MIMO Networking with Imperfect Channel State Information

0 downloads 0 Views 1MB Size Report
functions of channel coherence time based on Markov chain theory. ...... by using emerging technologies in multiuser MIMO communication or simply MU-MIMO. ...... Users learn the scheduling decisions from the indices of scheduled ...... To avoid deep fading due to the lack of diversity gain, opportunistic transmission.
Copyright by Kaibin Huang 2008

The Dissertation Committee for Kaibin Huang certifies that this is the approved version of the following dissertation:

MIMO Networking with Imperfect Channel State Information

Committee:

Jeffrey G. Andrews, Supervisor

Robert W. Heath, Jr., Co-Supervisor

Constantine Caramanis

Inderjit Dhillon

Sanjay Shakkottai

Sriram Vishwanath

MIMO Networking with Imperfect Channel State Information

by

Kaibin Huang, B.Eng., M.Eng.

Dissertation Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of

Doctor of Philosophy

The University of Texas at Austin May 2008

To my family

Acknowledgments I would like to thank my dissertation advisers, Jeffrey G. Andrews and Robert W. Heath, Jr., for their vision, guidance, and encouragement. Also, to my UT colleagues, especially Chan-Byoung Chae, Ramya Bhagavatula, Bishwarup Mondal, Wan Choi and Runhua Chen, for helpful discussions and friendships. I would also like to thank my parents, Hongjia Huang and Shunyu Chen, who are the source of my strength. Last but not least, I would like to thank my wife, Qingyuan Li. Without her support and sacrifice, completing my PhD study would be impossible.

Kaibin Huang

The University of Texas at Austin May 2008

v

MIMO Networking with Imperfect Channel State Information

Publication No.

Kaibin Huang, Ph.D. The University of Texas at Austin, 2008

Supervisors: Jeffrey G. Andrews, Robert W. Heath, Jr.

The shortage of radio spectrum has become the bottleneck of achieving broadband wireless access. Overcoming this bottleneck in next-generation wireless networks hinges on successful implementation of multiple-input-multiple-output (MIMO) technologies, which use antenna arrays rather than additional bandwidth for multiplying data rates. The most efficient MIMO techniques require channel state information (CSI). In practice, such information is usually inaccurate due to overhead constraints on CSI acquisition as well as mobility and delay. CSI inaccuracy can potentially reduce the performance gains provided by MIMO. This dissertation investigates the impact of CSI inaccuracy on the performance of increasing complex MIMO networks, starting with a point-to-point link, continuing to a multiuser MIMO system, and ending at a mobile ad hoc network. Furthermore, this dissertation contributes algorithms for efficient CSI acquisition, and its integration with beamforming and scheduling in multiuser MIMO, and with interference cancelation in ad vi

hoc networks. First, this dissertation presents a design of a finite-rate CSI feedback link for pointto-point beamforming over a temporally correlated channel. We address various important design issues omitted in prior work, including the feedback delay, protocol, bit rate, and compression in time. System parameters such as the feedback bit rate are derived as functions of channel coherence time based on Markov chain theory. In particular, the capacity gain due to beamforming is proved to decrease with feedback delay at least at an exponential rate, which depends on channel coherence time. This work provides an efficient way of implementing beamforming in practice for increasing transmission range and throughput. Second, several algorithms for multiuser MIMO systems are proposed, including CSI quantization, joint beamforming and scheduling, and distributed feedback scheduling. These algorithms enable spatial multiple access and multiuser diversity in a cellular system under the practical constraint of finite-rate multiuser CSI feedback. Moreover, this dissertation shows analytically that the throughput of the MIMO uplink and downlink using the proposed algorithms scales optimally as the number of users increases. Finally, the transmission capacity of a MIMO ad hoc network is analyzed for the case where spatial interference cancelation is applied at receivers. Most important, this dissertation shows that this MIMO technique contributes significant network capacity gains even if the required CSI is inaccurate. In addition, opportunistic CSI estimation is shown to provide a tradeoff between channel training overhead and CSI accuracy.

vii

Contents Acknowledgments

v

Abstract

vi

List of Tables

xv

List of Figures

xvi

Chapter 1 Introduction

1

1.1

Limited Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.2

Point-to-Point MIMO Links . . . . . . . . . . . . . . . . . . . . . . . . . . .

5

1.3

Multiuser MIMO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

1.4

Ad Hoc Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

1.5

Overview of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.6

Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Chapter 2 Beamforming with Limited Feedback

13

2.1

Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.2

Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3

System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.1

Forward Link . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.3.2

CSI Feedback Link . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 viii

2.4

2.5

2.6

2.7

Channel State Markov Chain . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4.1

Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.4.2

Construction by Simulation . . . . . . . . . . . . . . . . . . . . . . . 23

CSI Source and Feedback Bit Rates

. . . . . . . . . . . . . . . . . . . . . . 24

2.5.1

CSI Source Bit Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5.2

CSI Feedback Bit Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 25

Feedback Delay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.6.1

Feedback Throughput Gain . . . . . . . . . . . . . . . . . . . . . . . 27

2.6.2

Effect of Fixed Feedback Delay . . . . . . . . . . . . . . . . . . . . . 30

2.6.3

Effects of both Fixed and Variable Feedback Delay . . . . . . . . . . 32

Feedback Compression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.7.1

Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.7.2

Feedback Throughput Gain with Feedback Compression . . . . . . . 35

2.7.3

Extension: Block Feedback Compression . . . . . . . . . . . . . . . . 36

2.8

Extension to Higher-Order Channel State Markov Chains . . . . . . . . . . 38

2.9

Numerical Results and Design Example . . . . . . . . . . . . . . . . . . . . 39 2.9.1

Accuracy of Channel State Markov Chain . . . . . . . . . . . . . . . 39

2.9.2

CSI Source and Feedback Bit Rates . . . . . . . . . . . . . . . . . . 42

2.9.3

Feedback Delay and feedback throughput gain . . . . . . . . . . . . 43

2.9.4

Feedback Compression . . . . . . . . . . . . . . . . . . . . . . . . . . 45

2.9.5

Design Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

2.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.11 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.11.1 Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.11.2 Proof of Corollary 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.11.3 Proof of Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.11.4 Proof of Corollary 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.11.5 Proof of Proposition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ix

Chapter 3 SDMA with Limited Feedback: Joint CSI Quantization, Scheduling and Beamforming

56

3.1

Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2

Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.3

System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

3.4

Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.5

3.6

3.4.1

Limited Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.4.2

Joint Scheduling and Beamforming . . . . . . . . . . . . . . . . . . . 64

Asymptotic Throughput Scaling . . . . . . . . . . . . . . . . . . . . . . . . 66 3.5.1

Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5.2

Normal SNR Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

3.5.3

Interference-Limited Regime

3.5.4

Noise-Limited Regime . . . . . . . . . . . . . . . . . . . . . . . . . . 71

3.5.5

Non-Asymptotic Regimes . . . . . . . . . . . . . . . . . . . . . . . . 72

. . . . . . . . . . . . . . . . . . . . . . 69

Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.6.1

Effect of Increasing Channel Shape Feedback . . . . . . . . . . . . . 74

3.6.2

Comparison with ZF-SDMA and Dirty Paper Coding . . . . . . . . 74

3.6.3

Effect of SINR Quantization

. . . . . . . . . . . . . . . . . . . . . . 79

3.7

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3.8

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.8.1

Proof of Lemma 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.8.2

Proof of Lemma 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3.8.3

Proof of Lemma 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3.8.4

Proof of Lemma 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3.8.5

Proof of Proposition 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3.8.6

Proof of Lemma 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3.8.7

Proof of Proposition 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 87

3.8.8

Proof of Proposition 8 . . . . . . . . . . . . . . . . . . . . . . . . . . 88 x

Chapter 4 SDMA with Limited Feedback: Feedback Scheduling

90

4.1

Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.2

Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

4.3

System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4.4

Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.5

4.4.1

CSI Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.4.2

Feedback Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.4.3

Joint Beamforming and Scheduling . . . . . . . . . . . . . . . . . . . 97

Feedback Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 4.5.1

Feedback Thresholds . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

4.5.2

Overflow Probability for Feedback Channel . . . . . . . . . . . . . . 103

4.6

Analysis of Sum Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

4.7

Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 4.7.1

Performance Comparison . . . . . . . . . . . . . . . . . . . . . . . . 108

4.7.2

Effect of SINR Quantization

. . . . . . . . . . . . . . . . . . . . . . 113

4.8

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.9

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.9.1

Proof of Corollary 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.9.2

Proof of Proposition 9 . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.9.3

Proof of Lemma 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.9.4

Proof of Lemma 13 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

4.9.5

Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

4.9.6

Lower Bound for Asymptotic Sum Capacity . . . . . . . . . . . . . . 118

4.9.7

Upper Bound for Asymptotic Sum Capacity . . . . . . . . . . . . . . 120

4.9.8

Additional Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

Chapter 5 SDMA with Limited Feedback: Uplink Throughput Scaling 5.1

122

Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 xi

5.2

Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

5.3

System Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.4

Limited Feedback, Scheduling and Beamforming . . . . . . . . . . . . . . . 126 5.4.1

Expected SINR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5.4.2

Expected Throughput . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.4.3

Beamforming Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 129

5.5

Background: Analytical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.6

Throughput Scaling: High SNR . . . . . . . . . . . . . . . . . . . . . . . . . 133

5.7

5.8

5.9

5.6.1

Throughput Scaling for Orthogonal Beamforming . . . . . . . . . . . 133

5.6.2

Throughput Scaling for Zero-Forcing Beamforming . . . . . . . . . . 138

Throughput Scaling: Normal SNR . . . . . . . . . . . . . . . . . . . . . . . 140 5.7.1

Orthogonal Beamforming . . . . . . . . . . . . . . . . . . . . . . . . 140

5.7.2

Zero-Forcing Beamforming . . . . . . . . . . . . . . . . . . . . . . . 142

Throughput Scaling: Low SNR . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.8.1

Orthogonal Beamforming . . . . . . . . . . . . . . . . . . . . . . . . 143

5.8.2

Zero-Forcing Beamforming . . . . . . . . . . . . . . . . . . . . . . . 144

Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

5.10 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 5.11 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.11.1 Proof of Proposition 11 . . . . . . . . . . . . . . . . . . . . . . . . . 149 5.11.2 Proof of Lemma 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 5.11.3 Proof of Lemma 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 5.11.4 Proof of Lemma 19 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.11.5 Proof of Theorem 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 5.11.6 Proof of Theorem 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.11.7 Proof of Theorem 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

xii

Chapter 6 Spatial Interference Cancelation for MIMO Ad Hoc Networks159 6.1

Prior Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

6.2

Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

6.3

Network and Channel Models

6.4

6.5

6.6

6.7

. . . . . . . . . . . . . . . . . . . . . . . . . 162

6.3.1

Network Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

6.3.2

Channel Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

6.3.3

Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

Spatial Interference Cancelation: Algorithm and Model . . . . . . . . . . . 166 6.4.1

Perfect CSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166

6.4.2

Imperfect CSI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

Outage Probability and Transmission Capacity: Perfect CSI . . . . . . . . . 174 6.5.1

Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

6.5.2

Bounds on Outage Probability . . . . . . . . . . . . . . . . . . . . . 176

6.5.3

Asymptotic Transmission Capacity . . . . . . . . . . . . . . . . . . . 177

Outage Probability and Transmission Capacity: Imperfect CSI . . . . . . . 179 6.6.1

Bounds on Outage Probability . . . . . . . . . . . . . . . . . . . . . 179

6.6.2

Asymptotic Transmission Capacity . . . . . . . . . . . . . . . . . . . 181

6.6.3

Power Ratio between Residual and Other Interference . . . . . . . . 181

Simulation and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 6.7.1

Bounds on Outage Probability . . . . . . . . . . . . . . . . . . . . . 183

6.7.2

Scaling Laws of Transmission Capacity . . . . . . . . . . . . . . . . . 185

6.7.3

Transmission Capacity vs. Size of Antenna Array . . . . . . . . . . . 189

6.8

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

6.9

Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 6.9.1

Proof of Lemma 23 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

6.9.2

Proof of Lemma 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

6.9.3

Proof of Proposition 12 . . . . . . . . . . . . . . . . . . . . . . . . . 194

6.9.4

Proof of Lemma 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 xiii

6.9.5

Proof of Theorem 12 . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Chapter 7 Conclusion

199

7.1

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

7.2

Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201

Bibliography

203

Vita

220

xiv

List of Tables 2.1

Design example of a limited feedback beamforming system . . . . . . . . . . 49

6.1

Summary of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

xv

List of Figures 1.1

MIMO cellular system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

A point-to-point MIMO link with limited feedback . . . . . . . . . . . . . .

6

1.3

Ad hoc network

9

2.1

Limited feedback beamforming system . . . . . . . . . . . . . . . . . . . . . 19

2.2

CSI autocorrelation versus time separation for the normalized Doppler shift

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

fD T = {10−2 , 10−3 }. For the “Clarke’s model”, the autocorrelation function of each channel coefficient is given by the Clarke’s function; for the “Markov chain”, the autocorrelation of quantized CSI is modeled by a firstorder Markov chain. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 2.3

Normalized CSI source bit rate versus normalized Doppler shift for the CSI codebook size N = {64, 128, 256}.

2.4

. . . . . . . . . . . . . . . . . . . . . . . 42

Feedback outage probability versus the CSI feedback bit rate normalized by the CSI source bit rate; the normalized Doppler shift is fD T = {10−4 , 2 × 10−4 , 3 × 10−4 , 10−3 }, and the CSI codebook size is N = 128. . . . . . . . . 43

xvi

2.5

Comparison of the effects of CSI quantization and feedback delay on the ergodic throughput of a transmit-beamforming system. Three cases of CSI feedback are considered, namely 1) perfect, 2) quantized, and 3) quantized and delayed CSI feedback. The number of antennas in an array is L = 4, the SNR is 10 dB, the normalized Doppler shift is fD T = 10−3 , and the quantizer codebook size is N = 128. . . . . . . . . . . . . . . . . . . . . . . 44

2.6

The normalized feedback throughput gain and its approximation versus different combinations of the fixed and protocol delays; the normalized Doppler shift is (a) fD T = 10−4 and (b) fD T = 10−3 , and the CSI codebook size is N = 128. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

2.7

Feedback compression ratio versus different combinations of Doppler shift and the CSI codebook size; the normalized feedback bit rate is (a) Rf /Rs = 3 and (b) Rf /Rs = 5, the threshold for truncating low-probability channel state transitions is ² = 0.1.

2.8

. . . . . . . . . . . . . . . . . . . . . . . . . . . 47

Spectrum of an OFDMA system at the 2.5GHz band. The 10MHz bandwidth is divided into 8 subchannels of 1 MHz. Each subchannel is assigned to one user.

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.1

Downlink system with limited feedback

. . . . . . . . . . . . . . . . . . . . 61

3.2

Comparison between asymptotic and non-asymptotic throughput scaling laws for PU2RC for SNR = {0, 5, 30} dB, the codebook size N = 16, and the number of transmit antennas Nt = 2. . . . . . . . . . . . . . . . . . . . 73

3.3

Throughput of PU2RC for an increasing number of users U , SNR = 5 dB, and the number of transmit antennas Nt = 4. . . . . . . . . . . . . . . . . . 75

3.4

Throughput comparison between PU2RC and ZF-SDMA for an increasing number of users U , SNR = 5 dB, and the number of transmit antennas Nt = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

xvii

3.5

The average numbers of scheduled users for PU2RC and ZF-SDMA for SNR = 5 dB, and the number of transmit antennas Nt = 4. . . . . . . . . . 78

3.6

Throughput comparison between PU2RC and ZF-SDMA for an increasing number SNR; The codebook size N = 64 and the number of transmit antennas Nt = 4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

3.7

Comparison between the throughput of PU2RC and its upper bound achieved by dirty paper coding (DPC) and multiuser iterative water-filling for an increasing number of users U , SNR = 5 dB, and the number of transmit antennas Nt = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

3.8

The effect of SINR quantization for SNR = 5 dB, the number of transmit antennas Nt = 4, and the codebook size for channel shape quantization is N = 16. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.1

SDMA Downlink system with feedback thresholds . . . . . . . . . . . . . . 94

4.2

Average numbers of feedback users for OSDMA-TF . . . . . . . . . . . . . . 104

4.3

Feedback channel overflow probability for OSDMA-TF . . . . . . . . . . . . 106

4.4

Convergence of the lower bound in (4.33), 1 − Pβ , to one with the number of feedback bits per user log2 N . . . . . . . . . . . . . . . . . . . . . . . . . 109

4.5

Comparison between threshold feedback (OSDMA-TF) and all-user feedback: (a) sum capacity versus total number of users and (b) average number of feedback users versus total number of users for SNR = 5 dB, the size of the shape quantization codebook N = 8, and different numbers of antennas Nt = {2, 4}. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

4.6

Comparison between OSDMA-TF, OSDMA-LF, OSDMA-BS and OSDMA: (a) sum capacity versus total number of users and (b) average sum feedback rate versus total number of users for SNR = 5 dB, the number of antennas Nt = 2 and the feedback penalty factor α = 0.05.

xviii

. . . . . . . . . . . . . . 112

4.7

Capacity comparison between perfect and quantized SINR feedback for SNR = 5 dB, the number of antennas Nt = 2, and the size of the codebook for channel-shape quantization is N = 8.

. . . . . . . . . . . . . . . . . . . 114

5.1

Uplink SDMA system with limited feedback . . . . . . . . . . . . . . . . . . 125

5.2

The bins and balls model for multiuser feedback of quantized channel shapes132

5.3

Throughput comparisons between orthogonal and zero-forcing beamforming for uplink SDMA in (a) the high SNR regime and (b) the low SNR regime. The number of antennas at the base station is Nt = 2 and the quantizer codebook size is N = 8. The plotted values in brackets specify the SNR values in dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

5.4

Throughput comparisons between uplink SDMA with limited feedback, SDMA with random scheduling and uplink random access in [1]. The number of antennas at the base station is Nt = 2; the quantizer codebook size is N = 8; the SNR = 5dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

6.1

Effective channel and network models resulting from interference cancelation with (a) perfect CSI or (b) imperfect CSI for antenna arrays of three elements. The distance in the figures is proportional to the effective channel power. The data and interference links are plotted by using solid and dashed lines, respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

6.2

Normalized residual interference power for different transmitting node densities. The number of antennas per node is L = 4 and the path loss exponent is α = 4.

