m) = E{˜zilu (m)˜zilu (m)H }
(4.19)
for some fixed index m ∈ [0, N − 1], u ∈ [0, MR − 1]. It is shown in Appendix A.2 that: (Qilu (m))−1/2 (Diag(wimm ))1/2 = |Diag(HH (l)eu )|
(4.20)
H T wimm = [wH 0 (m; i)w0 (m; i), ..., wMT −1 (m; i)wMT −1 (m; i)] .
(4.21)
where
Thus, the magnitude of the diagonal ambiguity matrix of (4.12) can be computed via (4.20). Note that the value of |Diag(HH (l)eu )| in (4.20) is independent of m and i. Hence, we can select any m ∈ [0, N − 1] and i ∈ [0, M − 1] for this evaluation. For better accuracy, this can be evaluated as the average corresponding to different values of m and i. Finally, the channel matrix can be obtained within a constant unit modulus diagonal ambiguity, i.e., 4 ˜ H(k) = Cukl U†kl (Qilu (m))1/2 (Diag(wimm ))−1/2
= H(k)Diag[e− j arg Hu0 (l) , . . . , e− j arg HuMT −1 (l) ].
(4.22) (4.23)
51 Based on the obtained channel, the transmitted symbols of each user can be recovered within a phase ambiguity, or equivalently, within a rotation. The rotation can be resolved by transmitting at least one training symbol per user.
4.5
Implementation Issues The cross correlation Rikl in (4.8) can be estimated based on I received blocks as follows:
ˆ ikl = R
1
b I−i−1 M c
b I−i−1 M c+1
j=0
∑
yi+ jM (k)(yi+ jM (l))H .
(4.24)
In the channel estimate given in (4.11), the diagonal ambiguity involves the cross channels Hu0 (l) to Hu(MT −1) (l), hence l and u must be selected so that those values are different than zero. Since channel information is not available, the selection of those values will be based on information that is available at the receiver. ˆ lu (l) according to ˆ i for all possible values of i and l, and let us estimate the H Let us consider R ll Proposition 2 for all values of u ∈ [0, MR − 1]. We have ? ? ˆ lu (l) ≈ H(l)Diag[Hu0 H (l), . . . , HuM (l)] T −1
(4.25)
ˆ lu (l) ≈ [|Hu0 (l)|2 , . . . , |HuMT −1 (l)|2 ]. eTu H
(4.26)
and subsequently,
ˆ lu (l). Therefore, we can Thus we can estimate the magnitudes of Hu0 (l) to HuMT −1 (l) using eTu H select l and u such that |Hu0 (l)|, ...., |HuMT −1 (l)| are all non zero. If there are more than one choices for those values, we pick those for which the minimum possible value of |Hu0 (l)|, ....|HuMT −1 (l)| is maximized. We should also note that the freedom in selecting l relies on the matrices Ukl being invertible for all k for which H(k) needs to be estimated i.e., at least L such values if channel length is L. If we do not have freedom to choose l, we can only vary u. Notwithstanding this, the method will fail
52 if the channel has a deep fade on l, i.e. kHlu (l)k ≈ 0. 4.6
Selection of Precoding Matrices Let us first consider the criteria for the design of the precoding matrices. • Identifiability: In order to obtain the estimate of (4.12), Ukl must be full rank for a given l and k ∈ [0, N − 1]. We see that on any row of Ukl , the elements are derived from the precoding matrices of one source, and, on any column of Ukl , the elements are derived from precoding matrices of one block number. Hence, in order to have full rank, each source must vary the precoding matrices with the block number such that the variation of the inner product between the k-th row and the l-th row of the precoding matrices is unique. This can be compared with the precoding scheme employed in [5] where the authors use precoding unique in the cyclostationary domain for each antenna. • Decodability: Once obtaining the channel estimate and equalizing, we need to obtain the original decoded symbols. Hence Wip , p ∈ [0, MT − 1] must be full rank. • Power Conservation: The precoding scheme should keep the power constant as for an uncoded OFDM system. Hence,
trace(Wip (Wip )H ) =
σ2o N, ∀i ∈ [0, MT − 1] σ2
(4.27)
where σ2o is the symbol power in an uncoded OFDM system. We should note that there are many possible schemes that satisfy the above conditions. For an MR by MT system with M periodic precoding, we may need at most MT × M precoding matrices.
53 However, we can reduce this to MT matrices instead, by using the scheme given in Table 4.1, where we have made M = MT , the lowest possible value. Table 4.1: Assignment of precoding matrices with block number and source
source p 0 1 .. .
0 W0 WMT −1 .. .
1 W1 W0 .. .
block i . . . MT − 1 . . . WMT −1 . . . WMT −2 .. .. . .
MT − 1
W1
W2
...
MT W0 WMT −1 .. .
... ... ... .. .
W1
...
W0
In the scheme given in Table 4.1, we have used matrices W0 to WMT −1 between the sources in a cyclic manner. Hence, in this scheme, the matrices Ukl become circulant and for identifiability, the MT point DFT of any row of these matrices should have non-zero coefficients. An Example of Precoding Matrices Let us consider an extension of the single-user OFDM precoding scheme of [53], [77], where a linear combination of two symbols are transmitted on each carrier. We first select a fixed m from [0, N − 1]. Let us derive W0 based on its rows: wH 0 (k; i) =
αvH (k) + (−1)k βvH (m), k ∈ [0, N − 1], k 6= m γvH (m),
(4.28)
k=m
where vH (k) is the k-th row of an arbitrary unitary matrix V and α, β, γ ∈ C. Let us select the remaining MT − 1 matrices W1 to WMT −1 to be unitary, for instance, we can select them such that W p = IN , p ∈ [1, MT − 1]
(4.29)
In this scheme we only precode the first block and the remaining M − 1 blocks of each period are transmitted uncoded.
54 The channel H(k), k = 0, ..., N − 1 can then be estimated according to Proposition 1 based on
βγ?
0 Ukm = (−1)k . . . 0
0 βγ?
0
...
0
|γ|2
1 0 , k = 6 m, U = mm .. .. . . 1 . . . βγ?
1 |γ|2 ...
...
...
... ... . .. . . . . |γ|2
(4.30)
For identifiability, we need |γ|2 6= 1, β 6= 0, γ 6= 0,
(4.31)
and for block power conservation we need
|γ|2 + (N − 1)(|α|2 + |β|2 ) = N.
(4.32)
In spite its simplicity, this scheme does not allow us to choose the value of l at the receiver, since we must take l = m. Hence, if H(m) has no rows with all elements non zero, the method will fail. It is interesting to note that in this scheme, we can directly transmit pilots on the m-th carrier of the precoded blocks because symbols on those carriers are only scaled, i.e. no linear combination of symbols is performed.
4.7
Multiuser OFDM Systems with Space Time Coding In this section, we consider adaptation of the proposed method to suit Space-Time Block Coded
systems. Starting with the seminal work by Alamouti [3], STBC have been adopted from a symbol level implementation to a block level implementation in numerous ways. In this section we consider STBC pertaining to OFDM, as given in [42]. Let us first consider a single user STBC OFDM system, without any precoding. The transmitted blocks for each of the two antennae will be as given in Table 4.2 for a single user system.
55
Table 4.2: STBC for a single user system time 2i 2i + 1
antenna 0 d2i −(d2i+1 )?
antenna 1 d2i+1 (d2i )?
s0 y0
d0
s1 y1 s2
d1 s3
Figure 4.1: Block diagram of a 2 user STBC system. Each user will implement STBC independently.
At the receiving antenna, after OFDM demodulation, we have
y2i (k) (y2i+1 (k))?
=
H00 (k) (H01
(k))?
H01 (k) −(H00
(k))?
d 2i (k) d 2i+1 (k)
+ n(k)
(4.33)
which can be considered as a 2 by 2 multi-user OFDM system provided the data vectors d2i and d2i+1 are mutually uncorrelated. Extension of the above scheme for more than one user have appeared in [44] and references therein. Using this result, we can generalize the precoding scheme proposed in section II to a multi-user STBC OFDM system. However, for simplicity, we only consider the two user case. We pursue a combined linear precoding and STBC scheme as given in Table 4.3, which is the combined application of the schemes of Tables 4.1 and 4.2. The precoding cycle here repeats every 8 blocks. At the receiver, after OFDM demodulation, we have
yiST (k) = H(k)siST (k) + n(k), i = 0, 1, . . . , k ∈ [0, N − 1]
(4.34)
56
Table 4.3: Space-time coding for a single and 2 user system
time 0 1 2 3 4 5 6 7
source 1 s0 W0 d00 −(W3 d10 )? W1 d20 −(W0 d30 )? W2 d40 −(W1 d50 )? W3 d60 −(W2 d70 )?
source 2
s1 W3 d10 (W0 d00 )? W0 d30 (W1 d20 )? W1 d50 (W2 d40 )? W2 d70 (W3 d60 )?
s2 W2 d01 −(W1 d11 )? W3 d21 −(W2 d31 )? W0 d41 −(W3 d51 )? W1 d61 −(W0 d71 )?
s3 W1 d11 (W2 d01 )? W2 d31 (W3 d21 )? W3 d51 (W0 d41 )? W0 d71 (W1 d61 )?
2i+1 2i+1 where yiST (k) = [y2i (k))? , y2i (k))? ]T , siST (k) = [si0 (k), si1 (k), si2 (k), si3 (k)]T and 0 (k), (y0 1 (k), (y1
H00 (k) H01 (k) H02 (k) H03 (k) (H01 (k))? −(H00 (k))? (H03 (k))? −(H02 (k))? H(k) = H11 (k) H12 (k) H13 (k) H10 (k) (H11 (k))? −(H10 (k))? (H13 (k))? −(H12 (k))?
.
(4.35)
If di (see (4.5)) are independent for any given i, (4.34) can be considered as a 4 by 4 multiuser system and Proposition 2 can be applied to estimate H(k). However, (4.35) contains only 8 unknowns. Consider the reduced channel matrix:
H00 (k) H01 (k) H02 (k) H03 (k) ˜ H(k) = . H10 (k) H11 (k) H12 (k) H13 (k)
(4.36)
Since we have more equations than unknowns, we can obtain a least squares channel estimate as follows. Let us for some fixed u ∈ [0, 3], define
4 ˜ v (k, l) = R
R0kl (0, v) R0kl (2, v)
...
R3kl (0, v)
(R0kl (1, v))?
...
R3kl (2, v)
(R0kl (3, v))?
...
(R3kl (1, v))?
...
(R3kl (3, v))?
, v ∈ [0, 3]
(4.37)
57 to be a 2 by 16 matrix constructed from the elements of R0kl to R3kl in (4.8). The u, v element of matrix Rikl is given by Rikl (u, v). Let us also construct the matrices D0 = Diag[(H00 (l))? , (H01 (l))? , (H02 (l))? , (H03 (l))? ], D1 = Diag[H01 (l), −H00 (l), H03 (l), −H02 (l)], D2 = Diag[(H10 (l))? , (H11 (l))? , (H12 (l))? , (H13 (l))? ], D3 = Diag[H11 (l), −H10 (l), H13 (l), −H12 (l)]. Let us assume that H(l) is available through pilots. In Appendix A.3 we show that 4 [ ˜ ˜ v (k, l) [Dv Ukl PD?v U?kl ]† , k 6= l H(k) = R
˜ = H(k)
with
(4.38)
0 −1 0
1 P= 0 0
0 0 0
0
0 0 . 0 −1 1 0
(4.39)
The selection of l and u can be done according to section 4.5.
4.8
Simulation Results We next give test the performance of the proposed blind channel estimation method via simula-
tions. We consider a 2 by 2 multiuser system without and with space time coding in Sections 4.8.1 and 4.8.2, respectively. In Section 4.8.3, we provide comparisons with the method of [5]. In all cases we used 4-QAM sources without channel coding, unless stated otherwise. Once the channel was estimated, the symbols were recovered using zero forcing equalizer.
4.8.1
2 by 2 Multiuser System
We considered a time-varying 2 × 2 multi-user system. Each cross-channel was taken to be of length 3, and was generated based on the modified Jakes Model [12]. This model gives uncorrelated
58 channel taps. For our simulations we considered normalized Doppler frequencies fd 10−5 (fast fading), and 10−6 (slow fading). We generated 75 independent 2 × 2 channels, and for each channel realization we applied the proposed blind channel estimation method for 10 independent input realizations. The 75 channels were selected so that they all satisfied the non-singularity assumption, i.e., max(cond(H(k))) ≤ 10 1 We should note that the latter condition is not required for channel estimation but rather for symbol recovery. In practice, since forward error control is used, the channels do not have to satisfy that condition. For precoding we used the scheme of Section (4.6). We took W1 = I as in (4.29), and constructed W0 as in (4.28) with V = I and parameters α = j0.9, β = 0.432, γ = 1.1. The index m needed in (4.28) for precoding was chosen arbitrarily at m = 2 for all channels. The OFDM block length was N = 64 with a cyclic prefix of 8. The blind channel estimation method was applied as described in Proposition 2, followed by (4.22). In order to determine the magnitude of the diagonal ambiguity, (4.18) was evaluated for all carriers [0, N − 1] and then averaged. The residual constant phase ambiguity was resolved based on N/2 pilot symbols transmitted before the real data transmission. The obtained channel estimate was improved via denoising where perfect channel length information was assumed. The number of data blocks used in our simulation were 300 for the slowvarying case and 100 for the fast-varying case. These numbers resulted in the best performance for the proposed method. The average BER obtained based on all channel and input realizations is shown in Fig. 4.2 for different SNR levels. We should mention here that, using (4.17) for resolving the magnitude of the diagonal ambiguity did not change much the performance obtained with the blind method. The advantage of a blind method as opposed to a training-based one is less overhead. To see how much we would save in terms of overhead if we were to use the proposed blind method as opposed to a training based method and achieve the same BER performance, we conducted simulations as 1 cond(H(k))
= kH(k)ks kH† (k)ks , where k.ks is the spectral norm.
59 follows. For the training based method, we transmitted 2 blocks with known BPSK symbols, followed by 100 blocks unknown 4 − QAM symbols for the fast varying case. We will refer to the 100 blocks as one frame. The channel was estimated according to (4.13). Note that at least two blocks of pilots have to be transmitted for (4.13) to apply. Denoising was also applied here to improve the channel estimate. Although the channel changes over the frame, the channel estimate obtained based on the first two blocks was used to recover the entire frame. The corresponding BER is shown in Fig. 4.2. By experimenting with different numbers of blocks we found that 100 was the maximum frame size that would allow the training based method to achieve the same performance as the blind one. To demonstrate the latter, we also show in Fig.4.2.(a) the BER corresponding to frame length 120, where we can see that, at that frame size, the training method would require additional overhead to maintain the BER. In order for the training method to achieve the same performance with the blind method, it needs 4 times more overhead (2N for training compared to N/2 for blind) for channel estimation.
4.8.2
2 Users with STBC
We simulated a system with 2 independent users, who employ space time block coding [3]. Hence we have a system with 4 transmitters and 2 receivers. For comparison, we also simulated the users not employing STBC, i.e. a 2 by 2 system, using the same channels. We again used Jakes model with fd = 10−6 for generation of channels of length 3. Figure 4.3 shows the performance of the least squares semi blind channel estimate given in (4.38). 60 channels were used with 10 Monte Carlo runs for each channel. We used 400 OFDM blocks for channels estimation. We can see that for low SNR, the STBC performs better and this is due to diversity. However, for high SNR, channel estimation error prevents any improvement in the STBC performance at least when using a linear equalizer. On the other hand, the 2 by 2 system has less channel parameters to estimate and hence has less estimation error and outperforms the STBC method.
60 4.8.3
Comparison with Existing Methods
We compared the proposed method with the method given in [5], using the same channel as [5], i.e,
h11 = [0.4851, −0.4851, 0.7276]T , h12 = [0.32, 0.9387, −0.1280]T ,
(4.40)
h21 = [−0.3676, 0.8823, 0.2941]T , h22 = [0.2182, 0.8729, −0.4364]T and the same parameters as in [5], i.e. block size N = 12, cyclic prefix 4, and l = 2, again, arbitrarily chosen. We employed convolutional channel code of rate 1/2 similar to [5]. According to [5], the precoding results in an uneven power distribution among the OFDM blocks. The higher this unevenness is, the better the channel estimate will be, as seen on Fig. 5 of [5]. However, this has diminishing returns because under a constant energy constraint, the unevenness results in some blocks having very poor SNR which inevitably results in poor BER performance. On the other hand, the proposed method does not alter the power distribution between blocks and there is even freedom to vary the power distribution between individual carriers. The comparison results are given in Fig. 4.4. We see that the proposed method has better performance than that in Fig. 5 of [5] for high SNR.
4.9
Conclusions In this chapter, we have described in detail a novel blind channel estimation algorithm for MIMO
OFDM systems using linear precoding. It should be stressed that there is a great deal of freedom to select the precoding matrices. However, even though most precoding matrices could satisfy the requirements given in section 4.6, in practice, only a handful will perform well. Thus in the next chapter we will derive analytical performance figures of merit to differentiate between alternative precoding matrices and select best ones.