6.3

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

Outage probability for different transmitting node densities and perfect CSI. The size of the antenna array is (a) L = 2 and (b) L = 4

6.4

. . . . . . . 184

Outage probability for different node densities and imperfect CSI. The size of the antenna array is (a) L = 2 and (b) L = 4

xix

. . . . . . . . . . . . . . . 186

6.5

Comparison between asymptotic bounds on transmission capacity and the exact values obtained by simulation perfect CSI and the size of antenna array L = {2, 3, 4} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

6.6

Comparison between asymptotic bounds on transmission capacity and the exact values obtained by simulation for imperfect CSI, (a) the size of antenna array L = 2 and the normalized power of residual interference η = {−10, −15, −20} dB, and (b) L = 4 and η = {−10, −15, −20} dB.

6.7

. . . . . 188

Transmission capacity by simulation for different node densities and perfect CSI. The size of the antenna array is L = 4 and the outage constraint is ² = {10−1 , 10−2 , 10−3 }.

6.8

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

Transmission capacity by simulation for different node densities and imperfect CSI. The size of the antenna array is L = 4 and the outage constraint is ² = 10−2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191

xx

Chapter 1

Introduction Wireless communication is the fastest growing area in telecommunication due to the surge of cellular telephony and wireless Internet access. The vision for wireless communication is to achieve wireless broadband access at speeds comparable to those in wireline networks. Achieving this vision must overcome two main hurdles unique for wireless communication. First, the primary resource for wireless communication–spectrum–is scarce. Wireless communications of different users in the same area interfere with each other, and have to be placed in different channels defined in frequency, time or other domains. Consequently, the bandwidth needed for wireless access increases linearly with the product of the number of users in the same area and the data rate needed for each user. Therefore, wireless broadband access in a crowded area demands a huge block of spectrum and causes spectrum shortage given that usable spectrum is finite. Second, reliability of wireless communication is severely affected by fading. Fading refers to both path-loss for radio propagation, and the amplitude fluctuation of received wireless signals due to multiplepath radio propagation. Coping with fading relies on error-correcting coding of wireless signals that demands additional bandwidth. As a result, fading becomes a bottleneck for realizing wireless broadband access. These hurdles can be overcome by multiple-input-multiple-output (MIMO) technologies which are a key breakthrough in modern communication technologies. A MIMO 1

channel is created by employing arrays of antenna elements, called multi-antennas, at both a transmitter and a receiver. In networks, antenna elements in a MIMO channel can be distributed over multiple transmitters and receivers. Antenna elements at transmitters and receivers are called transmit and receive antennas, respectively. A MIMO channel can be represented by a matrix of channel coefficients, which correspond to multiple paths created by multi-antennas. The rank of a channel matrix is called the number of spatial degrees of freedom. MIMO technologies exploit the spatial degrees of freedom for coping with fading and multiplying data rates without requiring bandwidth expansion. These advantages of MIMO technologies have attracted intensive research and development in the past ten years or so. By now, MIMO technologies have been implemented in practice and widely adopted in latest wireless communication standards including IEEE 802.16 [2, 3] for broadband wireless access, IEEE 802.11n [4] for wireless local area networks, and 3GPP-LTE [5] for next-generation cellular telecommunication. Cellular systems are the most widely used wireless communication systems and currently support about two billion users worldwide [6]. In a cellular system, a geographic area is partitioned into cells. In each cell, a base station serves many wireless users and base stations are connected by high-speed cables. This cellular infrastructure makes use of path-loss for radio transmission over a distance, and enables reuse of spectrum in different cells without causing strong inter-cell interference. MIMO technologies will be widely adopted in next-generation cellular systems [3, 5]. Several types of MIMO systems considered in this dissertation, namely a point-to-point link, a downlink and a uplink, are simplified models of a MIMO cellular system illustrated in Fig. 1.1. In this system, a pointto-point MIMO link refers a single data link between the base station and a subscriber unit, where multi-antennas are employed at both ends of the link. A downlink models a single cell where a base station transmits to multiple subscriber units. Reversing the transmission direction in downlink gives the uplink. In this dissertation, we assume no base-station coordination, and interference from other cells appear as an additional source of noise in all systems considered here. 2

Figure 1.1: MIMO cellular system In MIMO systems, channel state information (CSI) refers to instantaneous information of the channel for data transmission or interference channels. CSI is useful for adapting transmission waveforms at transmitters, detecting data or interference cancelation at receivers, or scheduling at base stations. In particular, CSI at a transmitter, called transmit CSI, finds many applications. First, transmit CSI allows a data symbol to be mapped to transmit antennas after being multiplied with weights, which are selected for maximizing the power of the received symbol. This transmission strategy and the weight vector are called beamforming and a bemaforming vector, respectively. Beamforming effectively reverses the adverse effect of fading and increases the propagation distance over which wireless signals can be reliably received, called the transmission range. Second, transmit CSI enables preceding that maps multiple data symbols onto transmit antennas. The weight matrix for preceding is called a precoces. Comparing beamforming and preceding, the former supports a single data stream with maximum reliability and hence achieves the full diversity gain; preceding enables multiple data streams in space and attains spatial multiplexing gain. Third, transmit CSI allows adaptive modulation and power control for attaining the capacity of a time-varying channel. Finally, in ad hoc networks, transmit CSI of both data and interference channels enables a transmitter to avoid interference to unintended receivers. 3

In many MIMO systems, transmit CSI must be acquired by feedback from receivers to transmitters. In systems using time-division multiplexing, transmit CSI can be obtained by estimating the reverse-link channel from the receiver to the transmitter, since its realization is very close to that of the forward-link channel given hardware calibration [7]. In practice, CSI inaccuracy exists due to finite-rate CSI feedback and imperfect CSI estimation. Inaccurate CSI inevitably degrades the performance of MIMO systems. Moreover, due to multiuser and the multiplicity of MIMO channel coefficients, acquiring CSI can incur overwhelming training or feedback overhead. Addressing these issues, the theme of this dissertation is to design efficient algorithms for CSI acquisition and characterize the fundamental effects of CSI inaccuracy on the throughput of different types of MIMO systems including the point-to-point link, the cellular downlink and uplink, and the mobile ad hoc network. Considering MIMO systems with different complexity allows us to focus on different subsets of issues related to CSI inaccuracy and thereby maintain mathematical tractability. The simplicity of a point-topoint system facilitates the investigation of practical factors that cause CSI inaccuracy, such as feedback delay and the feedback protocol. For the cellular downlink and uplink, this dissertation focuses on combining scheduling and CSI feedback for coping with CSI inaccuracy. For the mobile ad hoc network, the research focus shifts towards spatial interference cancelation using imperfect CSI and the effects of CSI inaccuracy on the network spatial reuse efficiency. This dissertation provides methods and guidelines for transforming MIMO technologies from theory into practical applications. The remainder of this chapter presents background material and highlights contributions of this dissertation.

1.1

Limited Feedback

For most systems, transmit CSI is acquired through feedback by the receiver that performs channel estimation. Potentially, CSI feedback can incur excessive overhead due to the existence of multiple MIMO channel coefficients and hundreds of frequency sub-channels 4

as well as channel time variation. As shown by a series of results over the past few years, intelligent CSI quantization and compression can achieve very low-rate feedback without significantly compromising the advantages of transmit CSI. These encouraging results led to the emergence of an active research field known as limited feedback, focusing on the design of efficient CSI feedback techniques [8, 9]. The most common approach for limited feedback relies on codebook-based quantization for reducing CSI feedback into a small number of bits for each feedback instant. Designing codebooks is a popular theme in the area of limited feedback. The most important finding related to the codebook design is its equivalence with the classic geometry problem of packing on the Grassmannian or the Stiefel manifold [10, 11]. Beamforming and preceding codebooks designed based on such a relationship have been demonstrated to achieve low-rate CSI feedback with near optimal performance. Currently, designing limited feedback codebooks remains as an active research area. Besides quantization, feedback CSI can be further compressed by making use of channel correlation in three dimensions: time, frequency and space. A broadband channel usually exhibits high correlation over sub-channels in frequency, and simple sub-channel grouping dramatically reduces broadband CSI feedback [12]. A more sophisticated method involves CSI down-sampling and interpolation on manifolds as proposed in [12]. Channel correlation in space can be exploited by designing CSI quantizer codebooks based on channel statistics [13]. CSI compression in time is addressed in Chapter 2.

1.2

Point-to-Point MIMO Links

A point-to-point MIMO link consists of a transmitter, a MIMO channel and a receiver. This model has been the platform for designing most classical MIMO techniques including beamforming, spatial multiplexing, and space-time block coding (STBC). Beamforming and STBC aim at enhancing link reliability or in other words achieve diversity gain. Spatial multiplexing supports multiple spatial data streams and hence attains multiplexing gain. 5

M IM O C hannel

T rans m itter

R ec eiv er

CSI

F eedbac k C hannel Q uantiz er

Figure 1.2: A point-to-point MIMO link with limited feedback Spatial multiplexing can be realized by preceding, linear receiver or their combination. Adaptive MIMO transmission techniques such as transmit beamforming, preceding, and adaptive STBC all require CSI feedback and are referred to as closed-loop techniques. Limited feedback discussed in the preceding section allows closed-loop MIMO techniques to be implemented in practical point-to-point MIMO systems where CSI feedback makes use of the existing low-rate feedback channels for control signals. Fig. 1.2 illustrates a point-to-point MIMO link with limited feedback. By now, a rich set of limited feedback algorithms have been designed for efficiently quantizing CSI for different closed-loop MIMO techniques. Most prior work assumes the simplified block fading channels where temporal correlation in wireless channels is omitted. Considering a temporally correlated channel, CSI quantization is only one of several design issues for a CSI feedback link. Other issues include the feedback protocol, the bandwidth of the feedback channel, allowable feedback delay, and feedback compression in time. All above issues must be jointly considered to minimize feedback overhead and ensure accurate feedback CSI. Chapter 2 addresses these issues and provides a systematic method for designing a CSI feedback link for a point-to-point MIMO link.

6

1.3

Multiuser MIMO

Increasing the speeds of wireless access requires more spectrum, which, however, is extremely limited. For cellular systems in Fig. 1.1, this conflict can be largely resolved by using emerging technologies in multiuser MIMO communication or simply MU-MIMO. Research on MU-MIMO focuses on designing techniques for downlink which are more challenging than those for uplink. In downlink, MU-MIMO overlays multiuser data streams in the same bandwidth and time slot, effectively multiplying spectrum and alleviating the problem of spectrum shortage. Due to such spatial multiple access, MU-MIMO is also referred to as space division multiple access (SDMA). MU-MIMO requires the base station to employ multi-antennas but they are not mandatory for subscriber units. This feature is particularly attractive given the current market trend of shrinking cell-phone sizes, and the resultant difficulty in fabricating sufficiently-spaced multi-antennas into each phone. Due to this feature and its support for SDMA, MU-MIMO has been attracting active research [14] and adopted in latest communication standards such as the IEEE 802.16m [3] and the 3GPP-LTE standards [5, 15–17]. MU-MIMO (or SDMA) forms the theme of Chapters 3-5. For maximizing downlink throughput, the optimal design for SDMA is known as dirty paper coding [18]. This technique integrates multiuser CSI in coding at the base station such that multiuser interference is pre-canceled and transparent to a receiver. Combining DPC with power control has been proved by several groups of researchers to maximize the downlink throughput [19–22]. Despite being optimal, DPC requires non-causal channel information and hence is not directly realizable. There exist closeto-optimal techniques such as Tomlinson-Harashima preceding based on pre-interference cancelation, but they are sensitive to CSI inaccuracy [23]. More practical SDMA algorithms use transmit beamforming based on zero forcing, minimum mean squared error (MMSE), or channel eigen decomposition. In particular, zero-forcing beamforming completely avoids interference between SDMA users given perfect multiuser transmit CSI. 7

Nevertheless, CSI inaccuracy exists in practice due to either limited feedback or imperfect channel estimation. The CSI inaccuracy inherent in limited feedback causes residual interference between SDMA users even if zero-forcing beamforming is applied [24,25]. Multiuser interference becomes a bottleneck for increasing MU-MIMO throughput at high signal-to-noise ratios or for a relatively larger number of SDMA users. This places a more stringent requirement on CSI accuracy for MU-MIMO than that for point-to-point MIMO. To acquire highly-accurate CSI without incurring excessive overhead, limited feedback must be integrated with other techniques. Multiuser diversity refers to the degrees of freedom due to independent fading in different users’ channels. This concept was introduced in [26] for single-antenna downlink, where scheduling the user having the largest channel gain is shown to maximize capacity. The downlink throughput gain contributed by multiuser diversity is called multiuser diversity gain. For MU-MIMO, this gain is achieved by scheduling users with not only large channel gains but also small mutual interference. Thereby the stringent requirement on CSI accuracy is relaxed and as a result CSI feedback overhead decreases. To maximize multiuser diversity gain for MU-MIMO, limited feedback, beamforming and scheduling must be jointly designed, which is the topic for Chapters 3-5. In these chapters, we also quantify multiuser diversity gain by studying the sum-rate scaling laws. Furthermore, in Chapter 4, multiuser diversity is used in designing a distributed feedback algorithm for constraining sum feedback overhead.

1.4

Ad Hoc Networks

A cellular system has a fixed infrastructure and relies heavily on base stations at fixed locations for signal processing. The disadvantages of the cellular architecture include long deployment time, lack of flexibility, and vulnerability to single point of failure [6]. In contrast, an ad hoc network illustrated in Fig 1.3 has no fixed infrastructure and consists 8

Figure 1.3: Ad hoc network of nodes that are self-reconfigurable and fully distributed. Moreover, in this network, any pair of nodes can communicate with each other. Ad hoc networks represent the most general case of a wireless network. Such networks are robust, flexible and allow fast deployment. The above features make ad hoc networks attractive for a wide range of applications in situations such as military conflicts and natural disasters. Nevertheless, it is very challenging to design an ad hoc network that has high energy efficiency and throughput. An ad hoc network designed for maximum spatial reuse efficiency should be interference limited. Otherwise additional communication links can be added to the network without affecting existing ones. Therefore, interference between nodes limits the throughput of the ad hoc network. For a MIMO ad hoc network, spatial interference cancelation can significantly increase the network throughput. This technique from the multiuser detection family [27] decouples data and interference links by exploiting their spatial degrees of freedom. Spatial interference cancelation requires each receiver to estimate CSI of interference channels by using training signals broadcast by interferers1 . Imperfect CSI causes residual interference and compromises the effectiveness of spatial interference cancelation. In Chapter 6, we investigate the impact of imperfect CSI on the throughput of a MIMO ad hoc network with spatial interference cancelation. It is found that significant gains on network throughput can be attained by interference cancelation despite CSI inaccuracy. 1

For this case, limited feedback is unsuitable since it is impossible for each transmitter to send CSI to all unintended receivers.

9

1.5

Overview of Contributions

CSI overhead and inaccuracy remain as key hurdles in fully realizing the performance gains promised by many MIMO techniques. The goals of this dissertation are to design efficient methods for CSI acquisition in different types of MIMO systems and networks, and characterize the impact of CSI inaccuracy on throughput. Thereby this dissertation provides insights into transforming MIMO theory based on idealistic assumptions into practical applications. The main contributions of this dissertation are summarized as follows: 1. A new approach is proposed for designing the CSI feedback link of a point-to-point beamforming system with limited feedback. Various design factors are considered, including the feedback protocol, feedback bit rate, feedback delay and feedback compression. These factors are omitted in prior work for simplicity despite their importance for practical implementation. The proposed design approach based on Markov chain theory can be also extended to other limited feedback systems. 2. For beamforming, CSI feedback delay is shown analytically to reduce the throughput gain due to transmit beamforming at an exponential rate, which depends on channel coherence time. This result reveals the impact of feedback delay on throughput at high mobility. Therefore, feedback delay can overtake quantization as the main source of CSI inaccuracy and becomes the performance bottleneck for a closed-loop MIMO technique. 3. For beamforming, an algorithm is proposed for compressing feedback CSI by exploiting residual temporal correlation in feedback CSI. This algorithm compresses feedback CSI by truncating low-probability transitions between quantized CSI states. 4. For downlink SDMA with limited feedback, we design a joint CSI quantization, beamforming and scheduling algorithm, targeting the 3GPP-LTE standard [5]. In particular, the proposed algorithm uses a beamforming codebook of multiple or10

thonormal vector sets, which generalizes the existing design in [28]. SDMA using the proposed algorithm is proved to attain the optimal throughput scaling law. 5. A distributed CSI feedback algorithm is proposed for constraining the sum feedback rate in SDMA downlink. This feedback algorithm coupled with the proposed SDMA algorithm mentioned above is proved to retain multiuser diversity gain despite the sum feedback rate constraint. The proposed algorithm demonstrates that the redundancy in existing designs of multiuser CSI feedback can be exploited for dramatically reducing CSI overhead. 6. The throughput scaling laws are derived for uplink SDMA with limited feedback in different regimes of the signal-to-noise ratios (SNRs). At the normal and low SNRs, uplink throughput is proved to scale linearly with number of antennas at the base station and double logarithmically with the number of users. In other words, uplink SDMA achieves both multiplexing and multiuser diversity gains at the normal and low SNRs. At high SNRs, uplink throughput scales logarithmically with the number of users but loses multiplexing gain. This work suggests a set of guidelines for CSI feedback and scheduling for uplink SDMA. 7. Spatial interference cancelation is applied to increase the transmission capacity of a MIMO ad hoc network, defined as the maximum density of communication links under an outage constraint. Transmission capacity is analyzed by applying tools from stochastic geometry. For a small target outage probability, the transmission capacity is shown to grows following a power law for both perfect and imperfect CSI. The base of the power law is the target outage probability, and the exponent is the inverse of the antenna-array size if it is smaller than the pass-loss exponent. This result shows that even a few antennas at each receiver can increase transmission capacity dramatically. 8. An opportunistic CSI estimation algorithm is proposed for providing the CSI re11

quired for spatial interference cancelation. By opportunistically estimating CSI in the absence of strong interferers, this algorithm provides a trade-off between transmission power for training signals and the duration of CSI estimation.