61
0
10
Blind Training
120 blocks −1
BER
10
−2
10
100 blocks
−3
10
0
5
10
15
20
25
30
35
40
45
SNR/(dB)
(a) 0
10
Blind
−1
BER
10
−2
10
−3
10
−4
10
0
5
10
15
20
25
30
35
40
45
SNR/(dB)
(b) Figure 4.2: BER for (a) fast ( fd = 10−5 ) varying and (b) slow ( fd = 10−6 ) varying channels.
62
0
10
With STBC Without STBC
−1
BER
10
−2
10
−3
10
0
5
10
15
20
25
30
35
40
45
SNR/(dB)
Figure 4.3: BER comparison with and without STBC. Random channels using the Jakes model with fd = 10−6 were used in this simulation. 400 OFDM blocks were used in channel estimation. 0
10
Comparison Proposed Blind Method
−1
BER
10
−2
10
−3
10
2
4
6
8 SNR/(dB)
10
12
14
Figure 4.4: BER comparison with method of Bolcskei et al. 400 OFDM blocks were used in channel estimation.
63 Chapter 5. Precoder Design for Channel Estimation
5.1
Performance Metrics As seen in the previous chapter it is of paramount importance to design precoders that satisfy
conditions given in section 4.6 as well as give good performance. Since there is ample freedom to select the precoding matrices, it is paramount to derive criteria that will guide us in selecting good precoding schemes. The ultimate performance criterion it obviously the bit error rate. However, the BER is analytically intractable in most cases. The signal to interference ratio is not as accurate as the BER but is analytically tractable. Hence, we derive both the analytical BER and the SINR in this chapter. Since we need to estimate the channel first, the straightforward performance criterion is the channel estimation error. However, a low channel estimation error does not necessarily imply a good performance in terms of the BER. The reason for this is that the effective capacity will be reduced due to the non unitary precoding scheme [11]. Therefore, we first derive the channel estimation error which will be used in derivations of BER and SINR. The expressions derived in this section can be used to compare different precoding schemes and notably, can be used to design optimum precoding schemes. One could use these metrics to evaluate different precoding schemes, or even find the optimum ones that will minimize BER and SINR. Even though the optimization is not analytically possible, numerical schemes can be used. In section 5.5, we describe the application of genetic algorithms for finding optimal precoding matrices.
5.2
Channel Estimation Error
Proposition 3 Consider the channel estimation error on each carrier for some fixed l ∈ L : 4 ˆ ˆ lu (k) − Hlu (k))(Diag(HH (l)eu ))−1 , k ∈ [0, N − 1]. ξ(k) = H(k) − H(k) = (H
(5.1)
64 For σ2 σ2n it holds: E{ξ(k)} = δk,l
σ2n T † eu 1 U Diag(HH (l)eu )−1 σ2 M k,l
(5.2)
and E{ξ(k1 )H ξ(k2 )} = Diag(HH (l)eu )−H (U†k1 l )H Rk1 k2 U†k2 l Diag(HH (l)eu )−1
(5.3)
where the (i, j) element of Rk1 k2 is given as
Ri, j =
δi, j i T H(l)Ψ (l) trace(Ak1 ,k2 )I e u b I−i−1 M c+1
(5.4)
η4 − 2σ4 + Diag(Ak1 ,k2 (0, 0), . . . , Ak1 ,k2 (N − 1, N − 1)) ΨiH (l)HH (l) σ4 ! σ2n i 2 eu + 2 δk1 k2 Rll /σ + trace(Ak1 ,k2 )I σ and Ak1 ,k2 = ΨiH (k1 )HH (k1 )H(k2 )Ψi (k2 ). The proof is given in Appendix B.1.
ˆ If channel length information is available, e.g., L, the obtained channel estimate H(k), can be improved by taking an inverse DFT and enforcing the channel length. This process is referred to as denoising. Improvement can also be achieved if an upper bound on the length, instead of the actual length, is used. ˜ Let ξ(k) denote the denoised error corresponding to ξ(k). The relationship between the two can be obtained using the DFT. Moreover, the error variance after denoising can be given as H˜ ˜ E{(ξ(k)) ξ(k)} =
2π 1 −L+1 L−1 N−1 N−1 ∑ ∑ ∑ E{(ξ(m1))H ξ(m2)}e j N ((m1 −k)n1 +(m2−k)n2 ) N 2 n∑ 1 =0 n2 =0 m1 =0 m2 =0
(5.5)
where the values on the right hand side are taken from (5.3). 4 ˜ ˜ − 1)), formed using the denoised error. We also define the error Let ξ = Diag(ξ(0), . . . , ξ(N
65 4
variance as ζ = E{ξH ξ}. Since ξ is block diagonal, ζ will also be block diagonal. Using this result, we can estimate the mean squared error (MSE) given the channel as N−1 1 1 H ˆ ˆ trace E{(H(k) − H(k)) (H(k) − H(k))} = MSE = trace(ζ) ∑ NMT MR k=0 NMT MR 4
(5.6)
Before proceeding, we make several key assumptions about the error ξ for analytical tractability. • We assume that E{ξ} = 0, which is a reasonable assumption considering (5.2) where we have non zero mean only for k = l. • We assume that the normalized channel estimation error is small, i.e. kH(k)k kξ(k)k and is independent of noise. This is normally valid for high signalto-noise ratio (SNR) regimes, and can be considered as a secondary assumption to the one made in proposition 3 while deriving (5.3). If the normalized error is not small, it can be either due to the noise being high or due to the channel being not identifiable. In either case, we will have poor performance in terms of the BER and thus the analysis in this section would not be applicable. • We assume each block on the diagonal of ξ to have i.i.d. entries. In that case, E{ξH ξ} should be diagonal. Close scrutiny of (5.3) reveals that if the Ukl matrices are diagonal, ζ becomes diagonal, thus justifying our assumption. Hence, we assume Ukl matrices to be diagonal. This assumption holds in the example precoding scheme for k 6= l. In the following bit-error probability and SINR analysis, instead of the channel error covariance, ζ = E{ξH ξ}, we will need an expression of the from E{ξH Aξ}, where A will be some deterministic matrix of size NMR × NMR. The latter expression is related to ζ as described in Proposition 4.
66 4
Proposition 4 Let D(ζ, A) = E{ξH Aξ} where ζ = E{ξH ξ} and A is an NMR × NMR deterministic matrix. It holds:
D(ζ, A) =
where
1 Diag[trace(A0,0 )trace(ζ(0))I, . . . , trace(AN−1,N−1 )trace(ζ(N − 1))I] MT MR
A=
A0,0
A0,1
...
A0,N−1
A1,0 .. .
A1,1 .. .
... .. .
A1,N−1 .. .
AN−1,0 AN−1,1 . . . AN−1,N−1
and ζ = Diag[ζ(0), . . . , ζ(N − 1)].
(5.7)
(5.8)
Proof: Let A(k) be an MR × MR matrix. Let E{ξ(k)H ξ(k)} = ζ(k). If we assume ξ(k) (size MR × MT ) to have i.i.d. elements with variance ν2 and zero mean, then ζ(k) = ν2 MR I. Moreover, E{ξ(k)H A(k)ξ(k)} =
1 trace(A(k))trace(ζ(k))I MT MR
(5.9)
Let us consider the block diagonal matrix ξ now. Note that A is partitioned into matrices A p,q , p, q ∈ [0, N − 1] of size MR × MR . E{ξH Aξ} will be block diagonal with each block as given by
(5.9).
5.3
Analytical Bit Error Probability For the sake of simplicity, we ignore the block index i, and denote the transmitted symbols at
any block index by d. Using (4.5) and stacking up we can write
y = Fd + n
(5.10)
67 where y = [y(0)T , . . . , y(N − 1)T ]T is an NMR × 1 vector and n = [n(0)T , . . . , n(N − 1)T ]T . The matrix F represents the combined effect of the precoding and the channel. First, the elements of MT sources are precoded by MT matrices. Next, they are permuted into N different subcarriers. Finally, they are multiplied by the channel matrices on each subcarrier. Hence we have
F = HTW
(5.11)
where H = Diag(H(0), . . . , H(N − 1)) and W = Diag(W0 , . . . , WMT −1 ). The permutation matrix T has ones on row p column bp/MT c + (p mod MT )N, for 0 ≤ p ≤ NMT − 1. The reason behind the special structure of T is explained in Appendix B.2. Let G be the equalizer matrix constructed from the channel estimate. Ideally, we will have GF = I, however in practice this will not be the case due to channel estimation errors. Thus, our estimate for d will be dˆ = GFd + Gn ≈ GFd + F†n
(5.12)
where, assuming a zero-forcing equalizer, b †. G = W−1 T−1 H
(5.13)
Note that T−1 is the anti-permutation matrix of T and on the p-th row of T−1 = TH , a one appears on bp/Nc + (p mod N)MT for 0 ≤ p ≤ NMT − 1. Let the channel estimation error be ξ such that b = H+ξ H
(5.14)
G = W−1 T−1 (I + H† ξ)−1 H† ≈ W−1 T−1 (I − H† ξ)H†
(5.15)
Then we have
due to normalized error being small, i.e. kHk kξk.
68 ˆ ˆ i.e. d(p), Let us consider the p-th element in d, which is the estimate of d(p) in vector d. We assume all symbols are drawn from white, identical 4-QAM sources. Let us consider d(p) = √ (1 + j)σ/ 2, while all the other symbols in d act as zero-mean, i.i.d. noise sources with variance σ2 . Since NMT is large, we can apply central limit theorem to model as Gaussian, the additive effect of all other symbols towards the estimation of d(p). Moreover, since the noise is Gaussian, ˆ d(p) will also be Gaussian distributed. Hence we need to only estimate the first and second order ˆ statistics of d(p) to determine its p.d.f. From (5.12), we have dˆ = (I − W−1 T−1 H† ξTW)d + W−1T−1 H† n.
(5.16)
Let us consider the error ˆ − d(p) = eTp dˆ − d(p) e(p) = d(p) = eTp (I − W−1 T−1 H† ξTW)(d(p)e p + =
(5.17) NMT −1
∑
q=0,6= p
d(q)eq ) + eTp W−1 T−1 H† n − d(p)
−eTp W−1 T−1 H† ξTWd + eTp W−1 T−1 H† n.
We have E{e(p)} = 0 because E{ξ} = E{n} = 0 and moreover, due to the independence of ξ and n, we have σ2e = E{e(p)? e(p)} = E{dH WH TH ξH (H† )H T−H W−H e p eTp W−1 T−1 H† ξTWd} +E{nH (H† )H T−H W−H e p eTp W−1 T−1 H† n} = σ2 trace(WH TH D(ζ, (H† )H T−H W−H e p eTp W−1 T−1 H† )TW) +σ2n eTp W−1 T−1 H† (H† )H T−H W−H e p where D(., .) is obtained based on Proposition 4.
(5.18)
69 We should note here that the above analysis concerned a specific block i, the index of which was dropped for notational convenience. Since within M blocks the precoding matrices change, the bit error probability should be computed as the average over M blocks. The bit error probability of p-th symbol in the i-th block, the mean for all symbols of all users in the i-th blocks, and the latter averaged over M blocks are respectively (see Eq. (8.15) of [61]):
Pei (p) ≈ Q(
1 σ ), Pei = σe NMT
NMT −1
∑
Pei (p), Pe =
p=0
1 M−1 i ∑ Pe M i=0
(5.19)
where the last result is due to averaging over i from 0 to M − 1. 5.4
Signal to Interference Plus Noise Ratio (SINR) ˆ Let us next consider the SINR. Starting from (5.12), we can calculate the contribution to d(p)
from noise and all the other symbols. The noise contribution will be σ2n eTp GGH e p ≈ σ2n eTp F† (F† )H e p . ˆ Let us consider the contribution of d(q) to d(p): 2 ? T H H T ˆ E{kd(p)k q } = E{d(q) eq F G e p d(q)e p GFeq }
(5.20)
˜ −H e p eTp W−1 T−1 (I − H† ξ)TWeq } = σ2 E{eTq WH TH (I − ξH (H† )H )T−H W = σ2 δ pq + eTq WH TH D(ζ, (H† )H T−H W−H e p eTp W−1 T−1 H† )TWeq where D(., .) is obtained based on Proposition 4. Using superposition, we have for the i−th block
SINRip =
2} ˆ E{kd(p)k p
NM −1 2 } + σ2 eT (WH TH HH HTW)−1 e ˆ ∑q=0,6T = p E{kd(p)k p n p q
(5.21)
and the average SINR will be
SINRi =
1 NMT
NMT −1
∑
p=0
SINRip , SINR =
1 M−1 ∑ SINRi. M i=0
(5.22)
70 Using this result, the BER can be calculated as p Pe (p) ≈ Q( SINR p ). 5.5
(5.23)
Finding Optimal Precoders Using Genetic Algorithms Prior work on optimal precoding design have employed algebraic number theory [75], greedy
iterative refinement [55], algebraic coding theory [17], among many. It should be noted that most of these work were aimed not at channel estimation. On the contrary, most precoding methods are proposed to increase channel capacity or diversity. Moreover, most methods have assumed perfect knowledge of the channel at the receiver. However, in our case, we need the precoding matrices to allow channel estimation as well as to optimize performance. The selection of optimal precoding matrices that satisfy both requirements is a rather hard problem. We here start with the parametric structure of the precoding matrices used in [78], which allowed channel estimation, and determine the parameters that minimize BER. The minimizations can be achieved via genetic or evolutionary algorithms [46] to reduce the complexity. Genetic Algorithms (GA) are probabilistic algorithms that mimic the evolution of nature (Natural Selection) to find optimum solutions. In a GA, we start off with a number of possible solutions, that are randomly generated. We call this group of solutions the population and each solution an individual, borrowing the words used in nature. As the GA progresses, some individuals will die and new individuals will be created by the offspring of the current individuals which is called crossover. We directly apply Natural Selection to determine which ones will die and which ones will produce offspring. For this, we need a fitness function, which can be derived from the quantity we are optimizing. The individuals with a low fitness will be eliminated and the individuals with a high fitness will produce offspring. Moreover, we randomly change the properties of some individuals, which is called mutation hoping the change will produce a better individual. Genetic algorithms have been applied to numerous signal processing applications, as described in [63]. Genetic algorithms have been applied in code design as well [16, 27]. Classical GA, where
71 the information is represented as binary strings, have a close relationship with the design of these codes, which are again, binary strings (in Galois field). In this case mutation and crossover can be naturally applied as a random change of a bit in the string and concatenation of two strings. However, in our problem, we have to design codes in the complex field. Hence, we chose a structured approach using IEEE floating point numbers for representation of the data. Moreover, we have to redefine mutation and crossover operators, since the data is not a binary string. Further detail about the differences in these two schemes can be found in [70]. We need to find best values for α, β, ρ ∈ C. We kept ρ as a dependent variable in order to conserve the power. Hence we have an optimization problem with 4 variables (real and imaginary values of α and β) at hand. These four variables correspond to the genes of each individual. We minimized Pe in (5.19) and selected 1/Pe as our fitness function to be maximized. It should be noted that evaluation of Pe depends on the channel. Hence, in a closed loop system, current channel information can be used to obtain optimum code for subsequent blocks. This can be done at the receiver and only the parameters for code, α,β in this case, can be sent to the transmitter. The initial estimate can be obtained using training or using ad-hoc precoding. For an open loop system on the other hand, (5.19) can be evaluated considering the statistical properties of the channel, i.e. by using a statistical model. Hence we can determine the code that has best overall performance for open loop case. Since the estimated channel will have a diagonal ambiguity [78], the estimated channel cannot be used to evaluate Pe directly. However, if the matrices Uk,l are diagonal for all k, l then the expression (5.3) is independent of this diagonal ambiguity. Hence we can use the estimated channel for the evaluation of Pe . We see that the precoding scheme proposed in (4.6) has diagonal Uk,l for all k, l except k 6= l. We can ignore the minimization for k = l and evaluate Pe only for the other values of k, thus overcoming the need for pilots. The rudimentary algorithm we have used can be given as follows [63] and was implemented using PGAPack genetic algorithm library [35]. 1. Generate a random (subject to constraints) population of possible solutions.
72 2. Compute the fitness of each solution (e.g. using (5.19)) 3. Create new solutions by mutation of existing solutions and crossover of two solutions. 4. Replace a portion of the current population by the newly generated solutions, according to the fitness value of each solution. 5. If the conditions to end iterations have not being met, go to step 2, else stop and report best solution.
5.6
Simulations We here compare the analytical performance against simulation results. We have kept the SNR
at 30dB and used 40 static channels of length 3. We have calculated the NMSE for each channel as follows.
L−1 2 |hi j (l) − h[ 1 Mc −1 1 M−1 M−1 ∑l=0 i j (l)| NMSE = ∑ ∑ ∑ L−1 Mc m=0 M 2 i=0 j=0 |hi j (l)|2 ∑l=0
(5.24)
where Mc = 20 is the number of Monte Carlo runs and MT = MR = M = 2. We have presented the results in Fig. 5.2. We have presented the analytical BER results, derived from (5.19) and (5.23) in Fig. 5.1 against simulated results. From this we see a bias in the analytical results. However, there is close agreement in the relative change for different channels. In mutations, we changed the values of α and β by ±10%. In crossover we generated new individuals using parents (α1 , β1 ) and (α2 , β2 ), as (α1 , β2 ) and (α2 , β1 ). We also introduced a penalty scheme to satisfy the constraints. For instance, if the power constraint was violated, we made the fitness value negative. In other words, the cost function would give us −1/Pe instead of 1/Pe . To avoid degenerate solutions with α ≈ 0, we introduced an additional constraint as |α| ≥ 0.4. If this condition was violated, the cost function would give us −0.1/Pe instead of 1/Pe . In our simulations, we considered a channel of length 3. First we estimated the channel using the ad-hoc solution α = j0.95 and β = 0.2195, using 100 data blocks with pilots to eliminate the diagonal ambiguity. Next we used the channel estimate obtained to evaluate Pe . We used a population
73 of 200 and ran the genetic algorithm for 50 iterations. We obtained solutions using the genetic algorithm for SNR values from 0dB to 30dB. We compared this with the ad-hoc solution α = j0.95 and β = 0.2195, kept fixed for all SNR values. We have given simulation results with the two schemes in Fig. 5.4. We used 40 Monte Carlo runs to obtain the results and used 400 blocks for estimation. We see that for high SNR, the genetically optimized solution performs better. However, for low SNR, the effect of the optimization is negligible
5.7
Conclusions In this chapter we have derived performance criteria for the channel estimation method proposed
in chapter 4. We also applied numerical optimization using genetic algorithms to find optimal precoders.
74
0
10
BER, Simulated BER, Analytical BER, Analytical SINR −1
10
−2
BER
10
−3
10
−4
10
−5
10
−6
10
0
5
10
15
20 channels
25
30
35
40
Figure 5.1: Comparison of analytical and simulated BER and SINR for 40 different channels. 0
10
NMSE, Simulated NMSE, Analytical
−1
NMSE
10
−2
10
−3
10
0
5
10
15
20
25
30
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channels
Figure 5.2: Comparison of analytical and simulated NMSE for 40 different channels.
75
0
10
−1
10
−2
BER
10
−3
10
−4
10
−5
10
α=j0.9, β=0.432
−6
10
500
1000
1500 2000 Different Combinations of α,β
2500
3000
Figure 5.3: BER for different combinations of α, β.
0
10
ad hoc optimized −1
10
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BER
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10
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10
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SNR/(dB)
Figure 5.4: BER comparison of ad-hoc precoding scheme and the solution obtained using genetic algorithm for different SNR. BER below 10−6 was taken to be 10−6 .
76 Chapter 6. Multiuser cooperation in OFDM systems
6.1
Introduction Multiuser Cooperation has recently emerged as a candidate for improving the performance of
wireless communication systems. The pioneering work by [59, 60] showed that cooperation has the potential to increase the data rate. In [32], it is shown that in theory, the achievable diversity order achievable is equal to the number of users cooperating. Cooperation can be classified into decode-and-forward, and amplify-and-forward schemes. The former require each user to decode the relayed signal, thus is more demanding on each user. In [50] it was shown that decode and forward schemes perform better than other schemes in terms of Bit Error Rate (BER). Nevertheless, in some situations, due to complexity and available resources considerations it might be preferable to use simpler amplify-and-forward schemes. To avoid multiuser interference, and also to overcome hardware limitations that do not allow for simultaneous transmission and reception by same transceiver, cooperative systems usually rely on some form of orthogonality to transmit and receive signals from multiple users. For instance, in [32, 33], the authors consider time slotted systems. In [59, 60], orthogonality is achieved using orthogonal spreading codes. A different form of cooperation, where cooperative transmission of partitioned codewords (instead of relaying the received signal), has been studied in [28, 29, 30, 58]. Most reported work on cooperation has considered flat fading channels only [30, 59]. When multipath diversity is not available, and we are restricted to using single transmit/receive antenna per user, cooperation is the only way to increase diversity, or equivalently, improve BER performance. However, when multipath channels are involved, the multipath diversity can be exploited to reduce BER. In general, it is possible to achieve diversity equal to the number of fading paths. This is usually achieved by using a Rake Receiver or Maximal Ratio Combining[61]. The question that has not been addressed yet is whether cooperation can help even in cases where the multipath channel has been fully exploited in terms of diversity.
77 Dealing with multipath channels is not simple since it usually requires high complexity receivers. OFDM systems have recently gained popularity for their ability to handle multipath channels while requiring low complexity equalizers. As originally proposed [9], an OFDM system transmits each symbol along one of the available subcarriers. Thus, it was impossible to achieve diversity even in the presence of multipath channels. Several ways have been proposed in the literature for increasing diversity in OFDM systems. In [43], it is shown that a single user OFDM system with non-redundant block precoding can achieve diversity gain up to L, the number of channel fading paths. For maximum diversity, the performance gain is exploitable using a Maximum Likelihood (ML) decoder. Smaller than L diversity gains are also possible via subcarrier grouping [43], which results in reduced complexity decoding at the receiver. If one used redundant precoding or oversampling at the receiver, maximum diversity gain is still possible via a linear receiver [65] and [66]. In this chapter we propose a precoded multiuser OFDM scheme with cooperation. The precoding consists of intrablock precoding, for introducing multipath diversity, and interblock precoding for exploiting the time diversity introduced by cooperation. We study the potential advantages of the proposed scheme as compared to a multiuser OFDM scheme without diversity that exploits multipath diversity, and which uses the same transmission energy per information block. The findings of this chapter can be summarized as follows. • When both users face high signal-to-noise ratio (SNR) between each other and the receiver, the multiplexing gain of the OFDM system with cooperation with respect to the OFDM system without cooperation is one. In other words, the capacity does not improve by cooperation. On the other hand, when there is a difference in the SNR between the users the receiver, cooperation does improve capacity. • We show that a two-user precoded OFDM system with cooperation, with each user operating with a single transmit antenna, can achieve maximum diversity of order 2L. The corresponding precoded OFDM system without
78 cooperation would need to require two transmit antennae to achieve the same diversity. The chapter is organized as follows. In Section II we describe the proposed multiuser OFDM cooperation scheme. In Section III we derive the effective channel capacity and related results. In Section IV we analyze the performance in terms of diversity. Some preliminary ideas on extending the proposed scheme to more than two users is given in Section V. We give simulation results in Section VI before concluding in section VII. Notation: we use (.)T to denote transpose, (.)H to denote conjugate transpose. Vectors are denoted by bold lowercase letters as in x and matrices by bold uppercase letters as in X. D (x) denotes a diagonal matrix formed based on the elements of vector x. diag[X1 , ..., X2 ] a block diagonal matrix whose diagonal elements are the matrices X1 , ..., X2 .
6.2
Space Time Cooperation in a Multiuser OFDM System
6.2.1
Background
In a single-user OFDM system, at each index i ∈ Z , a block of symbols {sik , k = 0, ..., N − 1} is used to form an OFDM symbol si = [si0 , ..., siN−1 ]T . A norm preserving inverse N-point DFT is performed on si , and a cyclic prefix of length Ncp , Ncp ≤ N is pre-appended to the result, to form the symbols: 2π 1 N−1 tki = √ ∑ sik e j N kn , −Ncp ≤ n ≤ N − 1. N k=0
(6.1)
Those symbols are sent in a serial fashion through a noisy multipath channel. Over the time it takes to transmit the N + Ncp symbols corresponding to the i-th OFDM symbol, the channel is modeled as a linear time-invariant system with impulse response h(n), n = 0, ...L − 1. At the receiver, the first Ncp symbols are discarded. As long as L ≤ Ncp , the subsequently collected symbols, rki , k = 0, ..., N contain the contribution of the i−th OFDM symbol only. An N-point DFT is performed on those
79 symbols to yield the symbols xik , k = 0, ..., N − 1. For xi = [xi0 , ..., xiN−1 ]T it holds: xi = Hsi + ni
(6.2)
where H = D ([H(0), ..., H(N − 1)]T ), with L−1
H(k) =
∑ h(n)e− j
2π N kn
(6.3)
n=0
and ni represents noise. In a multi-user OFDMA system the users are assigned disjoint carriers. Let us consider a 2user system. The subcarriers are divided between users 1 and 2 according to the sets I 1 and I 2 , respectively. User 1 transmits over subcarriers in set I 1 , and receives over subcarriers in set I 2 , where I1
S
I 2 = {0, 1, ..., N − 1} and I 1
T
I 2 = {}. User 2 transmits over subcarriers in set I 2 , and receives
over subcarriers in set I 1 . Let |I | denote the cardinality of set I . We will next discuss a scenario where users both transmit and receive simultaneously using the same antenna, i.e., in full duplex mode. Since there could be practical hardware difficulties related to such scenario, we will later discuss an approach of using time division multiplexing to achieve full duplex operation. The only change this will introduce is to change the effective channel between a transmitter and a receiver, as seen later. We should stress that this will not change the following analysis or the conclusions drawn in this chapter . Thus we continue to present our method assuming full duplex operation for simplicity. Let sij be a |I j | × 1 vector denoting the i-th OFDM block of user j, and let xip denote the corresponding signal received by user p over the carriers in set I j . The time-domain multipath channel between j and p is denoted by h p j (n) n ∈ [0, L − 1], and in frequency domain is denoted by Hi j (k), k ∈ [0, N − 1]. Then we can rewrite (6.2) as xip = H p j sij + nip j where the channel matrix H p j is constructed as H p j = D [H p j (I j (1)), H p j (I j (2)), . . . , H p j (I j (|I j |)])
(6.4)
80 where I j (k) denotes the k-th element of the set I j , according to some predefined ordering. The noise added to the signal block transmitted by user j and received by user p is a vector given by n p j . 6.2.2
Proposed Scheme
Before the data blocks enter the OFDM system we will perform two types of precoding, interblock precoding and intra-block precoding. Inter-block precoding [38] will help exploit the temporal diversity that will be introduced by the cooperative retransmissions and is implemented as follows. If the uncoded blocks of user j are dij , di+1 j , the precoded blocks are
si1 s1i+1
= W1
di1 di+1 1
,
si2 si+1 2
= W2
di2 di+1 2
(6.5)
where W1 , W2 are unitary square matrices. Intra-block precoding will help exploit the multipath diversity [43]. Each OFDM block will enter the OFDM system as R j sij , where R j is the |I j | × |I j | precoding matrix of user j. In the sequel, for simplicity we will assume that R j is an orthonormal matrix. The full system with two users and the receiver (or base station) is shown in Fig. 6.1. The transmission of user 1 is illustrated in Fig. 6.2, where sij has undergone inter block precoding as given in (6.5). Each user will implement its own OFDM modulation scheme with N carriers as given in Fig. 6.2. Let ηi j denote the signal-to-noise ratio (SNR) at receiver i of the signal transmitted by user j. We make the following assumptions throughout this chapter. (A1) The transmitters have no knowledge of the channel and the receivers have perfect knowledge of the channel. (A2) We assume the channel to remain constant for a few OFDM blocks. We assume that all the channel taps hi j (n) are zero mean i.i.d. Gaussian with
81 unit variance. The taps hi j (n) and hiˆ jˆ(n) ˆ are uncorrelated for i 6= iˆ j 6= jˆ n 6= n. ˆ (A3) We assume the noise vector nip j to have identically distributed elements that are uncorrelated temporally and with the channel or data, and are circularly complex Gaussian distributed with zero mean. In practice, each user will have differently fading channels and different SNRs depending on the transmitter power employed. However, to keep the analysis simpler, we assume that only the noise power changes. The effect on the SNR will still be the same as in the real case. The i-th block of user j will consist of the user’s own transmission, and also data from the other user, m. Since each user is allocated a different set and number of carriers, the received block sent by user m, ( j 6= m) will be of size |I m | × 1 and will have to be mapped to a block of size |I j | × 1 before it is transmitted by user j. Let us represent this mapping by pre-multiplying the received block by a matrix P j . If |I 1 | = |I 2 | then the mapping matrix could be a permutation matrix, or simply the identity matrix. The proposed cooperation scheme (see Fig. 6.3) is implemented in cycles of 3 blocks (or slots): • In the i-th block, users do not cooperate. They transmit their own data R1 si1 and R2 si2 , respectively. This will be received as H21 R1 si1 + n21 and H12 R2 si2 + n12 , respectively, by the other user. i+1 • In the i + 1-th block, users transmit their own data R1 si+1 1 and R2 s2 plus
the received signal after it has been scaled and mapped from the incoming carriers to outgoing carriers. The scaling is dependent on the amount of power being allocated for cooperation. Under fixed energy constraint, allocating too much power to the forwarding signal would weaken the users own signal. We use α and β as the scalars to denote this scaling of signals. Hence, the amount of power
82 allocated for cooperation at users 1 and 2 will be proportional to α2 and β2 , respectively. 0
0
• In the i + 2-th block, users again transmit R1 si1 and R2 si2 as their own data, plus the received signal. Note that there is a component of si1 (si2 ) in the received signals at users 1 (2). In order to eliminate this, the precoding is modified as 0
R1 = R1 − αβP1 H12 P2 H21 R1
(6.6)
0
R2 = R2 − αβP2 H21 P1 H12 R2
(6.7)
for this block. The received signal during this block is ignored. The echo component gain αβP1 H12 P2 H21 R1 or (αβP2 H21 P1 H12 R2 ) can be tracked at each user by correlating the received signal at the i + 2-th block by the transmitted signal at the i-th time block. Similar echo cancellation is assumed in [59]. In the i + 3-th block, this cycle is repeated with two new data blocks. Hence the rate of transmission is 2/3. Let us consider the signal at the BS during three block intervals. The received signal on carriers in I 1 equals: yi1 = H01 R1 si1 + ni01
(6.8)
y1i+1 = H01 R1 s1i+1 + αH01 P1 H12 R2 si2 + αH01 P1 ni12 + ni+1 01
(6.9)
i i+1 i+2 y1i+2 = H01 R1 si1 + αH01 P1 H12 R2 si+1 2 + αβH01 P1 H12 P2 n21 + αH01 P1 n12 + n01 (6.10)
0
0
where on the i + 1-th block, (6.6) and (6.7) have been substituted for R1 and R2 . In vector form, we
83 have:
yi1
yi+2 1 y2i+1
where
H1 =
i s1 = H1 diag(R1 , R2 ) + n1 si+1 2
H01
0
H01
αH01 P1 H12
βH02 P2 H21
H02
,
ni01
i+1 i+1 i n1 = αβH01 P1 H12 P2 n21 + αH01 P1 n12 + n01 i+2 βH02 P2 ni+1 21 + n02
(6.11)
.
(6.12)
Similarly, the received signal over carriers in I 2 is: yi2 = H02 R2 si2 + ni02
(6.13)
y2i+1 = H02 R2 s2i+1 + βH02 P2 H21 R1 si1 + βH02 P2 ni21 + ni+1 02
(6.14)
i+1 i i+1 y2i+2 = H02 R2 si2 + βH02 P2 H21 R1 si+1 1 + αβH02 P2 H21 P1 n12 + βH02 P2 n21 + n02 (6.15)
or, in vector form:
yi2
yi+2 2 y1i+1
where
H2 =
i s2 = H2 diag(R2 , R1 ) + n2 si+1 1
H02
0
H02
βH02 P2 H21
αH01 P1 H12
H01
,
ni02
i+1 i+1 i n2 = αβH02 P2 H21 P1 n12 + βH02 P2 n21 + n02 i+2 βH01 P1 ni+1 12 + n01
(6.16)
.