1.6

Organization

The remainder of this dissertation is organized as follows. In Chapter 2, we design the CSI feedback link for a point-to-point beamforming system. Specifically, we derive the CSI source bit rate and the feedback bit rate, analyze the impact of feedback delay on throughput, and design an algorithm for feedback compression in time. Chapters 3-5 focus on SDMA with limited feedback. In Chapter 3, we propose and analyze a joint limited feedback, beamforming and scheduling design for downlink SDMA. In Chapter 4, targeting the same system, we present a distributed algorithm for constraining multiuser CSI feedback. This algorithm satisfies a sum feedback rate constraint, which is show to have an insignificant effect on downlink throughput. In Chapter 5, uplink SDMA is considered and the throughput scaling laws are derived for different SNR regimes. In Chapter 6, we analyze the capacity of a mobile ad hoc network with spatial interference cancelation. Both the cases of perfect and imperfect CSI are considered. The analysis for imperfect CSI is based on a proposed opportunistic algorithm for CSI acquisition. Finally, Chapter 7 concludes the technical content of this dissertation.

12

Chapter 2

Beamforming with Limited Feedback For a point-to-point MIMO link, transmit beamforming effectively alleviates the negative effect of channel fading by exploiting spatial diversity, and thereby increases system throughput. This chapter focuses on limited feedback beamforming systems over temporally correlated channels, and addresses a set of open design issues. Specifically, these issues include the information rate generated by a temporally-correlated channel, the required feedback bit rate, the effect of feedback delay on throughput and feedback CSI compression. By investigating these issues, the chapter provides a systematic method of designing the CSI feedback link of a limited feedback beamforming system.

2.1

Prior Work

For simplicity, most prior work on limited feedback adopts the block fading channel model, where each channel realization remains constant in one block and different realizations are independent [6]. This model omits the potential temporal correlation between channel realizations. Using this model, designing limited feedback reduces to a vector quantization problem [29]. Different methods for quantizing CSI have been developed such as line pack13

ing [10,11], combined parametrization and scalar quantization [30], subspace interpolation [12], and Lloyd’s algorithm [31]. Furthermore, different types of limited feedback systems have been investigated, including beamforming [10, 11], precoded orthogonal space-time block codes [32], precoded spatial multiplexing [33], and multiuser downlink [14]. Prior work focuses on analyzing and minimizing CSI inaccuracy due to quantization. The existing results do not account for CSI inaccuracy due to feedback delay. Related analysis is difficult since the common assumption of block fading omits channel temporal correlation. For the same reason, there exists no analysis on how channel coherence time influences the information rate generated by time-varying CSI, and the bit rate that a feedback channel must support. The aforementioned issues are addressed in this chapter. For practical systems, the block fading assumption in prior work is pessimistic since channel temporal correlation often exists and can be used for compressing feedback CSI by incremental feedback. Algorithms for feedback compression have been proposed in [30, 34, 35] that exploit channel temporal correlation. In [30], a MIMO channel is parameterized and the feedback of each parameter is compressed to be one bit. Nevertheless, the multiplicity of the channel parameters compromises the feedback efficiency. In [34], the feedback CSI is compressed to be one bit but requires the transmitter to periodically broadcast channel subspace matrices. Building on the preliminary results of the current work in [36], variable-length source codes such as Huffman code [37] are applied for CSI feedback compression in [35]. Despite being optimal, this approach may not suit practical applications where CSI feedback consists of fixed-length bit blocks [2, 5]. In a practical system with CSI feedback, feedback delay exists due to sources such as signal processing, propagation and protocols. The negative effects of CSI feedback delay on bit-error rate or information capacity have been observed in the literature of MIMO communication [38–44]. Specifically, this delay is found to decrease received signal power, and cause interference between spatial data streams. The negative impact of CSI feedback delay can be alleviated to some extent by channel prediction [44, 45]. Despite its importance for designing MIMO systems, there exists no simple relationship between 14

CSI feedback delay and throughput. This motivates the current work on deriving such a relationship in the context of limited feedback beamforming. The result of this work has been validated by measurement data from a MIMO prototype over an indoor channel [46]. The main approach of this work is to model quantized CSI as a finite-state Markov chain (FSMC), which allows the use of Markov chain theory as an analytical tool. A similar approach is common in modeling single-input-single-output (SISO) fading channels [47–51]. A FSMC model for fading captures the stochastic feature of wireless channels that is missing in block fading. Furthermore, the FSMC model is simple enough for allowing tractable performance analysis of communication systems. For these reasons, since it was proposed in [47], this model has be widely applied in wireless channel modeling, communication and networking. The FSMC models have been proposed for the satellite [52], indoor [53], Rayleigh fading [47, 49, 54], Nakagamni fading [55], and Rician fading [48] channels. The accuracy of FSMC models has been verified based on different criteria, including information capacity [56,57], packet errors [58], burst errors [59], and autocorrelation functions [48, 49]. Due to its accuracy and simplicity, the first-order FSMC channel models have been used in designing and analyzing adaptive video encoding [60], maximum a posteriori decoding [61], downlink rate control [62], downlink power control [63], automatic repeat request (ARQ) [64], and maximum likelihood detection [65]. Most prior work considers SISO channels. In contrast, this chapter focuses on MIMO beamforming with limited feedback and the design of CSI feedback link.

2.2

Contributions

This chapter presents a set of analytical results useful for designing a limited feedback beamforming system over a temporally-correlated channel. The contributions of this chapter are summarized as follows. 1. Quantized CSI is modeled as a first-order finite-state Markov chain, called channel state Markov chain, which allows using Markov chain theory for obtaining the key 15

results of this work. The stationary probabilities of the channel state Markov chain depend on the channel distribution and the CSI quantizer codebook; the transition probabilities are functions of channel coherence time. 2. Quantized CSI is treated as a Markov data source and the corresponding information bit rate, called CSI source bit rate, is derived in terms of the probabilities of the channel state Markov chain. In particular, the CSI source bit rate is shown to decrease linearly with the probability that the channel state remains unchanged over two consecutive data symbol durations. Note that this probability is larger for longer channel coherence time or vise versa. The CSI source bit rate provides a measure of the amount of information generated by the temporally-correlated channel. 3. CSI feedback can rely on a periodic or a random-access [66] feedback protocols, corresponding to separated or shared feedback channels for different users. Based on the periodic feedback protocol, the bit rate supported by a feedback channel, called CSI feedback bit rate, is derived under a feedback outage constraint, where an outage refers to multiple channel-state transitions within one feedback interval. This constraint guaranteers that CSI feedback is sufficiently frequent. The derived CSI feedback bit rate is useful for allocating bandwidth for the CSI feedback channel. 4. Define the feedback throughput gain as the difference in throughput between the cases of delayed CSI feedback and no feedback. Based on the theory of Markov chain convergence rate, the feedback throughput gain is shown to decrease at least exponentially with feedback delay and inversely with the feedback interval. The exponential rate increases with decreasing channel coherence time and vise versa. This result enables the joint design of the mobility speed, the bandwidth of the feedback channel, and feedback delay sources such as signal processing complexity and the propagation distance, under a constraint on the feedback throughput gain. 5. An algorithm is proposed for compressing feedback CSI by exploiting residual tempo16

ral correlation in feedback CSI. Note that the temporal correlation in CSI is largely reduced by the periodic feedback protocol. This algorithm compresses feedback CSI by truncating low-probability transitions between states of the channel state Markov chain. Moreover, this algorithm alternates compressed and uncompressed CSI feedback to prevent propagation of CSI errors due to such truncation. 6. Finally, extension of above results to higher-order channel state Markov chains is discussed. Simulation results are presented based on a MIMO channel with spatially i.i.d. Rayleigh fading. Moreover, it is assumed that scattering is uniform and dense, and receive antennas are omni-directional [6]. As a result, the temporal correlation function of each channel coefficient is the zeroth order Bessel function of the first kind. Several observations are made. First, the CSI source bit rate increases linearly with Doppler shift. Second, for a feedback outage probability of practical interest (e.g. 0.1), the ratio between CSI feedback and source bit rates is insensitive to a change in Doppler shift. Third, the feedback throughput gain normalized by its maximum value is observed to be closely approximated by the product between an exponential function of feedback delay and a function of the feedback interval. Fourth, the proposed CSI feedback compression algorithm can provide a feedback compression ratio more than 20%. Finally, a design example is presented for the CSI feedback link of a limited feedback beamforming system. This example demonstrates the joint application of the analytical results reported in this chapter. Notation: Capitalized and small boldface letters denote matrices and vectors, respectively. The superscript † represents the complex conjugate and transpose matrix operation. The operator [·]` gives the `th component of a matrix. Similarly, [·]`m returns the (`, m)th component of a vector.

17

2.3

System Description

The limited feedback beamforming system illustrated in Fig. 2.1 can be separated into the forward and the CSI feedback links, which are described in the following sub-sections. In this system, all signals are discrete-time and sampled at the sampling rate 1/T , where T denotes one sampling interval. The sample index is denoted by the subscript n.

2.3.1

Forward Link

The forward link refers to the data path in Fig. 2.1 from the input of the beamformer to the output of the maximum-ratio combiner. The nth received data symbol after maximum ratio combining, denoted as yn , is given as yn =

√ P wn† Hn fn xn + νn

(2.1)

where xn and νn are CN (0, 1) random variables modeling respectively the nth transmitted data symbol and the nth sample of the additive white Gaussian noise (AWGN) process, and P is transmission power. Let Nt and Nr denote the number of transmit and receive antennas, respectively. Then the Nr × 1 complex unitary vector wn represents the weights for maximum ratio combining, and the Nt × 1 complex unitary vector fn denotes the transmit beamforming vector. The Nr × Nt matrix Hn represents the nth realization of the MIMO channel. The random process {Hn } is assumed to be stationary and temporally correlated. The receiver continuously estimates the CSI sequence {Hn } by using pilot symbols sent by the transmitter. The estimated CSI is used for computing the maximum-ratio combining vector and the beamforming vector for feedback. This chapter considers the scenario where CSI quantization and feedback delay are the main sources of transmit CSI inaccuracy. Thus, CSI estimation is assumed perfect for simplicity. This assumption is commonly made in the literature of limited feedback (see e.g. [10, 11]). Based on this assumption, the maximum-ratio combining vector is computed as wn = Hn fn /kHn fn k. In 18

M IM O C hannel M ax- ratio c om biner

D ata

D ata

C odebook bas ed beam form er C hannel es tim ator

C S I quantiz er

F ix ed- length s ourc e enc oder

Periodic transmitter

F eedbac k C hannel

Figure 2.1: Limited feedback beamforming system the next section, the beamforming vector fn is selected from a codebook for maximizing the receive signal-to-noise ratio (SNR).

2.3.2

CSI Feedback Link

In Fig. 2.1, CSI feedback link refers to the CSI path from the input of CSI quantizer to the output of the feedback channel. To satisfy a finite-rate feedback constraint, CSI is quantized efficiently by using a Grassmannian codebook, denoted as F, designed for beamforming in [10, 11]. To maximize the receive SNR, the quantizer function, denoted as Qf (or Qi ), maps the channel matrix Hn to a beamforming vector in the codebook F (or its index denoted as In ) as follows [10, 11] fn = Qf (Hn ) = arg max kHn vk2 , v∈F

In = Qi (Hn ) = arg max kHn v` k2 . 1≤`≤N

(2.2)

The feedback of the index In , called the channel state hereafter, is sufficient for the transmitter to retrieve the selected beamforming vector fn from the codebook F. From (2.2), the channel states, {In }, are a sequence of alphabets of N letters. This alphabet sequence is encoded by using a B-bit fixed-length code, where B = log2 N [37]. The CSI bit rate at the output of the source encoder is derived in Section 2.5.1. 19

Intuitively, a more efficient alternative is to encode a long block of channel states by using variable-length codes such as Huffman code [37]. Nevertheless, block CSI encoding contributes additional feedback delay, which decreases throughput significantly as shown in Section 2.6. Furthermore, a variable CSI codeword length is unsuitable for typical limited feedback systems such as 3GPP-LTE [5], where an uniform number of bits are allocated for each instant of periodic CSI feedback. This also motivates the use of a periodic feedback protocol in this chapter. The periodic feedback protocol transmits latest encoded CSI to the feedback channel at fixed intervals. Thereby this protocol introduces variable CSI feedback delay as elaborated shortly. Let the integers K and D denote respectively a feedback interval and fixed feedback delay in samples, where D is contributed by sources including signal processing and propagation. Then CSI used for beamforming in successive symbol durations lags behind the corresponding channel states by D, (D + 1), · · · , (D + K − 1) samples repeated in a cyclic order. The CSI feedback bit rate for the periodic feedback protocol is derived under a feedback outage constraint in Section 2.5.2, and the effects of CSI feedback delay are investigated in Section 2.6. To simplify analysis, the feedback channel is assumed free of errors, which is typical in the literature of limited feedback (see e.g. [9–11]). This assumption is justified by the fact that beamforming feedback as control signals is usually well protected by using errorcorrection coding or high transmission power.

2.4

Channel State Markov Chain

In this section, the channel state sequence is modeled as a first-order finite-state Markov chain. The definition and the construction procedure of this Markov chain are presented in Sections 2.4.1 and 2.4.2, respectively. The channel state Markov chain is used as the primary analytical tool in the sequel.

20

2.4.1

Definition

As discussed in Section 2.1, the finite-state Markov chain models for fading have been established as valid models of wireless channels [47–52,54–59], and widely used in wireless communication and networking [60–65]. Following this common approach, a finite-state Markov chain is used for modeling partial CSI of a MIMO fading channel in this section. The resultant Markov chain is called the channel state Markov chain. Its parameters, namely the state space and the stationary and transition probabilities, are related to the channel statistics and the CSI quantizer codebook. The channel state Markov chain model is validated using simulation in Section 2.9.1. The channel state Markov chain models the time-variation of the channel state In in (2.2). Mapped from a stationary channel by the quantizer function in (2.2), {In } is a finite-sate stationary stochastic process. Motivated by the common approach in fading channel modeling [47–51], this process is modeled as a homogeneous finite-state Markov chain of order one with the state space I = {1, 2, · · · , N } [67]. The key property of this Markov model is that conditioned on the most recent state In−1 , In is independent of the past states In−2 , In−3 , · · · , thus Pr(In = `n | In−1 = `n−1 , In−2 = `n−2 , · · · ) = Pr(In = `n | In−1 = `n−1 ).

(2.3)

Give the above property, the probability of a transition from state m to ` is defined as P`m = Pr(In = ` | In−1 = m). Moreover, the stationary probability of state ` is defined as π` = Pr(In = `). For convenience, denote the N × 1 stationary probability vector as π with [π]` = π` , and the N × N transition probability matrix as P with [P]`m = P`m , which is known as the stochastic matrix [67]. The use of the first-order channel sate Markov chain simplifies notation and analysis. Depending on the channel correlation function and the quantizer codebook size, a higher-order channel state Markov chain may provide a more accurate CSI model. The extension to higher-order channel state Markov chains is discussed in Section 2.8. 21

The channel state Markov chain is assumed to be ergodic [67]. The ergodicity implies three properties. First, the states of the Markov chain are communicating, namely that a transition between every pair of states occurs within a finite duration. Second, each state is recurrent and thus the probability of returning to a same state is one. Third, each state is aperiodic. This property exists if all possible time durations (in samples) of leaving and returning to each state have the common divisor of one. The assumption of ergodicity for the channel state Markov chain is valid for typical continuous channel distributions such as Rayleigh or Rician [6]. This assumption provides the following property [67] lim PD = [π, π, · · · , π] .

D→∞

(2.4)

The probabilities π` and P`m depend on both the channel statistics and the CSI quantizer codebook. To specify these relationships, define a Voronoi cell based on the quantizer in (2.2) as [29] © ª V` = H ∈ CNr ×Nt | kHv` k ≥ kHvm k ∀ m ∈ I, m 6= `

(2.5)

where v` denotes the `th unitary vector in the codebook F. This set V` maps the channel to the `th state of the channel state Markov chain as follows

In = ` ⇐⇒ Hn ∈ V` .

(2.6)

Using the above relationship, the probabilities of the channel state Markov chain can be related to the channel statistics as

π` = Pr (H ∈ V` ) ,

P`m = Pr (Hn ∈ V` | Hn−1 ∈ Vm ) .

(2.7)

In general, (2.7) does not yield closed-form expression for the Markov chain probabilities except for the degenerate case of single antennas. Nevertheless, these equations are useful 22

for compute the probabilities by simulation as discussed in the next section.

2.4.2

Construction by Simulation

This section summarizes the procedure for estimating the stationary and transition probabilities of the channel state Markov chain by Monte Carlo simulation. Assume that a CSI quantizer codebook is given and the channel statistics are known. The simulation procedure is given below. 1. Generate a continuous sequence of (M + 1) channel matrices {Hn }M n=0 based on the channel probability distribution and temporal correlation functions. Generators for typical fading channels such as Rayleigh fading with uniform scattering are provided by system simulation softwares such as MatLabr . M 2. The corresponding channel state sequence {In }M n=0 is computed from {Hn }n=1 by

using the quantization function in (2.2); 3. The stationary and the transition probabilities are estimated as

π` = P`m =

M 1 X 1{In = `} M n=1 M X

1 π` M

1{In = `, In−1 = m}

(2.8)

n=1

where 1{Π} is the indicator function for the event Π. According to ergodic theory (cf. Lemma 1), such estimation is accurate if M is sufficiently large. The above procedure is used in Section 2.9 for generating numerical results. The analysis in other sections is independent of the specific parametric values of the channel state Markov chain.

23

2.5

CSI Source and Feedback Bit Rates

In this section, the overhead required for CSI feedback is analyzed based on the channel state Markov chain. Specifically, the CSI source and feedback bit rates are derived in Section 2.5.1 and Section 2.5.2, respectively.