(6.17)
By observing (6.11) and (6.16), one can clearly see the motivation behind using interblock precoding. By cooperation, we have effectively created two transmission paths for our information, given by (6.11) and (6.16). This is analogous to effectively employing two transmitters. Hence
84 we can exploit this by transmitting our information on both channels given by (6.11) and (6.16). In order to do this, we first precode the blocks i and i + 1 of the j th user as given in (6.5) to get sij and si+1 which are transmitted on (6.11) and (6.16). In addition to increasing diversity, the j interblock precoding will have the following effect on capacity. If the noise variance of ni01 and ni02 are significantly different, the paths (6.11) and (6.16) will have significantly different signal to noise ratios and thus significantly different channel capacities. Thus, by interblock precoding, we can increase the capacity of the effective channel with the lowest capacity, or the highest lever of noise. At the receiver we need combined decoding of (6.11) and (6.16). Let:
yi1 y1i+2 y2i+1 yi2 y2i+2 y1i+1
=
H01 R1 H01 R1 βH02 P2 H21 R1 H02 R2 H02 R2 αH01 P1 H12 R2
0
WH 0 11 αH01 P1 H12 R2 0 WH 22 H02 R2 0 0 WH 21 βH02 P2 H21 R1 H W12 0 H01 R1
di1 di+1 1 di2 di+1 2
+n
(6.18)
where
ni01
αβH P H P ni+1 + αH P ni + ni+1 01 1 12 2 21 01 1 12 01 i+2 βH02 P2 ni21 + n02 n= ni02 αβH P H P ni + βH P ni+1 + ni+1 02 2 21 02 2 21 1 12 02 i+1 i+1 αH01 P1 n12 + n01
H H W11 W21 , W1 = , W2 = . (6.19) H H W W 12 22
We can express (6.18) as y = Hd + n
(6.20)
85 where n is the noise vector with i.i.d. white entries with power spectrum N0 /2 and
H=
H01
0
H01
αH01 H12
γβH02 H21
γH02
γH02
0
γH02
γβH02 H21
αH01 H12
H01
with P1 = P2 = I and the scalar γ2 =
σ201 σ202
0 R 1 WH 11 H 0 R2 W22 0 R 2 WH 21 R 1 WH 0 12
(6.21)
is introduced the make the noise vector have i.i.d elements
with equal variance. If we do not have a multipath channel, we can avoid using intra block precoding because there is no exploitable multipath diversity. However, even in this case, it is advantageous to use interblock precoding because it will improve the overall channel capacity as it will be seen later. In other words, we can make R1 = R2 = I, P1 = I, P2 = I. Let W1 , W2 being (2 × 2) matrices, then, (6.18) can be reduced to N/2 equations
yi1 (k)
i+1 y2 (k) = yi2 (k) y2i+2 (k) y1i+1 (k) y1i+2 (k)
H01 (I 1 (k))
0
wH 11 H01 (I 1 (k)) αH01 (I 1 (k))H12 (I 2 (k)) 0 βH02 (I 2 (k))H21 (I 1 (k)) H02 (I 2 (k)) H02 (I 2 (k)) 0 0 H02 (I 2 (k)) βH02 (I 2 (k))H21 (I 1 (k)) wH 12 αH01 (I 1 (k))H12 (I 2 (k)) H01 (I 1 (k))
0 wH 22
wH 21 0
d1i (k)
+n i d2 (k) i+1 d2 (k) d1i+1 (k)
(6.22)
for k = 1, . . . , N/2. The inter block precoding is done in groups of 2 carriers so wH i j are row vectors corresponding to the rows of Wi . This is important in order to reduce the complexity of the maximum likelihood decoder at the receiver. We see that (6.18) has dimension 3N × 2N while (6.22) has dimension 6 × 4 (N/2 times). Hence the maximum likelihood decoding complexity reduces from O(22N ) to NO(24 )/2.
86 6.2.3
Multiuser OFDM without Cooperation
For comparison, let us also consider a two-user OFDM system with no cooperation with the same carrier allocation per user as the proposed scheme. The signals received by the base station over carriers I 1 and I 2 are: ˜ 1 si1 + n01 y˜ i1 = H01 R
(6.23)
˜ 2 si2 + n02 y˜ i2 = H02 R
(6.24)
and
˜ i is the precoding matrix of size |I i | × |I i |, used by user i. Note that here no inter block where R precoding is used because inter block precoding can only exploit temporal diversity. Thus unless the channel changes very fast, temporal diversity is not available in the non-cooperation case.
6.2.4
Maintaining Constant Transmission Energy Between Cooperative and Non-cooperative Schemes
requires 3 slots, as opposed to 2 In the cooperative OFDM scheme, transmission of sij and si+1 j slots in the non-cooperative scheme. To maintain the energy spent by the two scheme on these two block at the same level, we need to adjust the transmission power as follows. Let σ˜ 2i , σ2i be the signal power of user i without and with cooperation, respectively. For simplicity let us take σ˜ 1 = σ˜ 2 = σ˜ and σ1 = σ2 = σ. In the non-cooperative case, the transmission of 2 blocks requires energy equal to: ˜ 1R ˜H ˜ 2R ˜H ˜ 22 trace(R ˜ 21 |I 1 | + 2σ˜ 22 |I 2 |. 2σ˜ 21 trace(R 1 ) + 2σ 2 ) = 2σ
(6.25)
Under cooperation, the energy spent by user 1 to transmit 3 blocks equals: 2 2 H Γ1 = 3σ2 trace(R1 RH 1 ) + 2σ α trace((P1 H12 R2 )(P1 H12 R2 ) )
(6.26)
87 and the energy spent by user 2 equals 2 2 H Γ2 = 3σ2 trace(R2 RH 2 ) + 2σ β trace((P2 H21 R1 )(P2 H21 R1 ) ).
(6.27)
To ensure that the energy spent is same in cooperative and non cooperating cases, we need: H 2 2 H H 2σ˜ 2 (|I 1 | + |I 2 |) = 3σ2 (|I 1 | + |I 2 |) + 2σ2 α2 trace(P1 H12 HH 12 P1 ) + 2σ β trace(P2 H21 H21 P2 ).
(6.28) The amount of cooperation is limited by the minimum number of |I 1 | and |I 2 |. In order to maximize cooperation, we need to have |I 1 | = |I 2 | = N/2, or in other words, carriers are equally shared between the users. Then, P1 and P2 become square permutation matrices. Considering the properties of the permutation matrices we have N/2
2σ˜ 2 N = 3σ2 N + 2σ2 ∑ (α2 |H12 (I 2 (k))|2 + β2 |H21 (I 1 (k))|2 )
(6.29)
k=1
H because P1 PH 1 = I and P2 P2 = I.
The elements H12 (I 2 (k)) and H21 (I 1 (k)) are the diagonal elements of H12 and H21 but in no particular order. So far we treated the channel as deterministic. If we treat the channel taps as zero-mean Gaussian random variables, the magnitudes |Hi j (k)|2 will be i.i.d. Rayleigh distributed with E{|H21 (k)|2 } = 1. Then: 4
σ2 = E{σ2 } = 6.2.5
2 σ˜ 2 . 3 + (α2 + β2 )
(6.30)
Time Division Duplexing
The cooperation scheme described above was strongly dependent on the users being able to both receive and transmit simultaneously. However, in most situations this might be difficult. Nevertheless it is possible to effectively achieve full duplex operation by time division duplexing. Let us refer to Fig. 6.3 again. In time slot i, both users need to transmit their data vectors si1 and si2 . In
88 the original scheme both users transmit during the entire duration of time slot i (N symbols plus the cyclic prefix). However, we can allocate time for each user proportional to the length of the data vectors si1 and si2 or |I 1 | and |I 2 |. Instead of using OFDM block length N, each user will use OFDM block length equal to their data vector length. It should be noted that the cyclic prefix needs to be added to each user, and hence will increase the required time. Hence, during time slot i, user 1 will first transmit |I 1 | data symbols plus the cyclic prefix. Next, user 2 will transmit his own |I 2 | data symbols plus the cyclic prefix. During each transmission, all the other users will be in the receiving mode and there will not be any hardware limitations. The cooperation between two users with time division duplexing is given in Fig. 6.4. In contrast to Fig. 6.3, we see the division of each OFDM slot to two time slots in Fig. 6.4. Let us see how this will change the received signals. Instead of having a common OFDM block length of N, each user has its own OFDM block length or DFT length which is in general different from other users. Hence (6.4) will be reduced to (6.31) where Hi j (k) are the |I j | point DFT coefficients of channel hi j (n), because user j used an OFDM block length of |I j |. Moreover, Hi j = diag Hi j (0), Hi j (1), . . . , Hi j (|I j | − 1) .
(6.31)
Apart from this change in the effective channel, there will be no difference in the proposed scheme and hence the analysis and the conclusions will hold even for this case.
6.3
Channel Capacity In this section we derive an upper bound for the channel capacity of the OFDM scheme with
cooperation for a given cooperation level, and the OFDM scheme without cooperation. Our analysis is based on the following assumptions: (B1) The noise between the users is negligible compared to the noise between the users and the receiver, i.e., E{|n12 |2 } E{|n01 |2 } and E{|n21 |2 } E{|n02 |2 }. Since we are assuming equal transmission power for all users,
89 this implies the SNR between the users η12 , η21 is much higher compared to the SNR between the receiver and the users, η01 and η02 . (B2) Both users have equal number of carriers allocated, i.e. |I 1 | = |I 2 | = N/2 and P1 = P2 = I. (B3) The noise nikl (k) as a function of k is zero-mean, with variance (σikl )2 . Noises are uncorrelated between different i’s (slots), and between different 2 2 k, l (users). For simplicity we assume that (σikl )2 = (σi+1 kl ) = σkl .
Via (6.11) and (6.16) and the above assumptions the correlation matrices of n1 and n2 equals: 2 2 2 Rn1 = E{n1 nH 1 } ≈ diag[σ01 I, σ01 I, σ02 I]
(6.32)
Rn2 ≈ diag[σ202 I, σ202 I, σ201 I].
(6.33)
and
Then, based on (6.11) and (6.16) the channel capacities on the carriers of I 1 and I 2 , given the channel are [18]:
H R1 R1 C1 = log det σ2 H1 0
and
C2 = log det σ H2 2
Let η01 =
σ2 σ201
and η02 =
σ2 σ202
0 R2 RH 2
R2 RH 2
0
0
R1 RH 1
(6.34)
(6.35)
! H −1 H1 Rn1 + I
! H −1 H2 Rn2 + I .
denote the SNR values for each user as defined before.
At this point, we should elaborate on the effect of using precoding. Since interblock precoders R j are unitary, the channel capacity is unaffected. Without interblock precoding, we have to decode (6.11) and (6.16) separately because there is no sharing of information between the two. Hence, the lowest of C1 or C2 will act as a bottleneck in decoding (6.11) or (6.16). The lowest effective capacity per carrier in this case will be the minimum of C1 /N or C2 /N.
90 However, when we have interblock precoding, we have to decode (6.11) and (6.16) together, as given in (6.20). In that case, the total capacity will be C1 + C2 (see Appendix C.1). Then the effective capacity per carrier will be (C1 + C2 )/2N. Thus we see that by interblock precoding and joint decoding, we have increased the minimum capacity from minimum value of C1 /N or C2 /N to (C1 +C2 )/2N. We next give an upper bound for the channel capacities for the case where we have maximum cooperation.
Proposition 5 Under assumptions (A1-A3),(B1-B3), and for cooperation levels α and β, the ergodic capacity upper bound for C1 is:
E{C1 } ≤
(6.36)
1 N log (1 + η02 + η1 β2 ) Y1 2 η01
1 1 − 2 Y1 ) η01 η01 1 1 1 − 2 + 3 Y1 ) +(2α2 η201 + 2η201 + 2η201 η202 + η201 η02 + α2 β2 η201 η02 )( η01 η01 η01 2 1 1 1 +α2 η301 ( + − Y1 ) − η01 η201 η301 η401 +(3η01 + α2 η01 + 3η01 η02 + 2β2 η01 η02 + α2 β2 η01 η02 − β2 η201 )(
where Y1 = exp(
1 1 )Ei(1, ) η01 η01
(6.37)
where Ei(., .) is the incomplete exponential integral, i.e.
Ei(c, z) =
Z ∞ 1
e−tzt −c dt
(6.38)
Proof: See Appendix C.2.
Corollary: We can find a similar upper bound for C2 by interchanging η01 with η02 and α with
91 β in (6.36).
The non cooperating channel capacities are [18]: σ˜ H ˜ 1R ˜H C˜1 = log det( 2 H01 R 1 H01 + I), σ01 2
σ˜ H ˜ 2R ˜H C˜2 = log det( 2 H02 R 2 H02 + I) σ02 2
(6.39)
Using Jensen’s inequality (E{log x} ≤ logE{x}), upper bounds for non cooperative channel capacities become
E{C˜1 } ≤
σ˜2 N log(1 + 2 ), 2 σ01
E{C˜2 } ≤
σ˜2 N log(1 + 2 ) 2 σ02
(6.40)
˜ jR ˜ Hj = I and E{H0 j HH } = I. because R 0j 6.3.1
Comparison of Cooperative and Non-cooperative Schemes
Before comparing the two schemes, we should take into account that cooperation results in rate reduction. In the cooperative scheme, C1 +C2 is the capacity available to transmit 4 blocks using 3 time slots. On the other hand, in the non-cooperative scheme, C˜1 + C˜2 is the capacity available to transmit 2 blocks using 1 time slot. To account for the rate reduction in the cooperative scheme, we have reduced the user power (see section (6.2.4)), thus maintaining the energy spent per information symbol constant. First we consider the spatial multiplexing gain [86] acquired by cooperation. Spatial multiplexing gain is obtained by taking the limit of the capacity compared to an the capacity of an AWGN channel as the SNR goes to infinity. Hence, in this situation, we assume both users to have good channels or very high SNR values. Since both users have high SNR towards the receiver, we assume the SNR difference between the users is negligible. Proposition 6 Consider the proposed two-user OFDM scheme with cooperation defined in Section 6.2.2, and a two-user OFDM scheme without cooperation defined in Section 6.2.3, both of which spend the same amount of energy to transmit a fixed amount of information.
92 When both users have high SNR towards the receiver that we assume to be equal, and under assumptions (A1)-(A3) and (B1)-(B3), the multiplexing gain acquired by cooperation is 1. The proof is given in C.3. Thus we can conclude that if both users have high SNR values towards the receiver, the gain in channel capacity from cooperation is zero. In other words, the effective channel capacity will not increase by cooperation when both users have very good channels. Intuitively, the region of interest and applicability of cooperation is not when both users enjoy good channels. In fact it is the exact opposite where one user has a very good channel and the other user has a very poor channel. In order to examine this further, let us consider the excess capacity 4
available while cooperating, i.e., Ce = C1 +C2 − 2C˜1 − 2C˜2 . To see if we can draw any conclusions based on ergodic capacity bounds, we plot Ce as a function of η01 , η02 , and compare it against Ce computed based on simulated capacities. For the simulation results we evaluated (6.34), (6.35) and (6.39) for 50 randomly generated channels that satisfy our channel assumptions. The left column of Fig. 6.5 corresponds to the simulated result of the excess capacity. For the same values of α and β we evaluated Ce based on the bounds given in Proposition 5 and in (6.40). These results are given on the right column of Fig. 6.5. First, we investigate the effect of rate reduction due to cooperation. In order to do that, we have made both α = β = 0 and plotted the excess capacities. Note that making α = β = 0 will be equivalent to a rate reduction by 1/3 but without any cooperation. We see that mere rate reduction does not produce any excess capacity as seen in Fig. 6.5 (a) and (b). In fact, it will reduce the excess capacity as given by the negative contours. Next, we investigate the effect of cooperation with reduced rate and reduced power level on the effective capacity. In order to do that, we kept α = 0.5 and considered β = 0 and β = 0.5. We see both contour plots exhibit peaks when there is a significant difference between η01 and η02 , while for η01 = η02 , the excess capacity is at its lowest value. This is in agreement with the intuitive expectation that cooperation can improve performance when one user has a very good channel while the other user has a very poor channel. Furthermore, when α = 0.5 and β = 0, we see no
93 improvement in capacity by increasing η02 because user 2 does not cooperate. Finally, we see that the results given in Fig. 6.5 agree with Proposition 6 because along the main diagonal we do not see any gain in capacity by cooperation. As we take the limit in Proposition 6 we increase the power level of both users and this is equivalent in following η01 = η02 as η01 → ∞ on Fig. 6.5. In other words, we see no increase in capacity by cooperation in the case where both users have good channels or high SNR towards the receiver. Hence, in such a situation, we will not get any improvement in terms of capacity by cooperation. Thus, from the observations made in this section we can conclude that the upper bounds given in proposition 5 closely follow the simulated result and moreover, there is an increase in capacity by cooperation only when the users have significantly different SNR values towards the receiver.
6.4
Diversity It is shown in [43] that for a single user OFDM system, the maximum diversity gain achievable
with one transmit antenna is equal to the number of independent fading paths of the channel. In this section, we follow a similar procedure and study the diversity gain achieved by (6.18). Unlike multiplexing gain which is related to the data rate, diversity is related to the bit error rate performance [86]. Diversity is usually increased by adding more transmitters and receivers. We show that (6.18) achieves the full spatial diversity available, i.e. 2L without addition of transmitters.
Proposition 7 For a flat fading channel, the cooperative scheme given in (6.18) achieves full spatial diversity of 2 provided that the channels between the users H12 and H21 are given and are full rank and α, β 6= 0. If the channels between the users and receiver has L multipaths each, the combined diversity becomes 2L. The full diversity can be achieved either by intrablock precoding, or by interblock precoding or both. The proof is given in Appendix C.4. Remark: note that if H12 (I 2 (k)) = 0 or H21 (I 1 (k)) = 0, regardless of the value of α or β, we would
94 lose the full diversity for that particular carrier k. We can only prove the above proposition by keeping the interuser channel fixed. However, this does not mean we will not get any increase in diversity when we have a fading interuser channel. Practically, we must expect the interuser channel to be not in a deep fade. This is intuitively obvious because if we have a bad interuser channel, we will have a very low interuser SNR value that will undermine our performance gain by cooperation. Thus, unlike pure transmit diversity, where we always have a good (wired) channel between the transmitters, cooperation can exhibit the same performance only when the interuser channel is good. In OFDM, where we have multiple carriers, this means some carriers will enjoy the full diversity gain by cooperation while some carriers will not. It is possible to exploit the diversity offered by the multipath channel by linear constellation precoding as shown in [43]. If the channel between each user and the receiver has L multipaths, the maximum diversity gain achievable by each user without cooperation is L. However, following the discussion above, we see that if we cooperate in addition to linear constellation precoding, the maximum achievable diversity becomes 2L. We should note moreover that either intrablock precoding [43] or inter block precoding [38] or both can be used with maximum likelihood decoding at the receiver to exploit this diversity (see the proof of 7 for more details on this).
6.5
Determining the Level of Cooperation The key to improving performance of the cooperative system is determining the appropriate
values of α and β. For instance, using too much power for cooperation would decrease the power available for ones own signal, thus having a detrimental effect. Hence in this section we investigate the limits for selecting α and β and our objective is to minimize the BER. We can use the expressions derived in (6.36) to determine the best situations for cooperation in terms of capacity. However, since this is only an upper bound, analytical optimization will not give us the best results. Nevertheless, we can use this to get the region where we should limit α and β for best performance.
95 The fundamental assumption used in (6.36) was that α2 β2 E{|H01 P1 H12 P2 n21 |2 } + α2 E{|H01 P1 n12 |2 } E{|n01 |2 }
(6.41)
β2 E{|H02 P2 n21 |2 } E{|n02 |2 } Assuming E{|n12 |2 } = E{|n21 |2 } and taking expectation with respect to the channels, we get α2 β2 + β2
σ202 2 2 σ201 2 , α β + α σ212 σ212
(6.42)
and from these, we have upper bounds for cooperation power levels α2 and β2 . From Fig 6.3, in the i + 1-th block, the signal received from user 1 at user 2 is H21 (R1 si+1 1 + i+1 i+1 i αP1 (H12 R2 si2 + ni12 )) + ni+1 21 . After echo cancellation this becomes H21 R1 s1 + αH21 P1 n12 + n21 .
Since we use unitary interblock precoding, the precoding will have no effect on the capacities. Thus, the channel capacity from user 1 to 2 is (at worst case when the noise is maximum) when P1 = I N/2
C12 =
∑ log
k=0
σ2 |H21 (k)|2 +1 2 2 2 α σ12 + σ21
(6.43)
where E{|n12 |2 } = σ212 and E{|n21 |2 } = σ221 . After using Jensen’s inequality, E{C12 } ≤ N/2 log
σ2 + 1 α2 σ212 + σ221
(6.44)
and similarly for the channel from user 2 to 1,
E{C21 } ≤ N/2 log
σ2 β2 σ221 + σ212
+1
(6.45)
We should note that in an amplify and forward scheme since no decoding is done at either user, mathematically the capacities C12 and C21 will not form a real constraint. However, the conditions given in (6.42) are too loose for any practical use. Hence we choose to use the capacities C12 and C21 instead to find an upper bound for α and β. As a matter of fact this is equivalent to finding α
96 and β for a decode and forward cooperation scheme. By using the upper bound given in C12 and C21 , we will automatically satisfy (6.42). Let the transmission rates of users 1 and 2 be R1 and R2 respectively. Then we will have the capacity region as given in Fig. 6.7. The inequalities governing the rates R1 and R2 are
R1 ≤ C12 , R2 ≤ C21 , R1 + R2 ≤ C1 +C2
(6.46)
Hence, for given SNR values, we can determine the optimum values for α and β. With equal rates, R1 = R2 the maximum rates will be given by
R1 = R2 = max min(C12 ,C21 , (C1 +C2 )/2). α,β
(6.47)
and the optimum values of α, β can be found. In Fig. 6.6 we have given an example. We have plotted the contours of minimum value of C12 ,C21 , (C1 + C2 )/2 for different values of α and β for 10 randomly generated channels. The SNR values for users are η01 = 30dB and η02 = 10dB. We kept the interuser SNR at 40 dB. We see that the regions for best performance in terms of capacity according to Fig. 6.6 are α =∈ [0.2, 0.3] and β ∈ [0, 0.2]. Moreover, we see that if we make either α or β too high, we decrease capacity. We should also keep in mind that in order to exploit cooperation diversity, we need α, β 6= 0. Thus a practical selection would be α = 0.3 and β = 0.1. 6.6
Extension to more than Two Users In this section we consider the extension of the cooperative scheme to more than two users. Let
us consider M users using disjoint carrier allocation. In this situation also, two users can cooperate as described in the preceding sections. However, the question is selecting pairs of users for cooperation. As we have seen in the previous section, cooperation will improve performance in the situation where one user has a high SNR and the other user has a poor SNR. Moreover, the SNR between the users should also be high i.e. they should be co-located. Hence, given a set of users, a Voronoi diagram can be used to determine the nearest neighbors to
97 cooperate. Next, given a selection of neighbors, a user could choose the neighbor with the highest SNR for cooperation. In the example given in Fig. 6.8, there are 5 users called u1 , u2 , u3 , u4 , u5 . We need to find a partner for user u1 who has a poor SNR of 10dB. Even though user u4 has the highest SNR, we select user u2 with a SNR of 30 dB for cooperation because u2 is closer to u1 than u4 , as seen from the Voronoi diagram. If we have an odd number of users, i.e. 3 users, we can still cooperate in a manner similar to the one described in this chapter. We can do this if one of the users cooperates with the other two users. What we have presented is both the simplest and logical way of selecting pairs of users for cooperation. However, there may be better methods that can be investigated in future work.
6.7
Simulation Results In this section, we perform simulations to verify the validity of the theoretical results obtained
in the preceding sections. Our objective is to compare the performance of cooperative and noncooperative systems, and investigate the effect of various system parameters including the cooperating power levels and SNR. In section 6.4, we have seen that cooperation can increase the diversity provided that the interuser channel is fixed. Moreover, in section 6.3, we have seen that when there is a significant difference in the SNR of the users, cooperation increases the effective channel capacity. Thus we have an increase in both diversity as well as capacity due to cooperation. This should be reflected by improvement in system performance in terms of the BER. Hence, we investigate the effect of diversity alone, capacity alone and both diversity and capacity in this section. In our cooperation scheme, we have proposed both interblock [38] and intrablock [43] precoding. Thus in this section we first investigate the effect of these forms of precoding. Furthermore, we have a rate reduction due to cooperation. However, we have seen in Fig. 6.5 (a) (b) that mere rate reduction without cooperation will not increase channel capacity because the power is reduced as well. Thus we also investigate the difference in performance in mere rate reduction against rate reduction due to cooperation in the latter part of this section.
98 We simulated a 2 user system with the channels generated from the Jake’s model with normalized Doppler at 1 × 10−6 . At carrier frequency of 5.4 GHz, with symbol rate 1 × 106 symbols/sec, this corresponds to a speed of 0.2km/hour, which is virtually static. We specifically chose static channels in order to eliminate the temporal diversity created by rapid varying channels because we are only considering multipath and cooperation diversity. We simulated an OFDM system with block length N = 16 and prefix 4. The carrier allocation for each user was done randomly with equal number of carriers allocated to each user. The permutation matrices were kept as P1 = I,P2 = I. We have also reduced the power level for cooperative case in order to keep the energy spent as equal to the non cooperative case, as described in section 6.2.4. We should also note that all SNR values in this section refer to the signal to noise ratio for the non cooperative case because while cooperating, there is a reduction in the effective SNR due to transmission power reduction. We kept the interuser SNR at 30 dB in all the simulations unless stated otherwise.
Increase in Diversity by Cooperation We did simulations to find the maximum obtainable diversity by cooperation. For this, we generated 100 random channels of length L = 2 using the Jakes model as described above. Based on the theory, the maximum achievable diversity without cooperation is just 2. However, we can get better diversity of 4 by cooperation. The precoding was done as follows. For intrablock precoding as well as inter block precoding, we used 2 × 2 Hadamard matrices and grouped carriers into blocks of 2. For non cooperative comparison, only intra block precoding was done with carrier grouping of 2. We have kept the SNR difference of the two users at 0 dB in order to eliminate any gain in channel capacity due to cooperation as given in section 6.3. We have used 4-QAM modulation and used sphere decoding [71] at the receiver. We have given the results in Fig. 6.9 where we have plotted results for no cooperation, cooperation with only interblock precoding, cooperation with only intrablock precoding and cooperation with both
99 interblock and intrablock precoding. The cooperation power levels were adjusted by keeping α = β = 0.4. We see that combined interblock and intrablock precoding has the best performance because of the diversity of 4. The second best performance is by only intrablock precoding. Theoretically, intrablock precoding should enjoy a diversity of 4 as given in section 6.4. However, we should point out that the full diversity given in section 6.4 is achieved subject to a good interuser channel. Since OFDM has multiple carriers, it is difficult to guarantee a good interuser channel to all the carriers. Interblock precoding will improve the interuser channel and this is seen from Fig. 6.9 where combined interblock and intrablock precoding has best performance. We see that pure interblock precoding and no cooperation have the same BER performance and thus enjoy the same diversity which is expected from theory. We should also note that for low SNR, cooperative case performs worse due to reduction of the power level. Thus from this simulation we can conclude that the best performance is obtained by combined interblock and intrablock precoding.
Increase in Capacity by Cooperation We have kept the same 100 channels of length L = 2 used as in simulation 1 but considered varying SNR values for each user so that the SNR of user 2 (SNR02) is below the SNR of user 1 (SNR01) by 30, 20, 10 and 0 dB. According to theory, cooperation results in an increase in channel capacity when the SNR difference between the users is significant. We have used only interblock precoding by 2 × 2 Hadamard matrices for the cooperating case. In the non cooperating case, we have used intrablock precoding based on 2 × 2 Hadamard matrices. Both cooperative and non cooperative cases have the same diversity. We kept the cooperation power level of user 1, who has a good channel at α = 0.4. The cooperation power level of user 2, β was varied from 0 to 0.4 proportionally to his SNR value. Indeed, Fig. 6.10 shows that cooperation results in an improvement of performance when the SNR difference between users is greater than 10 dB. This can be attributed to the increase in channel capacity as given in section 6.3. However, since we have the same diversity, when the SNR
100 difference between the users is 0 dB, we do not see any improvement from cooperation.
Increase in both Diversity and Capacity by Cooperation In this simulation, we have considered the same configuration as in the preceding simulation except that, we have used combined interblock and intrablock precoding in the cooperation case. Thus, this will give us both an increase in diversity as well as an increase in capacity when the users have significantly different SNR values. We clearly see this in Fig. 6.11. When there is a significant difference in SNR of users, we see an improvement in performance from cooperation due to increase in capacity. Moreover, when there is no difference in SNR of users (i.e. 0 dB difference) we see an improvement in performance due to increase in diversity.
Effect of Rate Reduction due to Cooperation We also investigate the effect of the rate reduction on performance. Recall that for cooperation, we need a reduced rate of 2/3 of the full rate with no cooperation. We have considered cooperation with α = 0.4,β = 0.1 (rate 2/3), no cooperation (rate 1), and no cooperation with rate 2/3. In all cases we kept the energy constant. We see that from Fig. 6.12 that rate reduction with α = β = 0 have a degradation on performance without actual cooperation. The reason for this is the reduction of power due to the reduction of rate to keep the energy expenditure same. In this scenario, user 2 had a SNR 15dB below user 1. Thus we can conclude that the improvement in performance is not due to rate reduction but due to cooperation or the cooperative use of transmission power.
Effect of Interuser SNR Finally, we investigate how the inter user SNR affects the performance in a cooperating OFDM system. We have varied the inter user SNR and kept the SNR between the users and the receivers constant in Fig. 6.13. The SNR of user 2 was 20 dB below the SNR of user 1. The OFDM block length was N = 32 with equal carrier allocation between users. We see that the effect of interuser SNR is significant when it is below 10 dB. This underlines our assumption that the need for high
101 interuser SNR in order to fully exploit the advantages of cooperation.
6.8
Conclusions In this chapter we proposed a multiuser OFDM system that uses cooperation as a means to
improve system performance. We analyzed the performance of the system in terms of both channel capacity and diversity. In order to enhance system performance, we have considered both interblock and intrablock precoding. The main constraint of our method was to keep the energy spent to transmit a fixed amount of data equal to a system without cooperation. Due to the rate reduction in cooperative system, this implied reduction of the transmission power. Nevertheless, we have seen improvement in performance by cooperation in terms of channel capacity as well as diversity. In terms of diversity, we have shown that cooperation can double the diversity available given a fixed interuser channel. In terms of channel capacity, we have shown that there is an increase of effective channel capacity when the SNR of the users towards the receiver is significantly different from each other. Moreover, we have seen that when both users enjoy good channels and high SNR values, the increase in capacity by cooperation is negligible. In the limiting case, where both users have infinite power or SNR, the channel capacities becomes equal to the capacity of the non cooperating system. There is plenty of room for more research in this area. For example, the cooperation method proposed in this chapter is an amplify and forward scheme which uses simple scalar to determine the amount of cooperation. However, we have not determined the best amount of cooperation for a given scenario, which can be tackled in future work. Furthermore, decode and forward schemes that are superior to amplify and forward schemes should be considered in a similar setting. The design of space time codes for such systems is also an open problem. Finally, the extension of the proposed scheme to more than one user with combined time division and orthogonal frequency division multiplexing is yet to be considered.
102
h01 (n)
1
η01
η12 h21 (n)
h12 (n)
η02
0
h02 (n) η21 2 Figure 6.1: A 2 user OFDM system with one receiver
Transmitter si1 R1 si1
R1
N point I1
IFFT
Add Prefix
Map to Carriers Figure 6.2: An OFDMA transmitter
D/A
103
user1
user2 R2 si2
R1 si1 TimeSlot i H12
H21
H12 R2 si2 + ni12
H21 R1 si1 + ni21
i i R2 si+1 2 + βP2 (H21 R1 s1 + n21 )
i i R1 si+1 1 + αP1 (H12 R2 s2 + n12 )
TimeSlot i + 1 H12
H21
i+1 i i H12 R2 si+1 2 + βP2 (H21 R1 s1 + n21 ) + n12 i+1 i i H21 R1 si+1 1 + αP1 (H12 R2 s2 + n12 ) + n21 0
R1 si1 + αP1
TimeSlot i + 2
i i H12 R2 si+1 2 + βP 2 (H21 R1 s1 + n21 ) 0
R2 si2 + βP2
H12
i+1 + n12
i i H21 R1 si+1 1 + αP1 (H12 R2 s2 + n12 )
+ ni+1 21
H21
don0 t care Figure 6.3: Cooperative STBC OFDM scheme
don0 t care
104
user1 R1 si1
user2 H21 H21 R1 si1 + ni21 R2 si2
TimeSlot i H12 H12 R2 si2 + ni12
i i R1 si+1 1 + αP1 (H12 R2 s2 + n12 )
TimeSlot i + 1 H12
H21 i+1 i i H21 R1 si+1 1 + αP1 (H12 R2 s2 + n12 ) + n21 i i R2 si+1 2 + βP2 (H21 R1 s1 + n21 )
i + ni ) + ni+1 + βP (H R s H12 R2 si+1 2 21 1 1 2 12 21
0
R1 si1 + αP1
i+1 i+1 i i H12 R2 s2 + βP2 (H21 R1 s1 + n21 ) + n12 H21
TimeSlot i + 2
don0 t care 0 i i+1 i+1 i i R2 s2 + βP2 H21 R1 s1 + αP1 (H12 R2 s2 + n12 ) + n21 H12
don0 t care Figure 6.4: Cooperative STBC OFDM scheme with Time Division Duplexing
105
30
−2
30
−0.1
20
4
η02/(dB)
η02/(dB)
−2
−0.1
6
15
15
.1 −0
2
4
.5
10 −0.14
−0.5
−0.12 −0
.06
5
−2
5
−0
−1.5
−0.1
.1
08
−1
−0
10 − 0.
−0.12
−0.1 −0
0
5
10
15 η01/(dB)
20
25
0
30
−1
0
5
10
15 η01/(dB)
1
0
20
0
0
η02/(dB)
20
η02/(dB)
0
1
25
15
30
2
25
0.5
25
2
1.5
1
−2
20
3
0.5
1
−1.5 −2
(b) α = 0, β = 0 30
0
2
−1
−1.5 −2
(a) α = 0, β = 0 30
−0.5
−1 −1.5
−2 .5 −3
.08
0
−0.5
−1.5
25
−0.12
−0.1
20
−1
6 −0.1
25
15
0
−1
10
10
−1
−2
5
5
−3
0
−1
−1
0
0
5
10
15 η01/(dB)
20
25
0
30
(c) α = 0.5, β = 0.0 30
25
30
(d) α = 0.5, β = 0.0 0
2
1 0
1
0.5
25
−3 20
15 η01/(dB)
30
1
1.5
−3 10
5
−2
−2
−2
0
25
1 0 0
0
20
20
0.5
0
η02/(dB)
η02/(dB)
−1
0
15
15
5
−2
0. 0
1
0
10
5
1
2
0
−1
5
0.