2.5.1

CSI Source Bit Rate

The CSI source bit rate, defined as the average rate of encoded CSI bits, measures the amount of information generated by the temporally correlated MIMO channel. Based on fixed-length source coding as discussed in Section 2.3, the CSI source rate is the product of the CSI codeword length and the average rate of channel-state transition given as M B X 1{In 6= In−1 }. M →∞ M T

Rs = lim

(2.9)

n=1

Given the ergodicity assumption for the channel state Markov chain, Rs in (2.9) can be rewritten in terms of the Markov chain probabilities by applying the ergodic theorem in Lemma 1 from [68, Corolary 4.1]. Lemma 1 Given a function z : I × I → R that satisfies

PN PN `=1

m=1 |z(`, m)|π` P`m

< ∞,

the following convergence exists almost surely N M N X X 1 X z(`, m)π` P`m . z(In , In−1 ) = M →∞ M

lim

n=1

(2.10)

`=1 m=1

The main result of this section as summarized in Proposition 1 follows from (2.9) and Lemma 1. Note that the condition for applying Lemma 1 is checked to be satisfied by setting z(`, m) = 1{` 6= m). Proposition 1 The CSI source bit rate is

Rs =

N BPt X π` (1 − P`` ). T `=1

24

(2.11)

The above result shows that Rs decreases linearly with increasing probabilities {P`` }, which characterize the degree of channel temporal correlation. By definition, P`` is the probability that the channel state remains as ` for two consecutive samples. The values of {P`` } are close to one if the channel is highly correlated in time, or close to zero for fast fading. As shown by simulation in Section 2.9.2, Rs increases linearly with Doppler shift for spatially i.i.d. Rayleigh fading and uniform scattering.

2.5.2

CSI Feedback Bit Rate

The CSI feedback bit rate is defined as the maximum bit rate supported by the feedback channel. In this section, the CSI feedback bit rate is derived under a constraint on feedback outage probability. Recall that a feedback outage refers to the event that more than one channel-state transition occurs within one feedback interval. Let Z denote the number of channel-state transitions in a feedback interval of K samples. Moreover, let Po represent the feedback outage probability, thus Po := Pr(Z > 1). For a small real number δ > 0, the constraint Po ≤ δ ensures that CSI feedback is sufficiently frequent. The main result of this section is obtained and summarized in the in following proposition. Proposition 2 Under the feedback outage constraint Po ≤ δ, the CSI feedback bit rate is Rf = to

B KT

bit/s where the feedback interval in samples, K, is the largest integer subject N X

P``K π`

`=1

N N X K − P K )P π X (Pmm m` ` `` + ≥ 1 − δ. Pmm − P``

(2.12)

`=1 m=1 m6=`

Proof: See Appendix 2.11.1.

¤

To observe the dependence of the CSI feedback bit rate on channel temporal correlation, bounds on Rf can be obtained as a by-product of the proof for Corollary 1 and shown as log(maxa Paa ) ³ P log(1 − δ) − log 1 + m6=`

Pm` π` |Pmm −P`` |

25

´ ≤ Rf ≤

log(mina Paa ) . log(1 − δ)

(2.13)

The numerators of the above bounds suggest that Rf decreases with increasing probabilities {P`` }, which characterize the degree of channel temporal correlation as mentioned earlier. In other words, Rf is smaller for a more temporally correlated channel and vise versa, which is also observed for the CSI source bit rate given in Proposition 1. Note that this relationship is not proven to be strictly monotonic. The feedback interval K for the extreme cases of small and large target feedback outage probabilities are characterized in the following corollary of Proposition 2. Corollary 1 For a small target feedback outage probability δ → 0, the normalized feedback interval converges to one: K → 1; for a large probability δ → 1, K scales as

K = η log(1 − δ) + O(1)

where

1 log max` P``

≤η≤

(2.14)

1 log min` P`` .

Proof: See Appendix 2.11.2.

¤

For δ → 1, the first term η log(1 − δ) in (2.14) is dominant because it is asymptotically large. The factor η in (2.14) represents the sensitivity of K to the change on the feedback constraint δ. This factor depends on the degree of channel temporal correlation through the probabilities {P`` }.

2.6

Feedback Delay

This section focuses on the effects of feedback delay on the throughput of the limited feedback beamforming system. In Section 2.6.1, the feedback throughput gain with fixed feedback delay is defined and derived. In Section 2.6.2, fixed feedback delay is shown to reduce the feedback throughput gain at least at an exponential rate. Building on this result, an upper-bound on the feedback throughput gain is derived as a function of both fixed and variable feedback delay in Section 2.6.3. As mentioned earlier, fixed CSI feedback delay exists due to sources such as signal 26

processing and propagation. Furthermore, variable feedback delay is contributed by the periodic feedback protocol. Feedback delay can be reduced by channel prediction. Specifically, the prediction can be based on unquantized CSI and a standard algorithm such as minimum-mean-squared-error [45]. Alternatively, universal prediction can be applied on the channel state sequence at either the receiver or transmitter, which requires no knowledge of channel statistics [69]. Given channel prediction, feedback delay considered in this chapter refers to the effective delay after subtracting the channel prediction range. Detailed studies on how channel prediction errors affect the performance of MIMO adaptive transmission systems are outside the scope of this chapter but can be found in [45] and its references.

2.6.1

Feedback Throughput Gain

The feedback throughput gain is defined as the gain in ergodic throughput due to delayed CSI feedback with respect to the case of infinite feedback delay. In this section, only fixed feedback delay is considered and a corresponding upper bound on the feedback throughput gain is derived. This result is extended in Section 2.6.3 to include the effect of variable feedback delay. Let D denote fixed feedback delay in samples, and R represent ergodic throughput. Then the feedback throughput gain can be written as ∆R(D) = R(D) − R(∞). First, the ergodic throughput R(D) is derived as follows. With feedback delay, the input-output relationship for the limited feedback beamforming system in (2.1) can be rewritten as yn =

√ P gn (D)xn + νn

(2.15)

where the effective channel gain gn (D) = kHn f (In−D )k and f (In−D ) is the transmit beamforming vector chosen based on the delayed feedback CSI In−D . With only beamforming feedback, gn (D) is unknown to the transmitter, and hence constant transmission power P is optimal [6]. Given feedback delay, deriving the optimal strategy for transmit beam27

forming is difficult as it depends on the codebook, the channel stochastic distribution, and the channel prediction algorithm. Form simplicity, this chapter adopts the strategy of applying the codebook vector corresponding to In−D as the beamforming vector. Equivalently, f (In−D ) = Qf (Hn−D ) with the quantizer function Q(·) given in (2.2). Note that this sub-optimal beamforming strategy is optimal for zero feedback delay [10]. Using this strategy, the ergodic throughput of the effective scalar channel in (2.15) is obtained as [6] £ ¤ £ ¤ R(D) = E log2 (1 + gn2 (D)) = E log2 (1 + P kHn Qf (Hn−D )k2 ) .

(2.16)

To achieve this throughput, the channel code for forward-link data is assumed to be sufficiently long to attain channel ergodicity [6]. In other words, the code length and hence the decoding delay are much longer than channel coherence time. If a constraint on the decoding delay is required, the outage capacity is a more appropriate performance metric [70]. The ergodic throughput in (2.16) can be rewritten in terms of the probabilities of the channel state Markov chain as shown in the following lemma. Lemma 2 For fixed feedback delay of D symbol durations, the ergodic throughput is

R(D) =

N N X X

£ ¤ R`m PD `m πm

(2.17)

`=1 m=1

where ¤ £ R`m = E log2 (1 + P kHn vm k2 ) | Hn ∈ V` , Hn−d ∈ Vm . Proof: See Appendix 2.11.3.

(2.18) ¤

In general, the constants {R`m } do not have closed-form expressions and have to be estimated by using Monte Carlo simulation and following a procedure similar to that in Section 2.4.2. Second, consider the extreme cases of zero delay (D = 0) and infinite delay (D → ∞). The corresponding ergodic throughput is obtained and shown in the following 28

corollary of Lemma 2. Corollary 2 The ergodic throughput for zero feedback delay R(0) and infinite delay R(∞) are

R(0) = R(∞) =

N X

£ ¡ ¢¤ R`` π` = E log2 1 + P kHQf (H)k2

`=1 N X N X `=1 m=1

R`m π` πm =

N X £ ¡ ¢¤ E log2 1 + P kHv` k2 π` .

(2.19) (2.20)

`=1

Proof: See Appendix 2.11.4.

¤

A few remarks are in order. 1. For zero feedback delay, R(0) is equal to the ergodic capacity for limited feedback beamforming based on the block fading channel model [10, (26)], where feedback delay is omitted. 2. For infinite feedback delay, the beamforming vector at the transmitter is adapted to obsolete CSI independent of the current channel state, which is apparently suboptimal. Higher ergodic throughput than R(∞) may be achieved by using space time block coding or adapting the beamforming vector to channel statistics [71], both require no instantaneous CSI. Despite its sub-optimality, R(∞) serves as a reference value for computing the feedback throughput gain in this chapter. Targeting a sufficiently large feedback throughput gain in a system design retains most throughput gain due to limited feedback beamforming. 3. It can be observed from (2.19) and (2.20) that both R(0) and R(∞) are independent of the transition probabilities of the channel state Markov channel. This suggests that channel dynamics affect only the case of finite feedback delay as considered in the next section. Finally, from Lemma 2, Corollary 2 and the definition, the feedback throughput gain is readily written as shown in the following proposition. 29

Proposition 3 Assuming only fixed feedback delay, the feedback throughput gain is given as ∆R(D) =

N X N X

R`m

©£

PD

¤

ª − π πm l `m

(2.21)

`=1 m=1

where R`m is given in (2.18). An upper-bound on R(D) is derived in the next section. Other metrics: Besides the feedback throughput gain, other metrics can be defined for quantifying the effects of CSI quantization and feedback delay on the throughput of a transmit beamforming system. The ergodic capacity for perfect CSI feedback £ ¡ ¢¤ is Cideal = E log2 1 + P λ21 (H) where λ1 (H) represents the largest singular value of H. With respect to the case of perfect CSI feedback, the capacity loss due to CSI quantization, called quantization loss, can be defined as ∇q C = Cideal − R(0) where R(0) follows from (2.16). As shown in [10, 11], quantization loss is small given just a few bits of resolution for quantized CSI. Moreover, the capacity loss due to both CSI quantization and feedback delay can be written as ∇qd R = Cideal − R(D). Next, using the case of infinite feedback delay as the reference, the maximum throughput gain is defined as ∆max R = Cideal − R(∞). The feedback throughput gain defined earlier as ∆R(D) = R(D) − R(∞) takes into account of both CSI quantization and feedback delay. By definition, the feedback throughput gain ∆R(D) is the complement of the capacity loss ∇qd R in the sense that ∆R(D) + ∇qd R = ∆max C.

2.6.2

Effect of Fixed Feedback Delay

In this section, increasing fixed feedback delay is shown to reduce the feedback throughput gain at least at an exponential rate. This result is derived based on theory of Markov chain convergence rate. To state this theory, define the time reversal of the stochastic matrix ˜ as a matrix whose components are given as [67] P, denoted by P, πm P˜m` = P`m , 1 ≤ m, ` ≤ N. π` 30

(2.22)

˜ represents channel-state transitions in the reverse direction of As implied by its name, P ˜ = P, the channel state Markov chain is known as reversible P. For the special case of P [67]. Using the definition in (2.22), [72, Theorem 2.1] is restated as the following lemma. Lemma 3 For the ergodic channel state Markov chain, the following inequality holds ÃN !2 X ¯£ ¤ ¯ λD ¯ PD ¯ , − π ≤ ` `m πm

1≤m≤N

(2.23)

`=1

˜ where λ ∈ [0, 1] is the second largest eigenvalue of the matrix PP. By using Lemma 3, the main result of this section is derived and shown in the following proposition. Proposition 4 For feedback delay of D samples, the feedback throughput gain is upper bounded as √ 0 ≤ ∆R(D) ≤ α( λ)D where α =

PN

m=1

(2.24)

√ πm max` R`m with R`m given in (2.18), and λ ∈ [0, 1] is the second

˜ largest eigenvalue of the matrix PP. Proof: See Appendix 2.11.5.

¤

A few remarks are in order. 1. The eigenvalue λ is a key parameter characterizing the channel dynamics. A larger value of λ indicates longer channel coherence time and vise versa. 2. As observed from (2.24), the feedback throughput gain decreases at least exponen√ tially with the feedback delay. The decreasing rate is λ and thus depends on channel coherence time. ˜ 3. For a reversible channel state Markov chain with P = P, eigenvalue of the transition matrix P. 31

√ λ is the second largest

4. As observed from simulation results in Section 2.9.3, the upper bound in (2.24) is tight.

2.6.3

Effects of both Fixed and Variable Feedback Delay

In this section, the effects of both fixed and variable feedback delay on the feedback throughput gain are jointly considered. As discussed in Section 2.3.2, due to the periodic feedback protocol, CSI used for beamforming in successive symbol durations encounters variable feedback delay of D, (D + 1), · · · , (D + K − 1) samples repeated in a cyclic order. ˜ Let R(K, D) denote the ergodic throughput with both fixed and variable feedback delay. ˜ Thus, R(K, D) can be written as K−1 1 X ˜ R(k + D) R(K, D) = K

(2.25)

k=0

where R(·) is given in (2.17). Define the corresponding feedback throughput gain as ˜ = R(K, ˜ ∆R D) − R(∞). Then we have the following corollary of Proposition 4. Corollary 3 For fixed feedback delay of D samples and a feedback interval of K samples, the feedback throughput gain is upper bounded as √ √ D 1 − ( λ)K ˜ √ . 0 ≤ ∆R(K, D) ≤ α( λ) K(1 − λ)

(2.26)

where α and λ are identical to those in Proposition 4. √ The upper-bound in (2.26) can be decomposed into three factors α, ( λ)D and

√ 1−( λ)K √ . K(1− λ)

They characterize the effects of CSI quantization, fixed feedback delay and the periodic feedback protocol, respectively. ¯ ˜ Define the normalized feedback throughput gain as ∆R(K, D) = ∆R(K, D)/∆R(0), where the normalization factor ∆R(0) corresponds to the ideal case of zero feedback delay.

32

¯ Motivated by the result in Corollary 3, ∆R(K, D) can be approximated as h √ Ki 1 − ( λ) √ ¯ √ ∆R(K, D) ≈ ( λ)D × | {z } K(1 − λ) | {z } qD

(2.27)

qK

where the factors qD and qK separate the effects of the fixed and variable feedback delay on the feedback throughput gain. For a sanity check, the above approximated expression gives 1 for zero feedback delay K = D = 0, and zero for infinite delay K → ∞ or D → ∞. The approximation of the normalized feedback throughput gain in (2.27) is observed to be accurate from simulation results in Section 2.9. The result in (2.27) is useful for computing the allowable feedback delay under a constraint on the normalized feedback throughput gain as demonstrated by the design example in Section 2.9.5.

2.7

Feedback Compression

An algorithm is proposed in Section 2.7.1 for compressing CSI feedback by exploiting temporal correlation in feedback CSI. The effect of feedback compression on feedback throughput gain is analyzed in Section 2.7.2.

2.7.1

Algorithm

The proposed feedback compression algorithm is motivated by the existence of residual temporal-correlation in feedback CSI, which is largely reduced by the periodic feedback protocol (cf. Section 2.5.2). To characterize such residual redundancy, feedback CSI is modeled as a Markov chain obtained by down-sampling the channel stat Markov chain at feedback intervals. Consequently, the stochastic matrix for this down-sampled Markov (K)

chain is PK with the (`, m)th component denoted as P`m . The correlation between two consecutive instants of feedback CSI is reflected in an uneven distribution of the transition probabilities in PK . Specifically, conditioned on the previous instant of feedback CSI, the current one belongs to a subset of the Markov chain state space with high probability. To 33

define this subset, let the transition probabilities for the Markov chain state m, namely n o (K) N (K) P`m , be indexed according to the descending order of their values, thus P1m ≥ `=1

(K) P2m · · ·

(K)

≥ PN m . For a small positive real number ², the high-probability subset, called

the ²-neighborhood, of the mth Markov chain state is defined as  ¯  ¯X N N   X ¯ (K) (K) ˜ ≤`≤N¯ Nm (²) = N P ≥ 1 − ², P < 1 − ² . `m `m ¯   ¯ ˜ `=N

(2.28)

˜ +1 `=N

This ²-neighborhood groups most likely transitions from the Markov chain state m with total probability larger than (1 − ²). High channel temporal correlation results in small ²-neighborhoods and vise versa. The proposed algorithm compresses feedback CSI in alternate feedback intervals to prevent propagation of CSI errors due to lossy compression. The possibility of error propagation is due to backward dependence of CSI compression in time. Conditioned on the n ˜ th feedback channel state In˜ K = m that is uncompressed, the next one I(˜n+1)K is compressed by lossy source coding. Specifically, I(˜n+1)K is encoded into one of |Nm (²)| fixed-length codewords if I(˜n+1)K belongs to the ²-neighborhood of In˜ K , namely Nm (²) defined in (2.28). Otherwise, a codeword indicating CSI truncation is generated by the source encoder. It follows that compressing I(˜n+1)K requires a fixed-length source codebook having a size of [|Nm (²)| + 1] and a codeword length of Bm (²) = dlog2 (|Nm (²)| + 1)e bits. The above source coding algorithm compresses CSI since Bm (²) is usually much smaller than B = log2 N for the case of no compression. The number of bits for com˜ pressed CSI feedback should be uniform and thus is chosen as B(²) = max` B` (²). Finally, to decode both compressed and uncompressed CSI feedback, the transmitter stores all (N + 1) source codebooks including one for uncompressed and N for compressed CSI feedback. On decoding the codeword indicating CSI truncation, transmitter randomly chooses a beamforming vector from the codebook F. A few remarks are in order. ˜ feedback bits for 1. The proposed CSI compression algorithm alternates B and B(²) 34

successive feedback instants, and hence the CSI compression ratio is

∆Rf =

˜ B − B(²) . 2B

(2.29)

This ratio is evaluated by simulation in Section 2.9.4. 2. The proposed algorithm can be also applied for compressing other types of feedback CSI such as channel-gain feedback useful for power control and scheduling. Different types of compressed feedback CSI can be integrated such that the total number of CSI bits per feedback instant remains constant, which suits practical systems such as 3GPP-LTE [5]. 3. As mentioned earlier, the proposed feedback compression algorithm is preferred to the conventional lossless data compression algorithms, which use variable-length source coding and optionally block processing for higher compression efficiency [35, 37]. The reason is that variable-length CSI feedback is unsuitable for practical systems such as such as 3GPP-LTE [5]. Furthermore, block processing causes additional feedback delay that significantly decreases throughput as shown in Section 2.6.2. 4. The proposed feedback compression algorithm complements the periodic feedback protocol on further decreasing the CSI feedback rate by exploiting channel temporal correlation. In contrast, based on an aperiodic feedback protocol, the original version of this algorithm presented in [73] is the only feedback compression function in the CSI feedback link.