5
0
10
0
5
10
15 η01/(dB)
20
(e) α = 0.5, β = 0.5
25
2
1
1. 5
−2 0
30
0
−4
0
−1
−3
0
1
−2 5
10
15 η01/(dB)
20
25
30
(f) α = 0.5, β = 0.5
Figure 6.5: Excess capacity C1 + C2 − 2C˜1 − 2C˜2 (a),(c),(e) Simulated (b),(d),(f) Theoretical upper bounds
106
1
5.3465
5.3465 5.3465
0.9
5.4846
5.4846
0.8
5.4846 5.6226
5.6226
0.7
5.6226
0.6
5.7606
α
5.7606
5.760
0.5
6
5.8987
5.8987
0.4
5.89
0.3
87
0.2
5.8987 5.8987
0.1
0
5.7606
5.8987 5.7606
5.7606 0
0.1
0.2
0.3
5.6226 0.4 0.5 β
5.6226
0.6
0.7
0.8
5.4846 0.9
1
Figure 6.6: Minimum of C12 , C21 , (C1 +C2 )/2 for various α and β values
R1
R1 ≤ C12 R1 + R2 ≤ C1 +C2
R2 ≤ C21 0
R2 Figure 6.7: Capacity region for cooperation
107
u1 u2
10dB
30dB u3 20dB u4 40dB
u5
30dB
Figure 6.8: Voronoi diagram for users u1 , u2 , u3 , u4 , u5 . In each Voronoi cell, the SNR of each user at the receiver is given.
0
10
−1
10
−2
BER
10
−3
10
Cooperation, Intrablock Precoding No Cooperation
−4
10
Cooperation, Intrablock+Interblock Precoding Cooperation, Interblock Precoding
5
10
15 SNR01/(dB)
20
25
Figure 6.9: Performance of a 2 user OFDM system with and without cooperation for 4 QAM
108
0
10
−1
10
−2
10 BER
SNR02=SNR01, α=β=0.4
SNR02=SNR01−10dB, α=0.4, β=0.2
−3
10
SNR02=SNR01−20dB, α=0.4, β=0.1
−4
10
With Cooperation
SNR02=SNR01−30dB, α=0.4, β=0
No Cooperation −5
10
5
10
15
20
25
30
SNR01/(dB)
Figure 6.10: Performance of a 2 user OFDM system with and without cooperation for 4 QAM
0
10
−1
10
−2
10 BER
SNR02=SNR01, α=β=0.4 SNR02=SNR01−10dB, α=0.4, β=0.2
−3
10
SNR02=SNR01−20dB, α=0.4, β=0.1
−4
10
SNR02=SNR01−30dB, α=0.4, β=0
With Cooperation No Cooperation −5
10
5
10
15
20
25
30
SNR01/(dB)
Figure 6.11: Performance of a 2 user OFDM system with and without cooperation for 4 QAM
109
0
10
Cooperation, α=0.5,β=0.1 No Cooperation Cooperation, α=0.0,β=0.0
−1
BER
10
−2
10
−3
10
−4
10
0
5
10
15
20
25
30
SNR01 (dB)
Figure 6.12: Performance of cooperation and rate reduction
0
10
−1
10
η01=20dB
−2
BER
10
η01=25dB
−3
10
−4
10
η01=30dB
−5
10
0
5
10
15 SNR12/(dB)
20
25
30
Figure 6.13: Performance of cooperation with interuser SNR variation.
110 Chapter 7. Energy Efficient Channel Estimation in MIMO Systems
7.1
Energy Consumption in MIMO Systems The use of multiple input multiple output (MIMO) channels formed using multiple transmit/receive
antennas has been demonstrated to have great potential for achieving high data rates [18]. Of concern, however, is the increased complexity associated with multiple transmit/receive antenna systems. First, increased hardware cost is required to implement multiple RF chains. Second, increased complexity and energy is required to estimate large size MIMO channels. Energy conservation in MIMO systems has been considered in different perspectives. In [57] for instance, hardware level optimization is done to minimize energy. On the other hand, in [13],[62], energy consumption is minimized at the receiver by using low rank equalization. In [47] reducing the order of MIMO systems by selection of antennae is given as a viable option to minimize energy consumption both at the receiver and transmitter, without degrading the system performance. In [10] the transmission and circuit energy consumption per bit of information transmitted is analyzed. The authors claim in [10] that single input single output (SISO) (1 × 1) systems gives best performance over MIMO (2 × 2) systems for short range transmission. In this chapter we focus on MIMO channel estimation subject to delay and error constraints. We propose an antenna selection scheme for channel estimation that can minimize energy consumed both at the transmitter and the receiver. Note that antenna selection for data transmission [47] requires at least partial knowledge of the full channel matrix. Hence, the proposed scheme can be applied before the antenna selection is done for data transmission. We can summarize the novelty of the proposed scheme as follows. (i) we concentrate exclusively on the channel estimation phase unlike in [10] where the authors have considered the data transmission phase; (ii) we propose an antenna selection scheme to minimize energy during channel estimation unlike [47] where information theoretic performance during data transmission is considered for antenna selection; (iii) the proposed method can be applied independent of the hardware or
111 software used for channel estimation. In fact, the hardware and software can be optimized independently of the proposed method as in [57]. The rest of the chapter is organized as follows. In the next section we describe the generalized energy reduction scheme. After this we focus on minimizing energy at the transmitter and the receiver separately. Next we consider joint transmitter and receiver energy minimization. To illustrate our method we consider a scalar MIMO system of arbitrary size and give comparisons of energy and error variation for different channel estimation schemes obtained by varying the number of active transmit/receive antennas under a fixed delay and error constraint.
7.2
General Methodology In this section we describe the proposed method in a general sense.
h00
x0 (t) x1 (t)
xM−1 (t)
y0 (t) h10
h01 h(N−1)0
y1 (t)
yN−1 (t)
Figure 7.1: MIMO channel The fundamental property that we assume in our scheme is the modularity of hardware. For instance, when a complex hardware system is built, it is done in a modular way by assembling less complex blocks. Hence, a MIMO system can be considered as a collection of SISO systems, with respect to hardware. For instance, we assume a 4 by 4 MIMO system can operate as a 2 by 2 system by turning off some modules. Let us consider a MIMO system with M transmitters and N receivers as given in Fig. 7.1. We call the set of transmitters T and the set of receivers R. Their cardinalities, |T| and |R|, are M and N respectively. The objective is to estimate the channels hi j , 0 ≤ i ≤ N − 1, 0 ≤ j ≤ M − 1 in an energy efficient manner. The channel estimation requires the consumption of energy and time. We make the following assumptions:
112 A(1) We can ignore electromagnetic interaction between antenna elements. Thus, if we estimate hi j by having active only a subset of transmitters/receivers, the estimate will be the same as the estimate we would get for the same channel if all transmitters/receivers were active. A(2) The channels are frequency flat fading and during the training phase, the channels remain time invariant. We propose the following divide and conquer strategy. Instead of estimating all M by N channels at once, we estimate subsets of channels step by step. This seemingly gives an obvious reduction in complexity at the receiver. For instance, if we estimate all the channels at once, the complexity is O(NM 2 ), (inversion of an N by M matrix is approximately O(NM 2 )) assuming a matrix inversion is required. However, if we use only half the transmitters M/2 and all receivers N with two steps, the complexity is 2O(NM 2 /4) which cuts the complexity by half. However, such reduction does not consider the energy required for transmission and data acquisition and so we need more detailed models. Instead of estimating the full channel matrix at once (which we call the naive method), we propose to estimate the full channel matrix in K steps. On the k-th step, (k ∈ [1, K]) we select the transmitters given by the set Tk (⊆ T) and the receivers given by the set Rk (⊆ R) and estimate the channels between those transmitters and receivers. Let Pk be the power level of each transmitter at the k-th step, and lk denote the length of training data to be used in channel estimation. Moreover, let the noise power level at the receiver be σ2 . Hence, at the k-th step, the average SNR at the receiver will be proportional to Pk /σ2 . We assume all transmitters have the same fading level, i.e., each transmitter is approximately at the same distance from the receiver and the channel is flat fading. We will focus on minimizing the total energy consumption, both at the receiver and transmitter. We define the following functions. Let gT be the energy spent by all the transmitters. At the receivers, the energy consumption can be broken down into two components: the energy required to perform data acquisition and storage, which we denote by gI , and the energy needed to perform channel estimation or computations, which we denote by gC . In our formulation, gT , gI , and gC are
113 functions of the variables K, Tk , Rk , lk , Pk k = 1, . . . , K. For notational convenience, this dependence in not shown in the sequel. The total energy consumed can be given as
g = gT + gI + gC .
(7.1)
Our objective is to minimize g. Next we consider the constraints involved. • Avoiding trivial solutions: In order to estimate all the channels we need [
k=1,...,K
Tk ⊗ Rk = T ⊗ R
(7.2)
where ⊗ is the Cartesian product. In order to avoid trivial solutions we need Tk 6= φ, Rk 6= φ, k ∈ [1, K]
(7.3)
where φ is the null set. • Satisfying a channel MSE constraint: For acceptable performance, the mean channel estimation error (MSE) at each step εk should be below a minimum threshold, εk = εk (Tk , Rk , Pk , lk ) ≤ ε, k ∈ [1, K].
(7.4)
The exact expression for εk is dependent on the channel estimation method. If we consider the power level at each step, it should be higher than some threshold Pk for the channel estimation to work, and it should be lower than the maximum allowed by the transmitter P.
Pk ≤ Pk ≤ P, k ∈ [1, K]
(7.5)
114 • Satisfying a transmission delay constraint: The training length at step k should be above a certain threshold lk for the channel estimation to work and the total data length would be below the maximum delay allowed L. K
lk ≤ lk , k ∈ [1, K],
∑ lk ≤ L
(7.6)
k=1
Our objective is to find Tk ,Rk ,Pk and lk for k = 1, . . . , K subject to the above constraints (7.2), (7.3), (7.4), (7.5), (7.6), that minimizes g given in (7.1). This is an NP hard problem. However, we pursue simplified solutions in the following sections. Before we proceed, let us consider the feasibility of the problem. We see that all the parameters are bounded. Hence the feasibility region is bounded and in order to find feasible solutions, we should choose the limits ε and L in a suitable manner. For instance if we choose ε = 0 or L = 0 it is obvious that no solutions exist. Hence by increasing either or both of these values, we can increase the feasibility region. In other words, we can trade off energy with channel estimation error and delay.
7.3
Minimizing Energy at the Transmitter We make the following assumptions. B(1) We assume the energy spent for storage at the receiver is negligible compared to the energy required for transmission. This allows us to always make Rk = R. In other words, we use all receivers at all steps. B(2) The selection of Tk and Rk will change the error because the channels are different between each transmit/receive pair. However, we have no way of knowing this because that is what we are trying to estimate. Hence, the only variable that these sets have is their size, i.e. |Tk | and |Rk |. B(3) We select disjoint sets of transmitters, i.e. Tk are disjoint. In other words
115 each transmitter only transmit during one step. Proposition 8 The channel estimation error at the k-th step εk = c1
σ2 |Tk | Pk lk
(7.7)
where σ2 is the noise variance, c1 is a constant. The proof is given in Appendix D.1. The total energy spent by all the transmitters can be given as K
gT =
∑ c2 Pk lk |Tk |
(7.8)
k=1
where c2 is a constant. Due to Rk = R, and Tk being disjoint, we can simplify (7.2) as K
∑ |Tk | = |T| = M.
(7.9)
k=0
This is a standard integer partition problem. For instance if M = 4 the ways we can select the number of transmitters during the K steps are {4}(K = 1),{3, 1}(K = 2),{2, 2}(K = 2),{2, 1, 1}(K = 3) and {1, 1, 1, 1}(K = 4). Thus there are 5 possible ways in this case. If the number of possible ways of selecting |Tk | is p(M) for |T| = M, we have [25] π√(2/3)M e 1 . p(M) ≈ √ M 4 3
(7.10)
For small values of M, i.e. M ≤ 10, we can try all possible partitions to find the best one. Once we have enumerated Tk the problem reduces to K
min
∑ c2Pk lk |Tk |
Pi ,li ,i∈[1,K] k=1
subject to (7.4), (7.5), (7.6), where Tk , Rk and K are constants.
(7.11)
116 Proposition 9 Under assumptions A(1)-A(2) and B(1)-B(3), the channel estimation scheme that minimizes transmitter energy is to reduce the MIMO channel into a set of single input multiple output (SIMO) channels and transmit using one transmitter only at a time. Thus each time we estimate a SIMO channel. The minimum energy is σ2 M ε
(7.12)
σ2 2 M ε
(7.13)
g = c1 c2
as opposed to the energy of the naive method
g = c1 c2
The proof is given in appendix D.2. This result agrees with intuition since in this case there is reduced interference from other transmitters. However under different assumptions and different channel estimation schemes, we might get different results.
7.4
Minimizing Energy at the Receiver In contrast to the transmitter, the energy consumption at the receiver is due to data acquisition
and computation. From Appendix D.1 we see that the computational energy required will be gC,k = c3 |Tk ||Rk |lk2
(7.14)
where c3 is a constant. The energy required for data acquisition and storage will be proportional to the data length. Hence gI,k = c4 |Rk |lk
(7.15)
where c4 is a constant and the total energy will be K
gT =
∑ c3 |Tk ||Rk |lk2 + c4|Rk |lk
k=1
(7.16)
117 Our objective is to minimize gT subject to the (7.4), (7.5), (7.6) constraints. Proposition 10 Under assumptions A(1)-A(2) and B(2), the channel estimation scheme that minimizes the energy consumption at the receiver is to estimate each SIMO channel individually by using one transmitter and all receivers at each step. The minimum energy is 4 σ2 2 σ g = NM c3 c1 2 2 + c4 c1 P ε Pε
(7.17)
as opposed the the energy of the naive method 4 σ2 2 σ 2 g = NM c3 c1 2 2 M + c4 c1 P ε Pε
(7.18)
The proof is given in Appendix D.3.
7.5
Minimizing Energy Both at the Transmitter and Receiver From Propositions 9 and 10 we can conclude that the optimal scheme of channel estimation
for a MIMO system that minimize both transmitter and receiver energy consumption is to reduce the system into a set of SIMO channels and estimate each SIMO channel individually. In other words, instead of transmitting the training symbols from all transmitters simultaneously, we have to transmit them in a sequential manner by activating only one transmitter at a time. In order to satisfy the delay requirement, each transmitter will be active only for a fraction of the time it would have been active if all transmitters were transmitting simultaneously.
7.6
Numerical Example We considered an 8 × 8 system with SNR 20 dB. In Fig. 7.1 we show results of 4 possible
schemes for channel estimation. Our constraints are, maximum error ε = 10−3 and delay L = 56. In scheme 1, we used 56 symbols per each transmitter (total 448) and employed the naive method to estimate the 8 × 8 system. In scheme 2 we used Proposition 9 and used 7 symbols per each
118 transmitter (total 56) to estimate the 8 × 1 SIMO systems (8 times). In scheme 3 we transmitted 7 symbols from each transmitter (total 56) and again used the naive method. Finally in scheme 4 we used 14 symbols per each transmitter but reduced the system into 4, 8 × 2 systems (total 112) to estimate the channel in 4 steps. We see that scheme 1 has the lowest error but highest energy consumption. Scheme 3 has the lowest energy consumption and lowest delay, but the channel cannot be estimated because the training matrix does not have full row rank. Schemes 2 and 4 have intermediate performance in terms of error and energy, and we see that a trade off can be accomplished between error and energy. Although in this example schemes 2 and 4 have higher channel estimation error, the final conclusion can be drawn only after numerical evaluation of the performance in terms of the bit error rate. The constants P, c1 ,c2 ,c3 ,c4 can be calculated given the hardware or can be experimentally measured.
7.7
Conclusions Using a generic model for channel estimation error and energy consumption of a MIMO sys-
tem, we have shown that the optimal channel estimation scheme in terms of minimizing energy consumption is to convert the MIMO system into a set of SIMO channels by activating each transmitter individually and performing channel estimation on each SIMO system. However, the energy reduction comes at an increase in estimation error. We should emphasize that more accurate models for estimation error and energy consumption can be used to find the optimal scheme if the specific hardware and software is known. However, the general methodology will not change. In our formulation, we have assumed a homogeneous, uncorrelated set of transmitters and receivers with isotropic radiation patterns. There is room in this area for future work on adapting this method to a MIMO channel formed by a disparate set of transmitters and receivers with different power, computation and storage capabilities and also with different radiation patterns.
119
−3
x 10 3.5
Variation with M at N=4 Variation with N at M=4 3
2.5 MSE
J=30
2
J=50 1.5 J=70 1 2
4
6
8
10
12
14
16
18
20
N or M
Figure 7.2: MSE variation with N, M and J . We see that the MSE is independent of N, has a linear variation with M and is inversely proportional to J.
Table 7.1: MSE and energy for different channel estimation schemes for 50, random 8 by 8 channels . Scheme 1 2 3 4
MSE/10−3 0.2 1.0 ∞ 0.8
Energy c2 P448 + c3 (448)2 + c4 448 c2 P56 + c3 (56)2 + c4 448 c2 P56 + c3 (56)2 + c4 56 c2 P112 + c3 (112)2 + c4 448
120 Chapter 8. Conclusions
In this thesis we have considered several disparate areas of communications systems and have proposed possible improvements. However, we stress that for a real system a holistic approach should be taken where combined improvements should be sought. We should always keep in mind that by improving efficiency of one aspect, we might decrease the efficiency of another. For instance, blind channel estimation improves efficiency by eliminating the need for training. However, blind channel estimation may be computationally more intensive and also may result in poor BER performance due to increased channel error. The channel estimation methods proposed in this thesis were developed for communications systems with sinusoidal carriers. However, carrier free ultra wideband systems are becoming increasingly attractive for short range high speed communications. There is room to adapt the blind channel estimation method proposed in chapter 2 especially for such systems. OFDM is an efficient method to tackle multipath channels. The channel estimation method has significant room for improvement in terms of precoder design and in the number of carriers selected for channel estimation in frequency domain. Moreover, the cooperative transmission scheme proposed in chapter 6 is basically aimed at operation under multipath channels. However, we have not considered space time code design in this cooperation scheme. Neither have we considered the possibility of blind channel estimation. We should point out that this is an area with huge research potential. Finally, the divide and conquer strategy proposed in chapter 7 should not be limited to channel estimation. There are methods to both do channel estimation on one set of transceivers and do data transmission on another set of receivers. Moreover, in practical systems we have to take into account the correlation between transceivers in implementing such a divide and conquer scheme. Since in practice, antennae will not have isotropic radiation patterns, it is possible to consider the divide and conquer strategy taking into account the radiation pattern of each antenna.
121 Undoubtedly, wireless technology will be one of the driving forces of the twenty first century technological advances. Hence the opportunities to do research is endless and without such research being carried out, wireless will never become that driving force.
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132 Appendix A. Appendices of Chapter 4
A.1 Proof of Proposition 2 From (4.8) we have Rikl = σ2 H(k)Diag(wikl )HH (l) + δk,l σ2n IMR , i = 0, ..., M − 1
(A.1)
H T wikl = [wH 0 (k; i)w0 (l; i), ..., wMT −1 (k; i)wMT −1 (l; i)] .
(A.2)
where
Let us extract the u-th column of Rikl . This can be done by right multiplying Rikl by the column selection vector eu . Based on (A.1) it can be seen that: Rikl eu = σ2 H(k)Diag[HH (l)eu ]wikl + δk,l σ2n eu
(A.3)
MT −1 . It holds: The matrix Cukl is formed based on the u−th columns of matrices R0kl , R1kl , ..., Rkl
Cukl
=
1 0 T −1 eu ] [R eu , R1kl eu , ..., RM kl σ2 kl
(A.4)
T −1 = H(k)Diag[HH (l)eu ][w0kl , ..., wM ] + δk,l σ2n eu 1TMT kl
(A.5)
= H(k)Diag[HH (l)eu ]Ukl + δk,l σ2n eu 1TMT
(A.6)
Post multiplying by of U†kl (assuming Ukl have full row rank) yields (4.12).
133 A.2 Proof of (4.18)
ˆ † (m)Rimm (H ˆ † (m))H Qilu (m) = H lu lu = (Diag(HH (l)eu ))−1 Diag(wimm )(Diag(HH (l)eu ))−H
(A.7)
= |Diag(HH (l)eu )|−2 Diag(wimm )
(A.8)
where Rimm is given in (4.8) and wimm in (A.2).
A.3 Proof of (4.38) Starting from (A.1), let us consider a 4 by 4 system (MT = MR = 4). Let the generic channel matrix be
H 00 (k) H 01 (k) H 02 (k) H 03 (k)
H 10 (k) H 11 (k) H 12 (k) H 13 (k) H (k) = H (k) H (k) H (k) H (k) 20 21 22 23 H 30 (k) H 31 (k) H 32 (k) H 33 (k)
(A.9)
Then, the element of on row u column v of Rikl in (A.1) can be written as 3
Rikl (u, v) = σ2 ∑ wHj (k; i)w j (l; i)H u j (k)H v?j (l), u, v, i ∈ [0, 3]
(A.10)
j=0
However, the channel matrix has a special structure given in (4.35) due to STBC. Hence, we have ? (k) etc. Substitution of these values into (A.10) reduces the H 00 (k) = H00 (k) and H 11 (k) = −H00
number of unknowns. For instance, if we construct a row of 0, 0 elements of the matrices R0kl to R1kl , we have
[R0kl (0, 0), R1kl (0, 0), R2kl (0, 0), R3kl (0, 0)] = [H00 (k), H01 (k), H02 (k), H03 (k)]D0 Ukl
(A.11)
134 where D0 = Diag[(H00 (l))? , (H01 (l))? , (H02 (l))? , (H03 (l))? ]. Using (A.11), we can solve for the first row of (4.36). We can derive similar relationships for the other elements of the matrices R0kl to R1kl and generalize to (4.38), provided that we know H(l).
135 Appendix B. Appendices of Chapter 5
B.1 Proof of Proposition 3 First, we make the following observation. Let d be a vector of complex, random, i.i.d. elements di , with E{di } = E{di? } = 0, E{di2 } = E{di?2 } = 0, E{|di |2 } = σ2 , and E{|di |4 } = η4 . Then η4 , σ4 , ? ? E{di d j d p dq } = σ4 , 0,
i = j, p = q, p = i i = j, p = q, p 6= i
(B.1)
i 6= j, p = i, q = j otherwise
Secondly, let us consider C = E{ddH AddH }, where d is an N by 1 vector with random elements as described above and, A is an arbitrary deterministic matrix. Then, we can show C = σ4 A + σ4 trace(A)I + (η4 − 2σ4 )Diag(A(0, 0), A(1, 1), . . . , A(N − 1, N − 1))
(B.2)
Proof: We have C = E{(dH Ad)ddH }. If C = [Ci, j ] and A = [A(i, j)], Ci, j = E{(dH Ad)di d ?j } =
N−1
∑ eTp AE{(di d ?j d ?p )d}
(B.3)
p=0
For i = j, we have Ci, j = eTi Aη4 ei + σ4
N−1
∑
eTp Ae p = η4 A(i, i) + σ4
p=0,p6=i
N−1
∑
A(p, p)
(B.4)
p=0,p6=i
and for i 6= j Ci, j = eTi Aσ4 e j = σ4 A(i, j) Combining (B.4) and (B.5), we get (B.2).
(B.5)
136 Using (4.24) we get
ˆ ikl } E{R
=
1
b I−i−1 M c
b I−i−1 M c+1
p=0
∑
E{yi+pM (k)(yi+pM (l))H }
(B.6)
Using (4.5) we have
ˆ ikl } = E{R
1
b I−i−1 M c
b I−i−1 M c+1
p=0
∑
(B.7)
H(k)Ψi+pM (k)E{di+pM (di+pM )H }(Ψi+pM (l))H HH (l) i+pM i+pM H +E{n (k)(n (l)) } Because of the M periodicity in precoding Ψi+pM (k) = Ψi (k), ˆ ikl } = σ2 H(k)Ψi (k)(Ψi (l))H HH (l) + σ2n δkl I = Rikl + σ2n δkl I E{R
(B.8)
Next we have
ˆ j }= ˆ ik l )H R E{(R k2 l 1
1
I− j−1 b I−i−1 M cb M c
1
I− j−1 b I−i−1 M c+1 b M c+1
∑
∑
(B.9)
p=0 q=0 i+qM H
E{yi+pM (l)(yi+pM (k1 ))H yi+qM (k2 )(y
(l)) }
when i 6= j ˆ ik l )H R ˆ j } = E{(R ˆ ik l )H }E{R ˆj } E{(R k2 l k2 l 1 1 ≈ (Rik1 l )H Rk2 l + σ2n (δk1 l Rk2 l + δk2 l (Rik1 l )H ) j
j
(B.10)
137 when i = j
ˆ ik l )H R ˆ j }= E{(R k2 l 1
1 I−i−1 (b M c + 1)2
I−i−1 b I−i−1 M c b M c
∑
∑
(B.11)
p=0 q=0,q6= p
E{yi+pM (l)(yi+pM (k1 ))H }E{yi+qM (k2 )(yi+qM (l))H } +
b I−i−1 M c
∑
i+pM
E{y
i+pM
(l)(y
H i+pM
(k1 )) y
i+pM
(k2 )(y
p=0
H
!
(l)) }
= (Rik1 l )H Rik2 l + σ2n (δk1 l Rik2 l + δk2 l (Rik1 l )H ) 1 + I−i−1 (b M c + 1)2
b I−i−1 M c
∑
p=0
E{yi+pM (l)(yi+pM (k1 ))H yi+pM (k2 )(yi+pM (l))H } !
− E{yi+pM (l)(yi+pM (k1 ))H }E{yi+pM (k2 )(yi+pM (l))H }
We neglect terms containing powers of σn higher than two and using the fact that noise is complex circular white, get non-zero terms as
E{yi (l)(yi (k1 ))H yi (k2 )(yi (l))H }
(B.12)
= E{H(l)Ψi (l)di (di )H Ak1 ,k2 di (di )H (Ψi (l))H HH (l)} +E{H(l)Ψi (l)di diH Ψi (k1 )HH (k1 )ni (k2 )niH (l)} +E{H(l)Ψi (l)di niH (k1 )ni (k2 )diH ΨiH (l)HH (l)} +E{ni(l)diH ΨiH (k1 )HH (k1 )H(k2 )Ψi (k2 )di niH (l)} +E{ni(l)niH (k1 )H(k2 )Ψi (k2 )di diH ΨiH (l)HH (l)} i = H(l)Ψ (l) σ4 Ak1 ,k2 + σ4 trace(Ak1 ,k2 )I
+(η − 2σ )Diag(Ak1 ,k2 (0, 0), . . . , Ak1 ,k2 (N − 1, N − 1)) (Ψi (l))H HH (l) 2 i i i 2 +σn δk2 l Rlk1 + δk1 l Rk2 l + δk1 k2 Rll + σ trace(Ak1 ,k2 )I 4
4
where Ak1 ,k2 = ΨiH (k1 )HH (k1 )H(k2 )Ψi (k2 ) and the result in (B.2) was used in simplification.
138 Moreover, we have
E{yi (l)(yi (k1 ))H }E{yi (k2 )(yi (l))H }
= σ4 H(l)Ψi (l)Ak1 ,k2 ΨiH (l)HH (l) + σ2n δk2 l Rilk1 + δk1 l Rik2 l
(B.13)
Substituting (B.12) and (B.13) in (B.11), we get for i = j ˆ j } = (Rik l )H Rik l + σ2n (δk1 l R j + δk2 l (Rik l )H ) ˆ ik l )H R E{(R k2 l k2 l 2 1 1 1 1 + I−i−1 H(l)Ψi (l) σ4trace(Ak1 ,k2 )I b M c+1
+(η − 2σ )Diag(Ak1 ,k2 (0, 0), . . . , Ak1 ,k2 (N − 1, N − 1)) ΨiH (l)H HH (l) ! 2 i 2 +σn δk1 k2 Rll + σ trace(Ak1 ,k2 )I 4
4
(B.14)
Finally we formulate an estimate for error. We combine (4.24) and (5.1) to form ξ(k) =
1 ˆ0 ˆ M−1 − RM−1 )eu ]U† Diag(HH (l)eu )−1 [(Rkl − R0kl )eu , . . . , (R kl kl kl σ2
(B.15)
Using the result obtained in (B.8), we get (5.2). Next we consider the variance † H −1 ξH (k1 )ξ(k2 ) = (Diag(HH (l)eu ))−H (U† )H k1 l R Uk2 l (Diag(H (l)eu ))
(B.16)
where R = [Ri, j ] with Ri, j =
1 T ˆi ˆ j − R j )eu e (R − Rik1 l )H (R k2 l k2 l σ4 u k1 l
ˆ ik l − Rik l )H (R ˆ j − R j )} E{(R k2 l k2 l 1 1
iH iH 2 ˆ iH ˆ = E{R k1 l Rk2 l } − Rk1 l Rk2 l − σn δk1 l Rk2 l + δk2 l Rk1 l j
j
j
(B.17)
(B.18)
139 Hence using (B.10) and (B.14), δi, j i T E{Ri, j } = I−i−1 eu H(l)Ψ (l) trace(Ak1 ,k2 )I b M c+1
(B.19)
η4 − 2σ4 Diag(Ak1 ,k2 (0, 0), . . . , Ak1 ,k2 (N − 1, N − 1)) (Ψi (l))H HH (l) + σ4 ! σ2n i 2 + 2 δk1 k2 Rll /σ + trace(Ak1 ,k2 )I eu σ
which yields (5.3).
B.2 Construction of T Consider no precoding. We have the NMT data vector d. In the 0-th carrier, given by the locations 0, 1, . . . , MT − 1, the 0-th symbol of each user is transmitted. These symbols are on 0, N, 2N, . . . , (MT − 1)N locations of d. Similarly, on the k-th carrier, given by locations kMT , kMT + 1, . . . , (k + 1)MT − 1, the k-th symbol of each user is transmitted. These are on k, N + k, 2N + k, . . . , (MT − 1)N + k locations of d. This mapping generalizes to T.
140 Appendix C. Appendices of Chapter 6
C.1 Capacity after interblock precoding Let us rewrite (6.11) and (6.16) as
z1 = H1 s11 + n1
(C.1)
z2 = H2 s22 + n2
(C.2)
and
whose capacities are C1 and C2 respectively. After interblock precoding, we have
˜ n1 z1 H1 W1 d + = ˜2 n2 H2 W z2
(C.3)
which is equivalent to (6.20) where
˜ 1= W
WH 11
0
0
WH 22
˜ , W2 =
0
WH 21
WH 12
0
.
(C.4)
Then the combined capacity is
˜ ˜H H H 1 W1 W1 H 1 Cz = log det σ2 ˜ 2W ˜ H HH H2 W 1
1
−1 H H ˜ ˜ 0 H1 W1 W2 H2 Rn1 + I . −1 ˜ 2W ˜ H HH 0 R H2 W 2 2 n2
(C.5)
˜ 2W ˜ H = 0 and we have ˜ 1W ˜ H =W Because W1 and W2 is unitary, W 1 2
Cz = log det σ2
H1 HH 1
0
02
H2 HH 2
R−1 n1
0
0
R−1 n2
+ I = C1 +C2 .
(C.6)
141
C.2 Proof of Proposition 5 Let us first consider M = [A B C; D E F; G H J], a matrix expressed as block 3 by 3 matrix. Then, provided M is positive definite, we can write the determinant as [18] det(M) = det(A) det(E − DA−1B) det(J − GA−1 C − (H − GA−1B)(E − DA−1B)−1 (F − DA−1C)) (C.7) Using (C.7) for C1 in (6.34) and (6.35) gives us C1 = log det(A) det(E − DA−1B) det(J − GA−1 C − (H − GA−1B)(E − DA−1B)−1 (F − DA−1C)) (C.8) where H H A = η01 |H01 |2 + I, B = D = η01 |H01 |2 , C = η02 βH01 HH 21 P2 H02 (C.9) H H H H H E = η01 (|H01 |2 + α2 H01 P1 |H12 |2 PH 1 H01 ) + I, F = η02 (βH01 H21 P2 H02 + αH01 P1 H12 H02 ), H H H H G = η01 βH02 P2 H21 HH 01 , H = η01 (βH02 P2 H21 H01 + αH02 H12 P1 H01 ), H 2 J = η02 (β2 H02 P2 |H21 |2 PH 2 H02 + |H02 | ) + I.
The key fact about (C.9) is that all the matrices are diagonal because P1 = I and P2 = I. Hence we can rewrite (C.8) as C1 = log det(A−1 ) det (EA − DB)(JA − GC) − (HA − GB)(FA − DC)
(C.10)
142 After some substitutions we have
(EA − DB)(JA − GC)
(C.11)
= I + 2η01 |H01 |2 + α2 η01 |H01 |2 |H12 |2 + α2 η201 |H01 |4 |H12 |2 + η01 |H01 |2 + 2η201 |H01 |4 +α2 η201 |H01 |4 |H12 |2 + α2 η301 |H01 |6 |H12 |2 + η02 |H02 |2 + 2η01 η02 |H01 |2 |H02 |2 +α2 η01 η02 |H01 |2 |H02 |2 |H12 |2 + α2 η201 η02 |H01 |4 |H02 |2 |H12 |2 + η01 η02 |H01 |2 |H02 |2 +2η201 η02 |H01 |4 |H02 |2 + α2 η201 η02 |H01 |4 |H02 |2 |H12 |2 +α2 η301 η02 |H01 |6 |H02 |2 |H12 |2 + η02 β2 |H02 |2 |H21 |2 + 2η01 η02 β2 |H01 |2 |H02 |2 |H21 |2 +α2 β2 η01 η02 |H01 |2 |H02 |2 |H12 |2 |H21 |2 + α2 β2 η201 η02 |H01 |4 |H02 |2 |H12 |2 |H21 |2 and
(HA − GB)(FA − DC)
(C.12)
H 3 4 2 H H = β2 η201 |H01 |2 |H02 |2 |H21 |2 + αβη201 |H01 |2 |H02 |2 HH 12 H21 + αβη01 |H01 | |H02 | H12 H21
+αβη01 η02 |H01 |2 |H02 |2 H21 H12 + α2 η01 η02 |H01 |2 |H02 |2 |H12 |2 +α2 η201 η02 |H01 |4 |H02 |2 |H12 |2 + αβη201 η02 |H01 |4 |H02 |2 H12 H21 +α2 η201 η202 |H01 |4 |H02 |2 |H12 |2 + α2 η301 η02 |H01 |6 |H02 |2 |H12 |2 . The product A−1 (EA − DB)(JA − GC) − (HA − GB)(FA − DC) will be a diagonal matrix.