2.7.2

Feedback Throughput Gain with Feedback Compression

The CSI feedback compression algorithm proposed in the preceding section is lossy and hence inevitably incurs loss on the feedback throughput gain. To characterize this loss,

35

ˆ the ergodic throughput with feedback compression, denoted as R(K, D), is written as 1 (1 − ²) ² ˆ R(K, D) + R(∞) R(K, D) = R(K, D) + 2 2 2

(2.30)

where R(K, D) is the ergodic throughput without feedback compression. The three terms on the right-hand-side of (2.30) correspond to uncompressed, compressed, and truncated feedback CSI, respectively. Note that the probability for CSI truncation is ². The feedback throughput gain with feedback compression is obtained by using (2.30) and given in the following proposition. Also shown is the CSI feedback bit rate for CSI compression. Proposition 5 The proposed CSI feedback compression algorithm has the following properties. 1. The feedback throughput gain is ³ ²´ ˆ ∆R(K, D, ²) = 1 − ∆R(K, D) 2

(2.31)

where ∆R(K, D) corresponds to the case of no feedback compression. 2. The average CSI feedback bit rate is ˆ ˆ f (²) = B + B(²) . R 2KT

(2.32)

From (2.31), ² should be small so as to minimize the throughput loss due to lossy feedback ˆ f (²) CSI compression. Nevertheless, too small ² decreases compression efficiency since R converges to the feedback bit rate of uncompressed CSI, Rf , as ² reduces to zero. According ˆ f (²) is at least half of Rf . to (2.32), R

2.7.3

Extension: Block Feedback Compression

For the feedback compression algorithm proposed in Section 2.7.1, compressing every other feedback CSI instant is usually sufficient since temporal correlation in CSI is substantially 36

reduced by the periodic feedback protocol. Nevertheless, if the feedback interval is small, the residual correlation is significant and can be fully exploited by compressing CSI for multiple consecutive feedback instants, called block feedback compression. The algorithm in Section 2.7.1 can be extended to block feedback compression as follows. Let W denote the compression block length in feedback interval KT . Feedback compression specified in Section 2.7.1 repeats for a block of W feedback instants. Upon the first occurrence of feedback truncation in compressing a CSI block, the codewords indicating truncation are sent for the remaining feedback instants in this block. Uncompressed CSI feedback is performed between every two blocks of compressed CSI feedback to stop propagation of CSI errors due to feedback truncation. For block feedback compression, the ergodic throughput in (2.30) is rewritten as h i o 1 Xn 1 R(K, D) + (1 − ²)` R(K, D) + 1 − (1 − ²)` R(∞) . W +1 W +1 `=1 (2.33) W

ˆ R(K, D) =

Using (2.33), the feedback throughput gain is obtained as shown in the following lemma. Lemma 4 For block feedback compression, the feedback throughput gain is ˆ ∆R(K, D, ²) =

1 − (1 − ²)W +1 ∆R(K, D) (W + 1)²

(2.34)

and the average CSI feedback bit rate is ˆ ˆ f (²) = B + E B(²) . R (E + 1)KT

(2.35)

As observed from (2.34), if the temporal correlation in feedback CSI is high (² 1) = 1 − = 1−

N X £ ¤ Pr(Zn˜ = 0 | I(k−1)K = `) + Pr(Zn˜ = 1 | I(k−1)K = `) π` `=1 N X

P``K π`

= 1−

`=1

K−k Pmm Pm` P``k−1 π`

k=1 `=1 m=1 m6=`

`=1 N X



K X N X N X

P``K π`

¶ N X N K µ K P π X X Pmm P`` k m` ` − P`` Pmm `=1 m=1 m6=`

52

k=1

= 1−

N X

P``K π` −

N X N K − P K )P π X (Pmm m` ` `` . Pmm − P`` `=1 m=1 m6=`

`=1

Thus, by combining the above equation and the feedback constraint, the desired result follows.

2.11.2

Proof of Corollary 1

The left-hand-size of (2.12) attains its maximum value of one at K = 1. Thus, the result for δ → 0 follows. Next, consider the case of δ → 1 . An upper bound on K is obtained by considering the following upper-bound on the left-hand-size of (2.12) N X

P``K π` +

m6=`

`=1

=

N X

P``K π` +

`=1 N X

X (P K − P K )Pm` π` mm `` Pmm − P`` X [max(Pmm , P`` )K − min(Pmm , P`` )K ]Pm` π` |Pmm − P`` |

m6=`

X (maxa Paa )K Pm` π` |Pmm − P`` | m6=` `=1   X Pm` π`  ≤ (max Paa )K 1 + . a |Pmm − P`` | ≤

P``K π` +

m6=`

Therefore, the following constraint is the relaxation of that in (2.12)  (max Paa )K 1 + a

 X m6=`

Pm` π`  ≥ 1 − δ. |Pmm − P`` |

(2.40)

From the above constraint, an upper bound on K, denoted as K + , is obtained as

K+ =

³ P log(1 − δ) − log 1 + m6=` log (maxa Paa )

53

Pm` π` |Pmm −P`` |

´ .

(2.41)

To obtain an lower bound for K, the left-hand-size of (2.12) is bounded from below as N X

P``K π` +

`=1

X (P K − P K )Pm` π` mm `` Pmm − P``

m6=`



N X

³ ´K P``K π` ≥ min Paa .

`=1

a

(2.42)

Thus, the constraint (mina Paa )K ≥ 1 − δ is more stringent than that in (2.12). Based on this constraint, a lower-bound on K, denoted as K − , is obtained as K− =

log(1 − δ) . log(mina Paa )

(2.43)

Combining (2.43) and (2.41) gives the desired result.

2.11.3

Proof of Lemma 2

From (2.16)

R =

N N X X

¤ £ E log2 (1 + P kHn Qf (Hn−d )k2 ) | Hn ∈ V` , Hn−d ∈ Vm ×

`=1 m=1

Pr(Hn ∈ V` , Hn−d ∈ Vm ) = =

N N X X `=1 m=1 N N X X

¤ £ E log2 (1 + P kHn vm k2 ) | Hn ∈ V` , Hn−d ∈ Vm Pr(Hn ∈ V` , Hn−d ∈ Vm ) R`m Pr(In = `, In−d = m)

`=1 m=1

where R`m is defined in (2.18), vm is the mth member of the codebook F, and V` is the Voronoi cell.

2.11.4

Proof of Corollary 2

By substituting D = 0 into (2.17) N X £ ¤ R(0) = E log2 (1 + P kHn Qf (Hn )k2 ) | Hn ∈ V` π` . `=1

54

Given that ∪` V` = CNr ×Nt , the result in (2.19) follows from the above equation. Next, by substituting (2.4) into (2.17), the first equality in (2.20) is obtained. The second inequality in (2.20) follows from (2.7), (2.18) and the fact that ∪` V` = CNr ×Nt .

2.11.5

Proof of Proposition 4

From (2.21), the feedback throughput gain is upper-bounded as

R(D) ≤ ≤

N X N X

¯£ ¤ ¯ R`m ¯ PD `m − π` ¯ πm

`=1 m=1 µ N X

πm max R`m

m=1

`

¶X N

¯£ D ¤ ¯ P

`=1

The desired result is obtained from (2.44) and Lemma 3.

55

¯ ¯ − π ` `m

(2.44)

Chapter 3

SDMA with Limited Feedback: Joint CSI Quantization, Scheduling and Beamforming SDMA (or MU-MIMO) provides cost-effective solutions for high-rate communications in cellular networks. For downlink SDMA, joint optimizing limited feedback, downlink scheduling and beamforming alleviates the negative impact of CSI inaccuracy and achieves multiuser diversity gain besides enabling spatial multiplexing. Solving this optimization problem is highly complicated by involving several different system functions. In this chapter, a sub-optimal solution is presented. This solution achieves high downlink throughput and is proved to achieve the optimal throughput scaling for a large number of users.

3.1

Prior Work

In this chapter, we consider a practical scenario where partial CSI is acquired by the base station through quantized CSI feedback, known as limited feedback [8]. Quantized CSI feedback for point-to-point communications has been extensively studied recently (see e.g. [8,9] and the references therein). The effects of CSI quantization on a SDMA system have 56

been investigated in [24, 75, 76]. The key result of [24] is that the number of CSI feedback bits can be reduced by exploiting multiuser diversity. In [75], combined quantized CSI feedback and zero-forcing dirty paper coding are shown to attain most of the capacity achieved by perfect CSI feedback. In [76], it is shown that for a small number of users the number of CSI feedback bits must increase with the signal-to-noise ratio (SNR) to ensure that the throughput grows with SNR. This chapter addresses joint beamforming and scheduling for SDMA systems to maximize throughput, assuming backlogged users. A similar scenario but with bursty data and the objective of meeting quality-of-service (QoS) for different users is addressed in [77] and references therein. The optimal approach for our full-queue scenario involves an exhaustive search, where for each possible subset of users a corresponding set of beamforming vectors is designed using algorithms such as that proposed by Schubert and Boche in [78]. The main drawback of the optimal approach is its complexity, which increases exponentially with the number of users. This motivates the designs of more efficient SDMA algorithms. In [28], a practical SDMA algorithm, called opportunistic SDMA (OSDMA), is proposed, which supports low-rate beamforming feedback and satisfies the orthogonal beamforming constraint. As shown in [28], for a large number of users, an arbitrary set of orthogonal beamforming vectors ensures that the throughput increases with the number of users at the optimal rate. Nevertheless, for a small number of users, such arbitrary beamforming vectors are highly sub-optimal due to excessive interference between scheduled users. To reduce multiuser interference caused by sub-optimal beamforming vectors, an extension of OSDMA, called OSDMA with beam selection (OSDMA-S), is proposed in [79], where each mobile iteratively selects beamforming vectors broadcast by the base station and sends back its choices. Due to distributed beam selection, numerous iterations of broadcast and feedback are required for implementing OSDMA-S, which incurs significant downlink overhead and feedback delay. As a result, the throughput gains of OSDMA-S over OSDMA are marginal. 57

An alternative beamforming SDMA algorithm is proposed in [24], referred to as ZF-SDMA, where feedback CSI is quantized using the random vector quantization (RVQ) algorithm [76, 80] and greedy-search scheduling is performed prior to zero-forcing beamforming. A design similar to ZF-SDMA [17] has been proposed to the emerging 3GPP-LTE standard [5], which is the latest cellular communication standard. The drawback of ZFSDMA is its lack of robustness against CSI inaccuracy due to the separate designs of the limited feedback, scheduling and beamforming sub-algorithms. In industry, SDMA with orthogonal beamforming, under the name per user unitary and rate control (PU2RC) [16], has been proposed to the 3GPP-LTE standard. The main feature of PU2RC is limited feedback, where multiuser precoders or beamformers are selected from a codebook of multiple orthonormal bases. Based on limited feedback, PU2RC supports SDMA, scheduling, and adaptive modulation and coding. Because of its versatility and advanced features, PU2RC is one of the most promising solutions for high speed downlink in 3GPP-LTE. The importance of PU2RC for the next-generation wireless communication motivates the investigation of its performance in this chapter. In this chapter, we consider a simplified PU2RC system where scheduled users have single data streams, which are separated by orthogonal beamformers. In this case, PU2RC generalizes OSDMA [28] by allowing the beamforming codebook to contain more than one orthonormal basis. Such a generalization complicates the performance analysis of PU2RC because the resultant scheduler is more complicated. To be specific, the scheduler has to select an orthonormal basis from the codebook besides choosing a particular user for each codebook vector. Such a challenge motivates our use of a new analytical tool, namely uniform convergence in the weak law of large numbers [81], for analyzing the throughput of PU2RC instead of extreme value theory as applied in [28]. Theory of uniform convergence in the weak law of large numbers is also applied in our previous work [82] for analyzing the throughput of uplink SDMA with limited feedback. Despite using the same tool, the analysis in this chapter differs from [82] due to differences between the uplink and downlink. Specifically, the received data signal for the downlink 58

propagates through a single-user channel, but that for the uplink passes through multiuser channels. As a result, SINR feedback for downlink SDMA is infeasible for uplink SDMA, where SINR depends on multiuser CSI and is hence uncomputable at users. Consequently, downlink and uplink SDMA require different designs of scheduling algorithm. Thus, the joint beamforming and scheduling algorithm presented in this chapter is not applicable for uplink SDMA. Interestingly, despite the differences between the uplink and downlink, the asymptotic throughput scaling laws for downlink SDMA as derived in this chapter are found to be identical to those for uplink SDMA [82].

3.2

Contributions

The main contribution of this chapter is the analysis of the throughput scaling of PU2RC for an asymptotically large number of users U → ∞. Using the theory of uniform convergence in the weak law of large numbers, throughput scaling laws are derived for three regimes, namely the normal SNR , interference-limited and the noise-limited regimes. In the normal SNR regime, both the variance of noise and multiuser interference are comparable; in the interference-limited regime, multiuser interference dominates over noise; the reverse exists in the noise-limited regime. Our main results are summarized as follows. In the interference-limited regime, we show that the throughput scales logarithmically with U but does not increase with the number of transmit antennas Nt at the base station. In both the normal SNR and noise-limited regimes, we show that the throughput scales double logarithmically with U and linearly with Nt . This throughput scaling law shows that PU2RC achieves the optimal multiuser diversity gain as OSDMA in the normal SNR regime1 . Thereby, this result contradicts the intuition that using multiple orthonormal bases in the codebook splits the user pool and hence reduces the multiuser diversity gain. Using Monte Carlo simulations, the asymptotic throughput scaling laws are also found to hold in the non-asymptotic regime where U is finite. 1

The interference-limited and noise-limited regimes have not been considered for OSDMA in [28]

59

The asymptotic throughput analysis for PU2RC provides several guidelines for designing the scheduler to ensure optimal throughput scaling. First, in the interferencelimited regime, scheduling should use the criterion of minimum quantization error. Second, in the normal SNR regime, scheduled users should have both large channel power and small quantization errors. Third, in the noise-limited feedback, scheduling should select users with large channel power while the quantization error is a less important scheduling criterion. Numerical results are presented for evaluating the throughput of PU2RC and also comparing PU2RC with ZF-SDMA. Several observations are made. First, increasing the amount of CSI feedback (or the codebook size) can decrease the throughput for PU2RC if the number of users is small. Otherwise, more CSI feedback provides a throughput gain. Second, PU2RC achieves higher throughput than ZF-SDMA for large numbers of users but the reverse holds for relatively small numbers of users. Third, decreasing the codebook size causes a larger throughput loss for ZF-SDMA than that for PU2RC.

3.3

System Model

The downlink or broadcast system illustrated in Fig. 3.1 is described as follows. The base station with Nt antennas transmits data simultaneously to Nt active users chosen from a total of U users, each with one receive antenna. The base station separates the multiuser data streams by beamforming, i.e. assigning a beamforming vector to each of t the Nt active users. The beamforming vectors {wn }N n=1 are selected from multiple sets of

unitary orthogonal vectors following the beam and user selection algorithm described in Section 3.4.2. Equal power allocation over scheduled users is considered2 . The received signal of the uth scheduled user is expressed as r yu =

P †X h wn xn + νu , Nt u

u ∈ A,

(3.1)

n∈A

2

Note that equal power allocation is close to the optimal water-filling method if scheduled users all have high SINR.

60

Base Station

Mobiles

Beamforming

Scheduling

Downlink Channels

Feedback Channels

CSI Quantization

Figure 3.1: Downlink system with limited feedback where we use the following notation Nt number of transmit antennas and also number of scheduled users; hu (Nt × 1 vector) downlink channel; xu transmitted symbol with E[|xu |2 ] = 1; yu received symbol; † conjugate transpose matrix operation; wu (Nt × 1 vector) beamforming vector with kwu k2 = 1; A The index set of scheduled users; P transmission power; and νu AWGN sample with νu ∼ CN (0, 1). For the purpose of asymptotic analysis of PU2RC, we make the following assumption: Assumption 1 The downlink channel hu ∀ u = 1, 2, · · · , U is an i.i.d. vector with CN (0, 1) coefficients. 61

Given this assumption commonly made in the literature of SDMA and multiuser diversity [28, 76, 79, 83, 84], the channel direction vector hu /khu k of each user follows an uniform distribution. Assumption 1 greatly simplifies the throughput analysis of PU2RC in Section 3.5 but has no effect on the PU2RC algorithms in Section 3.4. Assumption 1 is valid for the scenario where wireless channels have rich scattering and users encounter equal path loss. Throughput analysis for a more complicated channel model is a topic for future investigation.

3.4

Algorithms

In this section, we propose the algorithms for PU2RC including (i) limited feedback by the mobiles and (ii) joint beamforming and scheduling at the base station. The principles for these algorithms have been described in the proposal of PU2RC [16] even though their details are not provided therein. The algorithms presented in the following sections are tailored for the system model in Section 3.3. The following discussion on algorithms serves two purposes: (i) to elaborate the operation of PU2RC and (ii) to establish an analytical model for the asymptotic throughput analysis in Section 3.5.

3.4.1

Limited Feedback

Without loss of generality, the discussion in this section focuses on the uth user and the same algorithm for CSI quantization is used by other users. For simplicity, we make the following assumption Assumption 2 The uth user has perfect CSI hu . This assumption allows us to neglect channel estimation error at the uth mobile. For convenience, the CSI, hu , is decomposed into two components: the gain and the shape. Hence, hu = gu su ,

u = 1, · · · , U,

62

(3.2)

where gu = khu k is the gain and su = hu /khu k is the shape. The uth user quantizes and sends back to the base station two quantities: the channel shape and the SINR. The channel shape su is quantized using a codebook-based quantizer [29] with a codebook comprised of multiple sets of orthonormal vectors in CNt . Let F denote the codebook, V (m) the mth orthonormal set in the codebook, and M the number of such S (m) and the codebook size is N = |F| = M N . For our design, sets. Thus, F = M t m=1 V the M orthonormal bases of F are generated randomly and independently using a method such as that in [85]. Following [10, 11], the quantized channel shape, represented by ˆ su , is the member of F that forms the smallest angle with the channel shape su . Mathematically, ˆ su = arg min d(v, su ), v∈F

(3.3)

where the distortion function d(v, su ) is given as ¯ ¯2 ¯ ¯ d(v, su ) = 1 − ¯v† su ¯ = sin2 (∠(v, su )).

(3.4)

It follows that the quantization error can be defined as ² = sin2 (∠(ˆ su , su )). It is clear that ² = 0 if |ˆ s†u su | = 1 and ² = 1 if ˆ su ⊥ su . The quantized channel shape ˆ su is sent back to the base station through a finiterate feedback channel [8,10]. Since the quantization codebook F can be known a priori to both the base station and mobiles, only the index of ˆ su needs to be sent back. Therefore, the number of feedback bits per user for quantized channel shape feedback is log2 N since |F| = N . The number of additional bits required for SINR feedback is discussed in Section 3.6. Besides the channel shape, the uth user also sends back to the base station the SINR, which serves as a channel quality indicator. For orthogonal beamforming, the SINR is given as [24] SINRu =

γρu (1 − ²u ) , 1 + γρu ²u 63

(3.5)

where γ =

P Nt

is the SNR, ²u the CSI quantization error, and ρu = khu k2 the channel

power. Since the SINR is a scalar and requires much fewer feedback bits than the channel shape, we make the following assumption: Assumption 3 The SINRu is perfectly known to the base station through feedback. The same assumption is also made in [28, 79]. The effect of SINR quantization on the throughput is shown to be insignificant using numerical results in Section 3.6.

3.4.2

Joint Scheduling and Beamforming

This section focuses on the joint scheduling and beamforming algorithm designed based on the principles of PU2RC. Having collected quantized CSI from all U users3 , the base station schedules Nt users for transmission and computes their beamforming vectors. To maximize the throughput, Nt scheduled users must be selected through an exhaustive search, which is infeasible for a large user pool. Therefore, we adopt a simpler joint scheduling and beamforming algorithm. In brief, this algorithm schedules a subset of users with orthogonal quantized channel shapes, and furthermore applies these channel shapes as the scheduled users’ beamforming vectors. The joint scheduling and beamforming algorithm is elaborated as follows. First, each member of the codebook F, which is a potential beamforming vector, is assigned (m)

an user with the maximum SINR. Consider an arbitrary vector, for instance vn , which is the nth member of the mth orthonormal subset V (m) of the codebook F. This vector can be the quantized channel shapes of multiple users, whose indices are grouped in a set n o (m) (m) defined as In = 1 ≤ u ≤ U : ˆ su = vn where ˆ su is the uth user’s quantized channel (m)

shape given in (3.3). From (3.3), In

can be equivalently defined as

n ³ ´ o In(m) = 1 ≤ u ≤ U | d su , vn(m) < d (su , v) ∀ v ∈ F and v 6= vn(m) . 3

(3.6)

For simplicity, we assume that the number of feedback bits per user is limited but not the total number of feedback bits from all users. Nevertheless, the sum feedback from all users can be reduced by allowing only a small subset of users for feedback, which is an topic addressed in a separate paper [86].

64

(m)

(m)

Among the users in In , vn

is associated with the one providing the maximum SINR,

which is feasible since the SNRs are known to the base station through feedback. The ³ ´ ³ ´ (m) (m) (m) index in and SINR ξn of this user associated with vn can be written as i(m) = arg max SINRu n (m)

u∈In

(m)

where the index set In

and ξn(m) = max SINRu , (m)

(3.7)

u∈In

and the function SINRu are expressed respectively in (3.6) and (m)

(3.5). In the event that In

(m)

= ∅, the vector vn

is associated with no user and the maxi-

(m)

mum SINR ξn

in (3.7) is set as zero. Second, the orthonormal subset of the codebook that ´ ³ P t (m) . maximizes throughput is chosen, whose index is m? = arg max1≤m≤M N log 1 + ξ n n=1 n ? oNt (m ) , Thereby, the users associated with this chosen subset, specified by the indices in n=1

are scheduled for simultaneous transmission using beamforming vectors from the (m? )th orthonormal subset. The above scheduling algorithm does not guarantee that the number of scheduled users is equal to Nt , the spatial degrees of freedom. For a small user pool, the number of scheduled users is smaller than Nt . This is desirable because it is unlikely to find Nt simultaneous users with close-to-orthogonal channels in a small user pool. In this case, having fewer scheduled users than Nt reduces interference and leads to higher throughput. As the total number of users increases, the number of scheduled users converges to Nt . Numerical results on the average number of scheduled users for PU2RC are presented in Section 3.6. Based on the preceding algorithm for joint beamforming and scheduling, the ergodic throughput for PU2RC is given as " R=E

max

1≤m≤M

Nt X

Ã

!#

log 1 + max SINRu (m)

(3.8)

u∈In

n=1

where SINRu is given in (3.5). The scaling of R with the number of users U as U → ∞ is analyzed in Section 3.5. 65

3.5

Asymptotic Throughput Scaling

In this section, we derive the scaling laws of the PU2RC throughput for an asymptotically large number of users. Auxiliary results required in the analysis are first presented in Section 3.5.1. Three SNR regimes, namely normal, interference-limited, and noise-limited regimes, are considered in Section 3.5.2 to 3.5.4, respectively. Finally, numerical results showing how the asymptotic throughput scaling laws apply in the non-asymptotic regime are presented in Section 3.5.5. The asymptotic throughput scaling laws derived in this section for downlink SDMA are observed to be identical to those for uplink SDMA [82]. This suggests duality between uplink and downlink SDMA in terms of asymptotic throughput.

3.5.1

Auxiliary Results

Two auxiliary results are provided in this section. In Section 3.5.1, the theory of uniform convergence in the weak law of large numbers is discussed, which is an important tool for the subsequent asymptotic throughput analysis. The other useful result related to the channel-shape quantization error is presented in Section 3.5.1. Uniform Convergence in the weak law of large numbers In this section, a lemma on the uniform convergence in the weak law of large numbers [81] is obtained by generalizing [87, Lemma 4.8] from R3 to CNt . This lemma is useful for analyzing the number of users whose channel shapes lie in one of a set of congruent disks on the surface of an unit hyper-sphere in CNt . Lemma 5 (Gupta and Kumar) Consider U random points uniformly distributed on the surface of an unit hyper-sphere in CNt and N disks on the sphere surface that have equal volume denoted as A. Let Tn denote the number of points belong to the nth disk. For every τ1 , τ2 > 0 Ã

! ¯ ¯ ¯ Tn ¯ Pr sup ¯¯ − A¯¯ ≤ τ1 > 1 − τ2 , 1≤n≤N U 66

U ≥ Uo

(3.9)

where

½ Uo = max

3 16c 4 2 log , log τ1 τ2 τ1 τ2

¾ (3.10)

and c is a constant. Proof: See Appendix 3.8.1.

¤

Quantization Error of Channel Shape The complementary CDF of the CSI quantization error ² is analyzed as follows. As defined in Section 3.4.1, ² = sin2 (∠(ˆ s, s)) where s and ˆ s are the original and the quantized channel shapes of an arbitrary user. From the quantization function in (3.3), the complementary CDF of ² is

à Pr(² ≥ δ) = Pr s ∈ /

[

! Bδ (v) ,

(3.11)

v∈F

© ª where 0 ≤ δ ≤ 1 and Bδ (v) = s ∈ ONt : |s† v|2 ≤ δ is a sphere cap on the unit sphere ONt . The CDF of ² for 0 ≤ δ ≤

1 2

has the simple expression as given in the following

lemma, but the derivation of CDF for

1 2

≤ δ ≤ 1 is difficult because the sphere caps

{Bδ (v) : v ∈ F } overlap. Lemma 6 The complementary CDF of ², Pr(² ≥ δ), for 0 ≤ δ ≤ £ ¤M Pr(² ≥ δ) = 1 − Nt δ Nt −1 ,

0 ≤ δ ≤ 12 ,

1 2

is given as

(3.12)

where M is the number of orthonormal bases in the quantization codebook F. In addition, ¡ ¢M Pr(² ≥ δ) ≤ 1 − δ Nt −1 ∀ 0 ≤ δ ≤ 1. Proof: See Appendix 3.8.2.

¤

Next, the following lemma provides an upper-bound for the quantity E[− log ²], which is useful for the throughput analysis in the sequel. The derivation of this result uses Lemma 6 and [76, Lemma 4]. 67

Lemma 7 Given a codebook of M orthonormal bases, the following inequality holds log M log Nt log M + 1 log Nt + ≤ E [− log ²] ≤ + (Nt − 1)Pα Nt − 1 (Nt − 1)Pα Nt − 1

(3.13)

where ² is the channel-shape quantization error and h iM Pα = 1 − 1 − Nt 2−(Nt −1) . Proof: See Appendix 3.8.3.

3.5.2

(3.14)

¤

Normal SNR Regime

In this section, the throughput scaling law of PU2RC is analyzed for the normal SNR regime, where the variance of noise and that of interference are comparable. For this regime, the SINR and throughput are given respectively in (3.5) and (3.8). As shown in the sequel, in the normal SNR regime, the throughput of PU2RC scales double logarithmically with the number of users and linearly with the number of antennas. This throughput scaling law is identical to those for ZF-SDMA [24] and OSDMA [28]. Therefore, these algorithms all achieve optimal multiuser diversity gain. The procedure for deriving the throughput scaling law for PU2RC is to first obtain an upper-bound for the throughput scaling factor and second prove its achievability. The achievability proof uses Lemma 5 on the uniform convergence in the weak law of large numbers. The above procedure is also adopted for the throughput analysis for other regimes in subsequent sections. For the normal SNR regime, the throughput scaling factor for PU2RC is upper bounded as shown in the following lemma. Lemma 8 In the normal SNR regime, the throughput scaling factor for PU2RC is upper bounded as R ≤ 1. U →∞ Nt log log U lim

68

(3.15)

Proof: See Appendix 3.8.4.

¤

Next, the upper-bound in (3.15) is shown to be achievable. Thereby, the throughput scaling law of PU2RC in the normal SNR regime is obtained as shown in the following proposition. Proposition 6 In the normal SNR regime, the throughput scaling law for PU2RC is

lim

U →∞

R = 1. Nt log log U

(3.16)

Proof: See Appendix 3.8.5.

¤

The proof uses Lemma 5 on the uniform convergence in the weak law of the large number. As shown in the proof, to achieve the throughput scaling law in (3.16), the quantization errors and channel power of scheduled users must scale with the number of users U as 1 log U

and log U , respectively. This suggests that a scheduler for the normal SNR regime

should schedule users with both small quantization errors and large channel power as U increases.

3.5.3

Interference-Limited Regime

In this section, the throughput scaling law of PU2RC is analyzed for the interferencelimited regime where interference dominates over noise. By omitting the noise term, the SINR in (3.5) for the interference-limited regime reduces to SINR(α) u =

1 −1 ²u

(3.17)

where the superscript (α) identifies the interference-limited regime. By substituting (3.17) into (3.8), the throughput for the interference-limited regime is written as " R

(α)

=E

max

1≤m≤M

Nt X n=1

69

Ã

1 log max (m) ²u u∈In

!# .

(3.18)

The scaling law of R(α) with U is obtained as follows. The upper-bound of the scaling factor of R(α) with U is shown in the following lemma. Lemma 9 In the interference limited regime, the throughput scaling factor is upper bounded as R(α) ≤ 1. log U

lim

(3.19)

U →∞ Nt Nt −1

Proof: See Appendix 3.8.6.

¤

This proof uses Lemma 7 in Section 3.5.1. Next, the equality in (3.19) is shown to be achievable. The main result of this section is summarized in the following proposition. Proposition 7 In the interference-limited regime, the throughput scaling law for PU2RC is lim

U →∞

R(α) = 1. Nt Nt −1 log U

(3.20)

Proof: See Appendix 3.8.7.

¤

Again, this proof makes use of Lemma 5 on the uniform convergence in the weak law of large numbers. By comparing Propositions 6 and 7, the throughput scales as

Nt Nt −1

log U in

the interference-limited regime but Nt log log U otherwise. The reason for this difference is that the asymptotic throughput is determined by the channel power (ρ) in the normal SNR and noise-limited regimes, but by the CSI quantization errors (²) of scheduled users in the interference-limited regime. In the normal SNR and noise-limited regimes, the asymptotic throughout can be written as Nt E[log ρ], where ρ scales as log U due to multiuser diversity gain. In the interference-limited regime, the asymptotic throughput is given as Nt E[− log ²] and the scaling law of ² is U

− N 1−1 t

.

A few remarks are in order. 70

1. The linear scaling factor in (3.20), namely Nt /(Nt − 1), is smaller than Nt , which is the number of available spatial degrees of freedoms. This indicates the loss in multiplexing gain for Nt ≥ 3 in the interference-limited regime. Such loss is not observed in the normal SNR (cf. Proposition 6) or noise-limited (cf. Proposition 8) regimes. 2. In the interference-limited regime, scheduling users with small channel-shape quantization errors is sufficient for ensuring optimal throughput scaling. The reason is that the SINR in (3.17) depends only on the quantization error. 3. In the interference-limited regime, the throughput scaling law for PU2RC is identical to that for ZF-SDMA [24, Theorem 2]4 .

3.5.4

Noise-Limited Regime

In this section, the throughput scaling law of PU2RC in the noise-limited regime is analyzed, where noise dominates over multiuser interference. By removing the interference term (γρu ²u ) in (3.5), the SINR for the noise-limited regime is given as SINR(β) u = γρu (1 − ²u )

(3.21)

where the superscript (β) specifies the noise-limited regime. By substituting (3.21) into (3.8), the corresponding throughput is written as ( R(β) = E

max

1≤m≤M

Nt X

"

#)

log 1 + max γρu (1 − ²u ) (m)

.

(3.22)

u∈In

n=1

The scaling law of R(β) with U for U → ∞ is obtained as shown in the following proposition. 4

Note that [24, (45)] gives the throughput scaling law for a single scheduled user. Multiplication of this result with Nt gives the identical throughput scaling law for PU2RC as shown in Proposition 7.

71

Proposition 8 In the noise-limited regime, the throughput for PU2RC scales as follows R(β) = 1. U →∞ Nt log log U lim

Proof: See Appendix 3.8.8.

(3.23)

¤

By comparing Proposition 6 and 8, the throughput scaling laws are observed to be identical for both the normal SNR and noise-limited regimes. Moreover, as reflected in the proof, to achieve the optimal throughput scaling law, scheduled users in the noise-limited regime are required to have channel power scaling as log U and quantization errors smaller than a constant dmin defined in (3.42). Thus, for the noise-limited regime, channel power is a more important scheduling criterion than quantization errors.

3.5.5

Non-Asymptotic Regimes

In preceding sections, the throughput scaling laws for PU2RC are derived for different asymptotic regimes characterized by an asymptotically large number of users (U → ∞). In this section, these asymptotic scaling laws are compared with their counterparts in the non-asymptotic regimes corresponding to a finite number of users (U < ∞). The purpose of such a comparison is to evaluate the usefulness of the asymptotic results derived in previous section for characterizing the throughput of practical PU2RC systems. For this purpose, Fig. 3.2 shows the throughput versus number of users curves for the SNR values of {0, 5, 30} dB, corresponding respectively to the noise-limited, the normal SNR and the interference-limited regimes. The range of the number of users is 1 ≤ U ≤ 140, the number of transmit antennas is Nt = 2 and the codebook size is N = 16. The above curves present the PU2RC throughput scaling laws in the nonasymptotic regimes. Also plotted in Fig. 3.2 are the curves defined by the asymptotic throughput scaling law

Nt Nt −1

log U for the interference-limited regime (cf. Proposition 7)

and Nt log log U for both the normal SNR and the noise-limited regimes (cf. Proposition 6 72

20

PU2RC (SNR = 30 dB) 18

Throughput (bps/Hz)

16 14

N log(U)/(N −1) t

t

12 10 8

PU2RC (SNR = 5 dB)

6

Ntloglog(U)

4 2

PU2RC (SNR = 0 dB) 0

20

40

60

80

100

120

140

Number of Users

Figure 3.2: Comparison between asymptotic and non-asymptotic throughput scaling laws for PU2RC for SNR = {0, 5, 30} dB, the codebook size N = 16, and the number of transmit antennas Nt = 2. and 8). As observed from Fig. 3.2, as the number of users increases, the non-asymptotic curve for SNR = 30 dB becomes parallel to the curve following the asymptotic throughput scaling law

Nt Nt −1

log U . Likewise, the non-asymptotic curves for SNR = 0 dB and 5 dB have

the same slopes as the corresponding asymptotic curve defined by Nt log log U . Therefore, the asymptotic throughput scaling laws also hold in the non-asymptotic regimes. Note that the gaps between the asymptotic and non-asymptotic curves are throughput constant factors that become insignificant in the asymptotic regimes (U → ∞).

3.6

Numerical Results

In this section, various numerical results are presented. In Section 3.6.1, the effect of increasing channel shape feedback on throughput is investigated. In Section 3.6.2, for an increasing number of users, the throughput of PU2RC is evaluated against that of ZF-SDMA in [24] as well as the upper bound achieved by dirty paper coding (DPC) and multiuser water filling [88]. For simplicity, Assumption 3 is made and thus the SINR 73

feedback is assumed perfect for all algorithms in comparison. In Section 3.6.3, the capacity loss due to the SINR quantization is characterized.

3.6.1

Effect of Increasing Channel Shape Feedback

For PU2RC, increasing channel shape feedback does not necessarily lead to higher throughput as shown in Fig. 3.3. In Fig. 3.3, the curves of PU2RC throughput versus the number of users U are plotted for different codebook sizes N . The SNR is 5 dB and the number of transmit antennas is Nt = 4. Fig. 3.3(a) and Fig. 3.3(b) show the small (1 ≤ U ≤ 50) and the large user ranges (1 ≤ U ≤ 200), respectively. As observed from Fig. 3.3(a), in the range of 4 ≤ U ≤ 22, increasing N decreases the throughput. The reason is that a larger codebook size divides the user pool because each user is associated with only one codebook vector (cf. Section 3.4.2). Consequently, increasing the codebook size reduces the probability of finding scheduled users with large channel gains and also associated with the same orthonormal basis in the codebook. Nevertheless, such an adverse effect of increasing the codebook size diminishes as the number of users increases. As shown in Fig. 3.3, for U ≥ 70, a larger codebook size results in higher throughput. The above results motivate the need for choosing an optimal codebook size for a given number of users.