Moreover, each diagonal element will be statistically independent of the other diagonal entries. Hence, we can find the expected value of the determinant (product) as the product of the expected
143 values of each diagonal entry. The expected value of each diagonal entry can be given as E{[A−1 (EA − DB)(JA − GC) − (HA − GB)(FA − DC) ]} ≈ (1 + η02 + η01 β2 )
(C.13)
1 Y1 η01
1 1 − 2 Y1 ) η01 η01 1 1 1 − +(2α2 η201 + 2η201 + 2η201 η202 + η201 η02 + α2 β2 η201 η02 )( + Y1 ) η01 η201 η301 1 1 1 2 − 2 + 3 − 4 Y1 ) +α2 η301 ( η01 η01 η01 η01 +(3η01 + α2 η01 + 3η01 η02 + 2β2 η01 η02 + α2 β2 η01 η02 − β2 η201 )(
We should note that the values η01 and η02 are not independent of the channel. However we have considered them to be independent to simplify the problem. Moreover, we have used E{Hi j (k)} = 0 and for a Rayleigh distributed random variable X 1 1/η01 1 1 }= e Ei(1, ),(C.14) 2 1 + η01 X η01 n 2 4 X 1 1 1/η01 1 X 1 1 1 1/η01 1 E{ }= − e Ei(1, ), E{ }= − − e Ei(1, ), 1 + η01 X 2 η01 η201 n 1 + η01 X 2 η01 η201 η301 n E{X } = 1, E{
E{
2 1 1 1 1/η01 1 X6 }= − + − e Ei(1, ) 1 + η01 X 2 η01 η201 η301 η401 n
Finally, using Jensen’s inequality (E{log x} ≤ logE{x} because log x is concave) we get (6.36).
C.3 Proof of Proposition 6 In order to prove this, first we find the limit
lim
σ2 →∞
C2 C1 = 2, =2 2 log σ log σ2
(C.15)
144 After rearranging terms we can write (6.36) as E{C1 } ≤ [C]k where [C]k
(C.16) =
Y1 (η01 2β2 − η02 2β2 + α2 − α3 ) + 1 − α2 + α3 η01
+η01 (2 + 2α2 − β2 − α3 ) + η02 (1 + 2β2 ) + η01 η02 (2 + α2 β2 ) + η201 2α3 is the capacity of a single carrier k. Since σ2 = η1 σ201 = η02 σ202 , we take the limit in terms of η01 instead of σ2 . Let us select γ2 =
η02 η01
=
σ201 . σ202
Hence we have
[C]k
(C.17) =
Y1 (η01 2β2 (1 − γ2 ) + α2 − α3 ) + 1 + α2 + α3 η01
+η01 ((2 + 2α2 − β2 − α3 ) + γ2 (1 + 2β2 )) + η201 (2α3 + γ2 (2 + α2 β2 )) We have ∂Y1 1 1 = − 2 Y1 ∂η01 η01 η01
(C.18)
Now we consider lim
log[C]k
η01 →∞ log η01
= lim
η01 ∂[C]k . ∂η01
η01 →∞ [C]k
(C.19)
We observe that Y1 = 0, m > 0 η01 →∞ ηm 01 lim
(C.20)
and hence the dominant terms in the limit of (C.19) will be η2 2(2α3 + γ2 (2 + α2 β2 )) log[C]k = lim 012 =2 η01 →∞ log η01 η01 →∞ η (2α3 + γ2 (2 + α2 β2 )) 01 lim
(C.21)
thus yielding a limit of 2. We see that in (6.34) and (6.35), the capacity per each user is half of each value, i.e. C1 /2 and C2 /2. Hence, the spatial multiplexing gain while cooperating is 2 × N/2 × 1/2
145 which is N/2. In comparison, we have the limit for non cooperation as log(1 + η˜01 ) = 1. η˜01 →∞ log η˜01
(C.22)
lim
and since each user has N/2 carriers allocated, the multiplexing order without cooperation becomes N/2. Thus, although cooperation results in rate reduction of 1/3, it achieves the same spatial multi
plexing gain of non-cooperation.
C.4 Proof of Proposition 7 First we show cooperation can achieve diversity of 2L when multipath channels of L order are 0
present. Following [64], the probability of d being detected when d is transmitted is 0
d 2 (y, y ) ) P(d → d |H01 , H02 , H12 , H21 ) ≤ exp(− 4N0 0
0
0
0
(C.23)
0
where d 2 (y, y ) = ky − y k2 , y = Hd and y = Hd . Then we have 0
0
0
d 2 (y, y ) = ky − y k2 = kH(d − d)k2 = kHek2 = eH HH He
(C.24)
0
where e = d − d is the error. Using e let
e1 =
R 1 WH 11
0
0
R 2 WH 22
e, e2 =
0
R 2 WH 21
R 1 WH 12
0
e
(C.25)
146 and using (6.21) let
˜ H1 =
H01
0
H01
αH01 H12
γβH02 H21
γH02
, H ˜2=
γH02
0
γH02
γβH02 H21
αH01 H12
H01
(C.26)
Then H ˜H ˜ ˜H ˜ kHek2 = eH 1 H1 H1 e1 + e2 H2 H2 e2
(C.27)
and by making e1 = [eT11 eT12 ]T , e2 = [eT21 eT22 ]T we have kHek2
(C.28)
2 2 2 2 2 H 2 2 2 H = eH 11 (2|H01 | + γ β |H02 | |H21 | )e11 + e11 (α|H01 | H12 + γ β|H02 | H21 )e12 2 2 2 2 2 H 2 H 2 2 +eH 12 (γ |H02 | + α |H01 | |H12 | )e12 + e12 (α|H01 | H12 + γ β|H02 | H21 )e11 2 2 H 2 2 2 2 2 H 2 +eH 21 (2γ |H02 | + α |H01 | |H12 | )e21 + e21 (γ β|H02 | H21 + α|H01 | H12 )e22 2 2 2 2 2 H 2 2 H 2 +eH 22 (|H01 | + γ β |H02 | |H21 | )e22 + e21 (γ β|H02 | H21 + α|H01 | H12 )e21
Using the fact that both H01 and H02 are diagonal matrices in (C.28), we can write kHek2 = h˜ H Gh˜
(C.29)
where h˜ = [h01 (0), . . . , h01 (L − 1), h02 (0), . . . , h02 (L − 1)]T and G a 2L by 2L matrix. We assume G to have full rank, conditioned on the inter-user channels H12 and H21 . Then the pairwise error probability is [64] 2L
P(d → d |H12 , H21 ) ≤ ∏ 0
i=1
1 1+
1 4N0 λi (G)
(C.30)
from which we see that for high SNR, the decay of the error probability has order 2L. This proves the achievable diversity. We also see that from (C.25) that in order to get this diversity it is sufficient to do either interblock of intrablock precoding, such that the matrices e1 and e2 do not become equal
147 to identity matrix. Next we consider the need to have α, β 6= 0. For this, we use (6.22). Using (6.22) and using the shortened notation H01 (I 1 (k)) = H01 , H02 (I 2 (k)) = H02 , H21 (I 1 (k)) = H21 , H12 (I 2 (k)) = H12 we have
HH H =
(2|H01 |2 + β2 γ2 |H02 |2 |H21 |2 )w11 wH 11 +(|H01 |2 + β2 γ2 |H02 |2 |H21 |2 )w12 wH 12
? + βγ2 |H |2 H ) (α|H01 |2 H12 02 21 H ×(w22 wH 11 + w21 w12 )
? ) (α|H01 |2 H12 + βγ2 |H02 |2 H21 H ×(w11 wH 22 + w12 w21 )
(γ2 |H02 |2 + α2 |H01 |2 |H12 |2 )w22 wH 22
+(2γ2 |H02 |2 + α2 |H01 |2 |H12 |2 )w21 wH 21 (C.31)
H assuming W1 = W2 and W1 WH 1 = W1 W1 = I.
eH HH He
(C.32)
H 2 2 2 2 2 H H = (2|H01 |2 + β2 γ2 |H02 |2 |H21 |2 )eH 1 w11 w11 e1 + (|H01 | + β γ |H02 | |H21 | )e1 w12 w12 e1 H H H ? )(eH +(α|H01 |2 H12 + βγ2 |H02 |2 H21 1 w11 w22 e2 + e1 w12 w21 e2 ) ? H H H +(α|H01 |2 H12 + βγ2 |H02 |2 H21 )(eH 2 w22 w11 e1 + e2 w21 w12 e1 ) H 2 2 2 2 2 H H +(γ2 |H02 |2 + α2 |H01 |2 |H12 |2 )eH 2 w22 w22 e2 + (2γ |H02 | + α |H01 | |H12 | )e2 w21 w21 e2
we have 0
d 2 (y, y )
2 2 2 H 2 2 2 H 2 2 H = |H01 |2 2|eH 1 w11 | + |e1 w12 | + α |H12 | |e2 w22 | + α |H12 | |e2 w21 | H H H H +2αReal(H12 (e1 w11 w22 e2 + e1 w12 w21 e2 )) 2 2 2 2 H 2 2 H 2 2 H 2 2 +|H02 | β2 γ2 |H21 |2 |eH 1 w11 | + β γ |H21 | |e1 w12 | + γ |e2 w22 | + 2γ |e2 w21 | 2 H H H H +2βγ Real(H21 (e2 w22 w11 e1 + e2 w21 w12 e1 ))
(C.33)
148 We see that if we choose α = 0 and β 6= 0 as an example, we lose the diversity for e1 = 0. Moreover, if H12 (I 2 (k)) = 0 or H21 (I 1 (k)) = 0, we would lose diversity for that particular k. Hence in order to get full spatial diversity we need both α, β 6= 0.
149 Appendix D. Appendices of Chapter 7
D.1 Channel estimation error and energy In this section we consider a MIMO system with frequency flat fading channels. We consider the least squares channel estimation using training symbols with all transmitters and receivers active (we call this the naive method). The basic equation can be given as y = Hx + v where y is a N by 1 vector, H is the N by M channel matrix, x is the M by 1 training vector and v is the N by 1 noise vector. Let us use J training blocks to estimate the channel. Grouping J blocks we have Y = HX + V where Y = [y1 , . . . , yJ ], X = [x1 , . . . , xJ ] and V = [v1 , . . . , vJ ]. For full rank condition of X we need J ≥ N. The channel estimation error is ˆ − H = VX† ξ=H
(D.1)
and the MSE MSE =
1 1 trace(ξξH ) = trace((XH X)† VH V) NM NM
(D.2)
We consider X having orthogonal rows and full row rank. Then XXH = I. However, since J > N there is no way to choose all columns orthogonally. Hence XH X will not be diagonal. Similarly, VH V will not be diagonal if J > M. If we consider any generic channel estimation scheme, we know that the channel estimation error is inversely proportional to the SNR and the data length while it is directly proportional to the interference i.e. the number of transmitters. Hence we formulate the error as MSE = cσ2
M JP
(D.3)
where σ2 is the noise power, P is the signal power and c is a constant. In order to verify above formulation, we have simulated random channels and have given the result in Fig. 7.2. By substituting J = lk , P = Pk , M = |Tk | we get (7.7).
150 Next we calculate the total number of computations required. If the rows of X are not orthogonal, the computation of the psuedoinverse X† requires approximately O(MJ 2 ) operations. The multiplication YX† requires O(MNJ 2) operations. Altogether we have an O(MNJ 2) computation assuming X† = XH .
D.2 Proof of Proposition 9 The Lagrangian (ignoring the lower bounds for Pk and lk ) is
L
=
K
K
k=1 K
k=1
∑ c2 Pk lk |Tk | + ∑ λ1,k (c1
σ2 |Tk | − ε) Pk lk
(D.4)
K
+ ∑ λ2,k (Pk − P) + λ3( ∑ lk − L) k=1
k=1
where λ1,k ,λ2,k ,λ3 are the multipliers. For optimality we need ∂L σ2 = c2 Pk |Tk | − λ1,k c1 2 |Tk | + λ3 = 0 ∂lk Pk lk
(D.5)
σ2 ∂L = c2 lk |Tk | − λ1,k c1 2 |Tk | + λ2,k = 0 ∂Pk Pk lk
(D.6)
and
We select a solution as follows. From (7.5), we select Pk = P. From (7.4) and (7.7) we select
lk = c1
σ2 |Tk | Pε
(D.7)
If Tk = T, |Tk | = M and from (7.6) we need σ2 M Pε
(D.8)
σ2 σ2 K |T | = c ∑ k 1 Pε M ≤ L Pε k=1
(D.9)
l1 = L ≥ c1 Hence K
∑ lk = c1
k=1
151 and we see that by selecting lk according to (D.7), (7.6) is automatically satisfied. Hence we have a feasible solution. Next we check its optimality. Since (7.6) is satisfied and active, we have λ3 > 0. From (D.5) we have λ1,k = (λ3 + c2 P|Tk |)
Plk2 c1 σ2 |Tk |
(D.10)
which is positive. Next from (D.6) we have λ2,k = λ3 lPk which is again positive. Hence the solution is optimal. By substitution of P = Pk and (D.7) in (7.8), we get (D.11). The minimum transmitter energy given the partition of T is gT = c1 c2
σ2 K ∑ |Tk |2 ε k=1
(D.11)
Thus we see that the partition that minimizes (D.11) consists of all ones, i.e. {1, 1, . . . , 1}. In other words, in order to minimize transmission energy, we should estimate channels selecting each transmitter individually. Substituting |Tk | = 1 into (D.11) we get (7.12) and substituting K = 1,|Tk | = M,and (D.8) we get (7.13).
D.3 Proof of Proposition 10 Note that there is no transmitter power term Pk in (7.16) and we can select Pk = P. Next we select the data length as in (D.7). By substitution into (7.16) we have σ4
K
g=
σ2
∑ c3c21 P2ε2 |Tk |3|Rk | + c4c1 Pε |Tk ||Rk |
(D.12)
k=1
and the only constraint (7.6) reduces (using (D.8)) to K
σ2
∑ c1 Pε |Tk | ≤ L,
k=1
K
or
∑ |Tk | ≤ M.
(D.13)
k=1
Minimizing (D.12) subject to (D.13) is a standard discrete programming problem. It is easy to see that in order to satisfy (D.13) we need to partition the transmitters disjointly. In that case, the partition that minimizes (D.12) is |Tk | = 1 for all k. In this case, we need to use all the receivers and the only possible partition for Rk is R. Hence we can conclude that the channel estimation
152 scheme that minimizes receiver energy consumption is to estimate each SIMO channel individually. Substituting |Tk | = 1 into (D.12) we get (7.17) and substituting K = 1,|Tk | = M,and (D.8) we get (7.18).
153 Vita
Sarod Bandara Yatawatta E DUCATION Drexel University, Philadelphia, PA, Ph.D. Electrical Engineering (Expected in December 2004) University of Peradeniya, Peradeniya, Sri Lanka, B.Sc. (Hons), Electrical and Electronic Engineering, August 2000 H ONORS
AND
AWARDS
InterDigital Travel Fellowship 2004 George Hill Jr. Fellowship 2003-2004 Ceylon Electricity Board Gold Medal and Prize for best performance, Electronic and Electrical Engineering 2000 Ranjan Herath Gunaratne Memorial Prize for best performance, Final Part I in Engineering 1999 University Scholarship for best performance, First Examination in Engineering 1998 Ananda Amarasinghe Memorial Prize for best performance, First Examination in Engineering 1998
S ELECTED P UBLICATIONS [1] Sarod Yatawatta, Athina Petropulu and Riddhi Dattani, “Blind channel estimation using fractional sampling,” IEEE Transactions on Vehicular Technology, vol. 53, no. 2, pp. 363-371, March 2004. [2] Sarod Yatawatta and Athina Petropulu, “Blind channel estimation in MIMO OFDM systems,” Under review, IEEE Transactions on Signal Processing, Submitted in December 2003. [3] Sarod Yatawatta and Athina Petropulu, “A multiuser OFDM system with cooperation,” Under review, IEEE Transactions on Signal Processing, Submitted in December 2004. [4] Sarod Yatawatta, Athina Petropulu and Charles Graff, “Energy efficient MIMO channel estimation,” IEEE International Conference on Acoustics, Speech and Signal Processing, ICASSP 2005, Philadelphia, March 2005.
P ROFESSIONAL A FFILIATIONS
AND
ACTIVITIES
Student Member IEEE and ACM Reviewer, IEEE Transactions on Signal Processing, IEEE Transactions on Communications, and IEEE Transactions on Vehicular Technology, 2004