3.6.2

Comparison with ZF-SDMA and Dirty Paper Coding

Presently, PU2RC and ZF-SDMA [15, 17, 24] are two main solutions for multiuser MIMO downlink for 3GPP-LTE. In this section, their performance is compared using numerical results. Moreover, the throughput of PU2RC is evaluated against the upper-bound achieved by dirty paper coding. In Fig. 3.4, the throughput of PU2RC is compared with that of ZF-SDMA for an increasing number of users. The number of transmit antenna is Nt = 4 and the SNR is 5 dB. Moreover, the codebook sizes N = {4, 8, 16, 32} for channel shape quantization are considered. As in [24], the threshold 0.25 is applied in the greedy-search scheduling for 74

8 N=8 N = 32 N = 64

Throughput (bps/Hz)

7

6

5

4

3

2

5

10

15

20

25

30

35

40

45

50

Number of Users, U

(a) Small numbers of users

10.5 N = 64 N = 32 N = 16 N=8 N=4

10

Throughput (bps/Hz)

9.5 9 8.5 8 7.5 7 6.5 6 5.5 5

0

50

100

150

200

Number of Users

(b) Large numbers of users

Figure 3.3: Throughput of PU2RC for an increasing number of users U , SNR = 5 dB, and the number of transmit antennas Nt = 4.

75

ZF-SDMA. Fig. 3.4(a) and Fig. 3.4(b) show respectively the small (1 ≤ U ≤ 35) and the large (1 ≤ U ≤ 200) user ranges. As observed from Fig. 3.4(a), for a given codebook size (either N = 16 or N = 64), PU2RC achieves higher throughput than ZF-SDMA for a relative large number of users but the reverse holds for a smaller user pool. Specifically, in Fig. 3.4(a), the throughput curves for PU2RC and ZF-SDMA cross at U = 19 for N = 16 and at U = 27 for N = 64. For a sufficiently large number of users, PU2RC always outperforms ZF-SDMA in terms of throughput as shown in Fig. 3.4(b). Furthermore, compared with ZF-SDMA, PU2RC is found to be more robust against CSI quantization errors. For example, as observed from Fig. 3.4(b), for U = 100, the throughput loss for PU2RC due to the decrease of the codebook size from N = 64 to N = 16 is 0.3 bps/Hz but that for ZF-SDMA is 1.5 bps/Hz. The above observations are explained shortly. In summary, these observations suggest that PU2RC is preferred to ZF-SDMA for a large user pool but not for a small one. To explain the observations from Fig. 3.4, the average numbers of scheduled users for PU2RC and ZF-SDMA are compared in Fig. 3.5 for an increasing number of users. It can be observed from Fig. 3.5 that PU2RC tends to schedule more users than ZFSDMA. First, for a small number of users, interference between scheduled users can not be effectively suppressed by scheduling, and hence more simultaneous users result in smaller throughput. This explains the observation from Fig. 3.4(a) that PU2RC achieves lower throughput than ZF-SDMA due to more scheduled users. Second, for a large user pool, the channel vectors of scheduled users are close-to-orthogonal and interference is negligible. Therefore, a larger number of scheduled users leads to higher throughput. For this reason, PU2RC outperforms ZF-SDMA for a large number of users as observed from Fig. 3.4(b). Last, with respect to ZF-SDMA, the better robustness of PU2RC against CSI quantization errors is mainly due to the joint beamforming and scheduling (cf. Section 3.4.2). Note that beamforming and scheduling for ZF-SDMA are performed separately [24]. Fig. 3.6 compares the throughput of PU2RC and ZF-SDMA for an increasing SNR. The number of transmit antennas is Nt = 4 and the codebook size is N = 64. As observed 76

8

Throughput (bps/Hz)

7

6

5

4

3 ZF−SDMA: N = 64 ZF−SDMA: N = 16 PU2RC: N = 16 PU2RC: N = 64

2

1

5

10

15

20

25

30

35

Number of Users

(a) Small Numbers of Users

10

PU2RC

Throughput (bps/Hz)

9

8

7

6

ZF−SDMA PU2RC: N = 64 ZF−SDMA: N = 64 ZF−SDMA: N = 16 PU2RC: N = 16

5

4

0

50

100

150

200

Number of Users

(b) Large Numbers of Users

Figure 3.4: Throughput comparison between PU2RC and ZF-SDMA for an increasing number of users U , SNR = 5 dB, and the number of transmit antennas Nt = 4.

77

Average Number of Scheduled Users

4

PU2RC 3.5

3

ZF−SDMA

2.5

2

1.5

1

5

10

15

20

25

30

35

40

45

Number of Users

Figure 3.5: The average numbers of scheduled users for PU2RC and ZF-SDMA for SNR = 5 dB, and the number of transmit antennas Nt = 4. from Fig. 3.6, for the number of users U = 20, PU2RC achieves lower throughput than ZF-SDMA over the range of SNR under consideration (0 ≤ SNR ≤ 20 dB). Nevertheless, for larger numbers of users (U = 40 or 80), PU2RC outperforms ZF-SDMA for a subset of the SNRs. Specifically, the throughput versus SNR curves for PU2RC and ZF-SDMA crosses at SNR=7 dB for U = 40 and at SNR=18 dB for U = 80. The above results suggest that in the practical range of SNR, PU2RC is preferred to ZF-SDMA only if the user pool is sufficiently large. Fig. 3.7 compares the throughput of PU2RC with an upper bound achieved by dirty paper coding (DPC) and multiuser iterative water-filling [88]. A smaller number of antennas Nt = 2 is chosen to reduce the high computational complexity of iterative water-filling for a large number of users. Hence, each user has a 2 × 1 multiple-inputsingle-output (MISO) channel. Moreover, SNR = 5 dB and the channel shape codebook size is N = {2, 4, 8, 16}. As observed from Fig. 3.7, the gap between the throughput of PU2RC and its upper bound narrows as the number of users U or the codebook size N increases. At U = 200 and N = 16, PU2RC achieves about 85% of the sum capacity of 78

12 PU2RC ZF−SDMA

11

U = 80

Throughput (bps/Hz)

10 9 8

U = 40 7 6

U = 20 5 4

0

5

10 SNR (dB)

15

20

Figure 3.6: Throughput comparison between PU2RC and ZF-SDMA for an increasing number SNR; The codebook size N = 64 and the number of transmit antennas Nt = 4. DPC.

3.6.3

Effect of SINR Quantization

In this section, using numerical results, a small number of bits for SINR feedback is found sufficient for making the capacity loss due to SINR quantization negligible. For PU2RC, Fig. 3.8 compares the cases of perfect and quantized SINR feedback. For quantizing SINR, a scalar quantizer using a squared-error distortion function is employed [29]. Moreover, the quantizer has a simple codebook containing evenly spaced scalars in the SINR range corresponding to a probability of 99%. The number of transmit antenna is Nt = 4, the SINR is 5 dB and the codebook size for channel shape quantization is N = 16. As observed from Fig. 3.8, 2 bits of SINR feedback per user causes only marginal loss in throughput with respect to the perfect SINR feedback. Such loss is negligible for 3-bit feedback. Therefore, a few bits of SINR feedback from each user is almost as good as the perfect case, which justifies Assumption 3.

79

9

DPC 8

Throughput (bps/Hz)

7 6

PU2RC (N = 2, 4, 8, 16)

5 4 3 2 1 0

0

50

100

150

200

Number of Users

Figure 3.7: Comparison between the throughput of PU2RC and its upper bound achieved by dirty paper coding (DPC) and multiuser iterative water-filling for an increasing number of users U , SNR = 5 dB, and the number of transmit antennas Nt = 2.

10

Throughput (bps/Hz)

9

8

7

SINR Feedback = ideal (dashed line), 3, 2, 1 bit

6

5

4

3

0

50

100

150

200

Number of Users

Figure 3.8: The effect of SINR quantization for SNR = 5 dB, the number of transmit antennas Nt = 4, and the codebook size for channel shape quantization is N = 16. 80

3.7

Summary

This paper presents asymptotic throughput scaling laws for SDMA with orthogonal beamforming known as PU2RC for different SNR regimes. In the interference limited regime, the throughput of PU2RC is shown to scale logarithmically with the number of users but does not increase with the number of antennas. In the normal SNR or noise-limited regimes, the throughput of PU2RC is found to scale double logarithmically with the number of users and linearly with the number of antennas at the base station. Numerical results showed that PU2RC can achieve significant gains in throughput with respect to ZF-SDMA for the same amount of CSI feedback.

81

3.8 3.8.1

Appendix Proof of Lemma 5

Lemma 4.8 in [87] can be generalized from R3 to CNt as follows. [87, Lemma 4.8] concerns N congruent disks on the surface of a sphere in R3 , and its derivation relies on two results: the first one is the stereographic projection [89] that one-to-one maps a point on the surface of the sphere to a point on a plane both in R3 ; the second is that the Vapnik-Chervonenkis dimension of a set of disks on a plane in R3 is three [87]. An unit hyper-sphere in CNt can be treated as one in R2Nt [11]. Thereby, the stereographic projection also exists between an unit hyper-sphere and a hyper-plane in CNt [90]. Next, following the same procedure as in [87, Lemma 4.6], the Vapnik-Chervonenkis dimension of a set of disks on a hyperplane in CNt is shown to also three. Based on the two results obtained above for CNt , the remaining steps for proving Lemma 5 are identical to those for [87, Lemma 4.8] and are thus omitted.

3.8.2

Proof of Lemma 6

Since the orthonormal bases in the codebook F are independently and randomly generated, the complementary CDF (3.11) can be equivalently expressed as

Pr(² ≥ δ) =

M Y

¢ ¡ Pr s ∈ / ∪v∈V (m) Bδ (v) ,

(3.24)

m=1

where V (m) denotes the mth orthonormal basis in F. Given that s is isotropically distributed on the unit sphere, (3.24) can be re-written in terms of the volume of sphere caps [29] Pr(² ≥ δ) =

M Y © ª 1 − vol[∪v∈V (m) Bδ (v)] . m=1

82

(3.25)

Since the sphere caps {Bδ (v)} | v ∈ V (m) } are non-overlapping for δ ≤

1 2

and the volume

of each sphere cap is vol[Bδ (v)] = δ Nt −1 as obtained in [28], we can obtain from (3.25) M Y ¡ ¢ Pr(² ≥ δ) = 1 − Nt δ Nt −1 ,

0 ≤ δ ≤ 12 .

(3.26)

m=1

The desired result in (3.12) follows from the last equation. Moreover, from (3.25) and ª Q QM © Nt −1 , which gives the for v ∈ F, Pr(² ≥ δ) ≤ M m=1 {1 − vol[Bδ (v)]} = m=1 1 − δ inequality in the lemma.

3.8.3

Proof of Lemma 7

The minimum of M i.i.d. Beta (Nt , 1) random variables, denoted as {β1 , β2 , · · · , βM }, has the following CDF [76] µ Pr

min βm

1≤m≤M

¶ ≥ b = (1 − bNt −1 )M .

(3.27)

From (3.27) and Lemma 6 1

NtNt −1 ² ∼ = min βm , 1≤m≤M

²≤

1 2

(3.28)

where ∼ = represents equivalence in distribution. The above equivalence results in the following equality (a) ·

µ ¶¸ 1 Nt −1 E − log Nt ²

·

≤ (a)

=



µ ¶ 1 Nt −1 E − log Nt ² |0≤²≤ · µ ¶ E − log min βm | 0 ≤ 1≤m≤M

1 2

¸

min βm

1≤m≤M

E [− log (minm βm )] ¶. µ 1 Nt −1 1 Pr 0 ≤ minm βm ≤ 2 Nt

83

1 1 ≤ NtNt −1 2

¸

(3.29)

As shown in [76, Lemma 4] h ³ ´i log M + 1 log M ≤ E − log min βm ≤ . m Nt − 1 Nt − 1

(3.30)

By combining (3.27), (3.29), and (3.30), the desired inequality follows.

3.8.4

Proof of Lemma 8

From (3.5) and (3.8) !# 1 + γρu R = E max log max (m) 1≤m≤M u∈In 1 + γρu ²u n=1 " Ã !# Nt X ≤ E max log 1 + γ max ρu "

1≤m≤M

≤ E

Ã

Nt X

"N t X

(3.31)

(m)

u∈In

n=1

Ã

log 1 + γ max

!#

max ρu

1≤m≤M u∈I (m) n

n=1

·

µ ¶¸ = Nt E log 1 + γ max ρu .

(3.32)

1≤n≤U

The following result is well-known from extreme value theory (see e.g. [28, (A10)]) ¯ µ¯ ¶ ¶ µ ¯ ¯ 1 ¯ ¯ Pr ¯ max ρu − log U ¯ < O(log log U ) > 1 − O . 1≤u≤U log U

(3.33)

From (3.33) and (3.32) µ R



Nt E {log [1 + γ log U − γO(log log U )]} Pr " Ã !# µ ¶ U X 1 Nt E log 1 + γ ρu O log U

¶ max ρu ≤ log U − O(log log U ) +

1≤n≤U

u=1

(a)



µ

Nt E {log [1 + γ log U − γO(log log U )]} + Nt log (1 + γNt U ) O

1 log U

¶ (3.34)

where (a) is obtained by using Jensen’s inequality. The desired inequality follows from (3.34). 84

3.8.5

Proof of Proposition 6

Define a set of disks on the unit hyper-sphere as n o Bn(m) (d) = s ∈ CNt | ksk2 = 1, 1 − |s† vn(m) |2 ≤ d

1 ≤ m ≤ M, 1 ≤ n ≤ Nt

(3.35)

(m)

where d is the radius of Bn (d). Furthermore, define the user index sets Tˆn(m) =

½ µ ¶¾ 1 (m) 1 ≤ u ≤ U | su ∈ Bn log U (m)

where the disk Bn 1 , 2(log U )Nt −1

1 ≤ m ≤ M, 1 ≤ n ≤ Nt

(3.36)

is defined in (3.35). By applying Lemma 5 with τ1 = τ2 = A =

we obtain that à |Tˆn(m) |

Pr



!

U

>1−

Nt −1

(log U )

1

∀ U > Uo

2 (log U )Nt −1

(3.37)

(m) (m) where Uo is in (3.10). Let U1 denote a sufficiently large integer such that Tˆn ⊂ In . (m)

From (3.8) and (3.5) and by replacing In

R

!# 1 + γρu , U ≥ U1 E log max ˆn(m) 1 + γρu ²u u∈ T n=1 "N Ã !# t X 1 + γρu E log max , U ≥ U1 1 (m) 1 + γρ u log U u∈Tˆn n=1 "N Ã !# t 1 + γ maxu∈Tˆ (m) ρu X n E log , U ≥ U1 1 + logγ U maxu∈Tˆ (m) ρu "N t X

≥ (a)

≥ ≥

(m) with Tˆn

Ã

n=1

(3.38)

n

(m) where the inequality in (a) holds because u ∈ Tˆn ⇒ ²u ≤

1 log U

according to the definition

in (3.36). From (3.37) and (3.38)

R ≥ E "

"N t X

à log

1+

n=1

1−

1 + γ maxu∈Tˆ (m) ρu γ log U

1 2 (log U )Nt −1

#

n

maxu∈Tˆ (m) ρu

! |

|Tˆn(m) |

n

∀ U > max(U1 , Uo ). 85



U (log U )Nt −1

# ×

From the last inequality and (3.33), "

Ã

˜ − O(log log U ˜) log U R ≥ Nt E log 1 + ˜ + O(log log U ˜ )] 1 1/γ + [log U log U · µ ¶¸Nt 1 , ∀ U > max(U1 , Uo ) 1−O log U ˜= where U

U . (log U )Nt −1

!# Ã 1−

1 2 (log U )Nt −1

! ×

It follows from the last inequality that

lim

U →∞

R ≥ 1. Nt log log U

(3.39)

The desired result is obtained by combining (3.39) and Lemma 8.

3.8.6

Proof of Lemma 9

From (3.18)

R

(a)

≤ E

"N t X

à − log

n=1

!# min

min ²u

1≤m≤M u∈I (m) n

· µ ¶¸ = Nt E − log min ²u .

(3.40)

1≤u≤U

In the above equation, min1≤u≤U ²u follows the same distribution as the quantization error for an enlarged codebook having M U orthonormal bases. Therefore, from (3.40) and Lemma 7 R(a)

Nt ≤ Nt − 1

(

log U + log M + 1 ¤M U + log Nt £ 1 − 1 − Nt 2−(Nt −1)

) .

(3.41)

The desired upper bound of the throughput scaling factor follows from the last inequality. n £ ¤M U o → 1 as U → ∞. Note that 1 − 1 − Nt 2−(Nt −1)

86

3.8.7

Proof of Proposition 7

Define the minimum distance of the codebook F as

dmin = min 0

v,v ∈F

1 − |v† v0 |2 . 4

(3.42) (m)

Moreover, similar to (3.36), define the index set of the users in the disk Bn

(dmin ) (cf.

(3.35)) as n o Tn(m) = 1 ≤ u ≤ U | su ∈ Bn(m) (dmin ) , (m)

By the definitions in (3.6) and (3.42), su ∈ Bn

1 ≤ m ≤ M, 1 ≤ n ≤ Nt .

(3.43)

(dmin ) ⇒ u ∈ Im,n . Using this fact, a

throughput lower bound follows by replacing Im,n in (3.8) with Tm,n "

Nt X

Ã

1 ≥ E max log max (m) ²u 1≤m≤M u∈Tn n=1 "N !# Ã t X 1 ≥ E . log max (m) ²u u∈Tn

R(a)

!#

(3.44)

n=1

1

t −1 By applying Lemma 5 with τ1 = τ2 = U − 2 and A = dN min , the numbers of users belonging

to the index sets (3.36) satisfy ¶ µ ¯ ¯ 1 1 ¯ (m) ¯ Nt −1 Pr min ¯Tn ¯ ≥ dmin U − U 2 ≥ 1 − U − 2 , m,n

∀ U ≥ Uo

(3.45)

where Uo is defined in (3.10). From (3.44) and (3.45) " R(a) ≥ Nt E − log

Ã

! min ²u (m)

u∈Tn

# t −1 | |Tn(m) | ≥ dN min U − U

87

1 2

³ ´ 1 1 − U−2 ,

U ≥ Uo .

By applying Lemma 7 R(a)

" # ³ ´ − 12 t −1 1 log M + log dN + log U + log(1 − U ) Nt min + log Nt 1 − U − 2 , ≥ Nt − 1 Pα

U ≥ Uo (3.46)

where Pα is modified from (3.14) as 1 t −1 h iM (dN 2 min U −U ) −(Nt −1) Pα = 1 − 1 − Nt 2

(3.47)

It follows from the last inequality that

lim

U →∞

R(a) ≥ 1. Nt Nt −1 log U

(3.48)

Combining (3.48) and (3.19) gives the desired throughput scaling law for the interferencelimited regime.

3.8.8

Proof of Proposition 8

From (3.22) and since 0 ≤ ²u ≤ 1 " R(β) ≤ E

max

1≤m≤M

Nt X

!#

à log 1 + γ max ρu

n=1

(n)

.

(3.49)

u∈Im

In (3.31) in Appendix 3.8.4, the above upper-bound is also used for bounding the PU2RC throughput in the normal SNR regime. Therefore, the upper-bound for the throughput scaling factor as obtained in Appendix 3.8.4 is also applicable for the present case, hence R(β) ≤ 1. U →∞ Nt log log U lim

(3.50)

Next, the above upper bond is shown to be achievable as follows. By replacing the

88

(m)

index set In

(m)

in (3.22) with its subset Tn

R

(β)



E

(N t X n=1

(a)



E

(N t X

defined in (3.43)

"

#)

log 1 + γ max ρu (1 − ²u ) (n)

u∈Tm

"

#)

log 1 + γ(1 − dmin ) max ρu (n)

u∈Tm

n=1

(N t X

#

"



¶ 1 ≥ E ≥ −1 1− , U ≥ Uo log 1 + γ(1 − dmin ) max ρu | (n) U u∈Tm n=1 ¶ n h io µ (c) 1 t −1 ≥ Nt E log 1 + γ(1 − dmin ) log(dN U − 1) + γ(1 − d )O(log log U ) 1 − × min min U · µ ¶¸Nt 1 1−O , U ≥ Uo . log U (b)

|Tm(n) |

(m)

The inequality (a) follows from the definition of Tn

t −1 dN min U

in (3.43). The inequality (b) follows

from (3.45). The inequality (c) is obtained by using (3.33). It follows from (c) that R(β) ≥ 1. U →∞ Nt log log U lim

Combining (3.50) and (3.51) gives the desired throughput scaling law.

89

(3.51)

Chapter 4

SDMA with Limited Feedback: Feedback Scheduling For a MIMO downlink, multiuser CSI feedback enables joint SDMA and scheduling for supporting multiple data streams to scheduled users and achieving multiuser diversity gain. Unfortunately, in the case where all users share a common feedback channel, the sum feedback rate can rapidly become a bottleneck for the MIMO downlink with a large number users. This chapter addresses the following questions: How to design a SDMA downlink with a bounded sum feedback rate? Does this sum feedback rate constraint significantly affect the system performance?

4.1

Prior Work

The sum feedback rate of a downlink system can be reduced by applying a feedback threshold, where users below the threshold do not send back CSI. This feedback reduction algorithm was first proposed in [91] for a downlink system with single-input-single-output (SISO) channels, where only users meeting a signal-to-noise-ratio (SNR) threshold are allowed to send back SNR information for scheduling. The algorithm proposed in this chapter is shown to reduce the sum feedback rate significantly. Nevertheless, the sum 90

feedback rate for this algorithm increases linearly with the number of users even though their ratio, namely the normalized sum feedback rate, is constrained. To further reduce the sum feedback rate, the feedback reduction algorithm in [91] is modified in [92] to have an adaptive threshold. The drawback of this modified algorithm is the feedback delay due to multiple rounds of feedback and also the additional feedback cost incurred by this process. In [93, 94], for both SISO and multiple-input-single-output (MISO) channels, combining a feedback threshold and one bit feedback per user is shown to achieve the optimal growth rate of the sum capacity with the number of users. A common problem shared by these feedback reduction algorithms is that the sum feedback rate increases linearly with the number of users, placing a burden on the uplink channel if the number of users is large. To constrain the sum feedback rate, an approach combining a feedback threshold and contention feedback is proposed in [66] for SISO channels, where feedback users contend for the use of a common feedback channel. By extending this approach to MIMO channels, a SDMA algorithm is proposed in [95], which nevertheless has limitations for practical implementation. First, the number of simultaneous users supported by space division is limited by the number of receive antennas for each user, which is usually very small. Second, every user must inefficiently perform zero-forcing equalization even though only a small subset of users is scheduled for transmission. These limitations motivate us to consider a more practical downlink system. In the literature of SDMA with transmit beamforming, the sum feedback rate constraint has not been considered as most work focuses on feedback reduction for individual users. For the opportunistic SDMA (OSDMA) algorithm proposed in [28], the feedback of each user is reduced to a few bits by constraining the choice of a beamforming vector to a set of orthogonal vectors. The sum capacity of OSDMA can be increased by selecting orthogonal beamforming vectors from multiple sets of orthogonal vectors, which motivates the OSDMA with beam selection (OSDMA-BS) [79] and the OSDMA with limited feedback (OSDMA-LF) [96] algorithms1 . These two algorithms assign beamforming vectors 1

Limited feedback refers to quantization and feedback of CSI [8, 9]

91

at mobiles and the base station, respectively. Existing SDMA algorithms share the drawback of having a sum feedback rate that increases linearly with the number of users. This motivates us to apply a sum feedback rate constraint on SDMA. In this chapter, the topic of feedback protocol design is not treated but briefly discussed in Section 4.5.2 since our focus is on developing algorithms for constraining the average sum feedback rate. We note that this is a topic of current research (see e.g. [66, 95]).

4.2

Contributions

We propose an multiuser CSI feedback algorithm for upper bounding the average sum feedback and hence satisfying the sum feedback rate constraint. This constraint is enforced by using two feedback thresholds for selecting feedback users, which gives the name of the algorithm: OSDMA with threshold feedback (OSDMA-TF). First, a feedback threshold on all the users’ channel power selects users with large channel gains for feedback. Second, a threshold on all the users’ channel quantization errors prevents CSI quantization from stopping the growth of the sum capacity with the number of users [24,96]. The key differences between OSDMA-TF and existing algorithms are summarized as follows. Contrary to the sum feedback reduction algorithms for SISO channels [91–94,97] and other OSDMA algorithms with finite-rate feedback [28,79,96], OSDMA-TF satisfies the sum feedback rate constraint. Among downlink algorithms enforcing this constraint, OSDMA-TF has the advantage of supporting simultaneous users compared with the SISO contention feedback algorithm in [66] and the advantage of having simple receivers for subscribers compared with the MIMO contention feedback algorithm in [95]. The main contributions of this chapter are the OSDMA-TF algorithm, the design of feedback thresholds for enforcing the sum feedback rate constraint, and the analysis of the impact of this constraint on the sum capacity. First, the OSDMA-TF sub-algorithms for CSI quantization at users, selection of feedback users using thresholds and joint beam92

forming and scheduling at a base station are proposed. Second, the feedback thresholds on users’ channel power and channel quantization errors are designed such that the sum feedback rate constraint is satisfied. Third, from an upper-bound, the feedback overflow probability is found to decrease approximately exponentially with the difference between the allowable and the average sum feedback rates. Fourth, it is shown that the growth rate of the sum capacity with the number of users can be made arbitrarily close to the optimal one by having a sufficiently large sum feedback rate. Last, OSDMA-TF is compared with several existing SDMA algorithms and is found to be capable of achieving higher sum capacities despite the sum feedback rate constraint. The main conclusion of this chapter is that the proposed SDMA algorithm allows a sum feedback rate constraint to be applied on a SDMA downlink without causing any appreciable negative impact.

4.3

System Model

The downlink system illustrated in Fig. 4.1 is described as follows. A base station with Nt antennas transmits data simultaneously to Nt scheduled users chosen from a total of U users, each with one receive antenna. The base station separates the multi-user data streams by using beamforming, i.e. assigning a beamforming vector to each of the t Nt scheduled users. The beamforming vectors {wn }N n=1 are selected from multiple sets

of unitary orthogonal vectors following the beam and scheduling algorithm described in Section 4.4.3. The received signal of the nth scheduled user is expressed as

yn =

Nt √ X P h†n wi xi + νn ,

n = 1, · · · , Nt ,

i=1

where we use the following notation Nt number of transmit antennas and also number of scheduled users; hn (Nt × 1 vector) downlink channel; P transmit power; 93

(4.1)

Subscribers

Base Station

Beamforming

Scheduling

DL Channels

CSI Quantization Shared Feedback Channel

Feedback Thresholds

Figure 4.1: SDMA Downlink system with feedback thresholds wn (Nt × 1 vector) beamforming vector with kwn k2 = 1; xn transmitted symbol with |xn | = 1; yn received symbol; and νn AWGN sample with νn ∈ CN (0, 1). We assume that each user quantizes his/her CSI and sends it back following a feedback algorithm to be discussed in Section 4.4.2. Furthermore, all users share a common feedback channel. Therefore, it is necessary to constrain the average sum feedback rate. Let B denote the number of bits sent back by each feedback user and K the number of feedback users. It follows that the instantaneous sum feedback rate is BK. Since B is a constant and K a random variable, the constraint of the average sum feedback rate can be written as (Sum Feedback Rate Constraint) BE[K] ≤ R,

(4.2)

where R is the sum feedback rate constraint. To simplify the analysis of the proposed algorithms in Section 4.4, we make the following assumption about the multi-user channels: Assumption 4

The downlink channel hu is an i.i.d.

CN (0, 1).

94

vector whose coefficients are

Given this assumption, which is commonly made in the literature of SDMA and multiuser diversity [28,76,79,83,84], the channel direction vector of each user follows a uniform distribution, which greatly simplifies the design of feedback thresholds in Section 4.5.1 and capacity analysis in Section 4.5.1.

4.4

Algorithms

OSDMA-TF is comprised of (i) CSI quantization at the subscribers, (ii) selection of feedback users using feedback thresholds, and (iii) joint beamforming and scheduling at the base station. The algorithms in (i) and (iii) are essentially identical to those proposed in Chapter 3 with simple modifications to accommodate the new feedback selection algorithm in (ii). Form completeness, all algorithms are discussed as follows.

4.4.1

CSI Quantization

Without loss of generality, the discussion in this section is focused on the uth user and the same algorithm for CSI quantization is used by other users. For simplicity, we assume: Assumption 5 The uth user has perfect receive CSI hu . This assumption allows us to neglect channel estimation error at the uth mobile. For convenience, the CSI, hu , is decomposed into two components: the gain and the shape, which are quantized separately. Hence, hu = gu su where gu = khu k is the gain and su = hu /khu k is the shape. The channel shape su is quantized and sent back to the base station for choosing beamforming vectors. The channel gain gu is used for computing SINR, which is also quantized and sent back as a channel quality indicator. Due to the ease of quantizing SINR that is a scalar, we make the following assumption: Assumption 6 The SINR is perfectly known to the base station through feedback. The same assumption is made in [24, 28, 76, 79]. As shown by numerical results in Section 4.7.2, quantized feedback of SINR does not affect sum capacity significantly. The 95

assumption AS 6 allows us to focus our discussion on quantization of the channel shape. Quantization of the channel shape su is the process of matching it to a member of a set of pre-determined vectors, called a codebook. Different from [76, 96] where code vectors are randomly generated, we propose a structured codebook constructed as follows. The codebook, denoted as F, is comprised of M sub-codebooks: F = ∪M m=1 Fm , each of which is comprised of Nt orthogonal vectors. The sub-codebooks F1 , F2 , · · · , FM are independently and randomly generated for example using the method in [85]. Through the joint beamforming and scheduling algorithm to be discussed in Section 4.4.3, one of the sub-codebooks is chosen by the base station as the set of beamforming vectors for downlink transmission such that sum capacity is maximized. Given a codebook F thus generated, the quantized channel shape, denoted as ˆ su , is the member of F that forms the smallest angle with the channel shape su [10]. Mathematically, ¯ ¯ ¯ ¯ ˆ su = Q(su ) = arg max ¯f † su ¯ , f ∈F

(4.3)

where Q represents the CSI quantization function. We define the quantization error as (Quantization Error) δu = sin2 (∠(ˆ su , su )).

(4.4)

It is clear that the quantization error is zero if ˆ su = su .

4.4.2

Feedback Algorithm

To satisfy the sum feedback rate constraint (4.2), we propose a threshold-based feedback algorithm, which allows only users with good channels (high SINRs) to send back their CSI to the base station. The SINR for the uth user is a function of the channel power ρu = khu k2 and the quantization error δu in (4.4) (see also [96]): SINRu =

1 + P ρu − 1. 1 + P ρu δu 96

(4.5)

Therefore, the feedback algorithm employs two feedback thresholds for feedback user selection: the channel power threshold, denoted as γ, and the quantization error threshold, denoted as ². It follows that the uth user meets the feedback criteria if ρu ≥ γ and δu ≤ ². The thresholds γ and ² are designed in Section 4.5.1 such that the sum feedback rate constraint in (4.2) is satisfied. Given that the uth user meets the feedback thresholds, the quantized channel shape ˆ su is sent back to the base station through a finite-rate feedback channel [8, 10]2 . Since the quantization codebook F can be known a priori to both the base station and mobiles, only the index of ˆ su needs to be sent back. Therefore, the number of feedback bits per user is log2 N since |F| = N .

4.4.3

Joint Beamforming and Scheduling

Among feedback users, the base station schedules a subset of users for downlink transmission using the criterion of maximizing sum capacity under the constraint of orthogonal beamforming. To facilitate the discussion of joint beamforming and scheduling, we group feedback users according to their quantized channel shapes by defining the following index sets: Im,n= {1 ≤ u ≤ U| ρu ≥ γ, δu ≤ ², Q(su ) = fm,n }, where

1 ≤ m ≤ M,

(4.6)

1 ≤ n ≤ Nt , and Q(·) is the quantization function in (4.3)

and fm,n ∈ F is the nth member in the mth sub-codebook Fm ⊂ F. The base-station adopts a two-step procedure for joint beamforming and scheduling. First, the user with maximum SINR is selected from each index set defined in (4.6). Therefore, for the index set Im,n , the selected user has a SINR equal to maxu∈Im,n SINRu and is associated with the code vector fm,n . Second, from these selected users, the base station schedules up to Nt users for downlink transmission under the constraint of orthogonal beamforming. Due to this constraint, only users associated with code vectors from the same sub-codebook (cf. 2

The feedback of SINR is ignored due to AS 6.

97

Section 4.4.1) are allowed for simultaneous transmission, which divides the selected users into M sub-groups. Among these sub-groups, the one giving the maximum sum capacity is scheduled for downlink transmission. It follows that the resultant sum capacity can be written as

" C=E

max

m=1,··· ,M

Nt X

# log2 (1 + max SINRu ) , u∈Im,n

n=1

(4.7)

where the two “max” operators correspond to the two steps in the procedure for joint beamforming and scheduling. In the event that Im,n is empty, we set maxu∈Im,n SINRu = 0 in (4.7).

4.5

Feedback Design

In Section 4.5.1, the feedback thresholds for OSDMA-TF (cf. Section 4.4.2) are designed as functions of the number of users under the sum feedback constraint in (4.2). Even if this constraint is satisfied, it is likely that the instantaneous sum feedback rate exceeds the maximum allowable feedback rate of the feedback channel and hence causes an overflow. In Section 4.5.2, an upper-bound for the feedback overflow probability is derived, which is useful for designing the maximum feedback rate for the feedback channel.

4.5.1

Feedback Thresholds

The feedback probability of each user is derived as a function of the feedback thresholds. Subsequently, since the sum feedback rate is proportional to this probability, we can thus derive the feedback thresholds for a given sum feedback rate. The feedback probability of a user is defined as the probability that the user’s channel power and quantization error meet the respective thresholds. We focus on the feedback probability of a single user since the channels of different users are i.i.d. given AS 4 and hence the feedback probabilities of different users are identical. For simplicity, we omit the user index, hence the subscript u, for all notation in this section. Given AS 4, the channel power ρ = khk2 and the channel direction s = h/khk are independent. It 98

follows that the two events, namely the channel power and quantization error thresholds are met, are independent. Therefore, we can derive their probabilities separately. First, the probability that the channel power ρ of a user meets the power threshold is obtained as

Z Pγ = Pr{ρ ≥ γ} =



γ

fρ (ρ)dρ,

(4.8)

where fρ (ρ) is the chi-squared PDF function given as fρ =

ρNt −1 e−ρ . (Nt − 1)!

(4.9)

Second, the probability for meeting the quantization error threshold, denotes as P² , is obtained. To this end, we define a set for each member of the codebook F as Vn = {kvk = 1 | 1 − |vH fn |2 ≤ ²},

n = 1, 2, · · · , N.

(4.10)

Intuitively, the set Vn can be viewed as a “cone” with the radius ² and the axis fn . The probability, P² , can be defined in terms of these sets as P² = Pr{s ∈ ∪n Vn }.

(4.11)

By applying the union bound and using the symmetry of different users’ channels,

P² ≤

N X

Pr{s ∈ Vn } = N Pr{s ∈ Vn }.

(4.12)

n=1

By denoting 1 − |sH fn |2 as δ and from (4.10), P² ≤ N Pr{δ ≤ ²} = N ²Nt −1 ,

99

(4.13)

where we use the following result from [76], Pr{δ ≤ ²} = ²Nt −1 .

(4.14)

From (4.8) and (4.13), the feedback probability for each user is given in the following lemma. Lemma 10 The feedback probability for each user is given as Z Nt −1

Pv = Pγ P² ≤ N ²

γ



fρ (ρ)dρ.

(4.15)

The sum feedback rate, denoted as R, can be expressed as R = E[K]B where E[K] denotes the average number of feedback users and B the number of bits sent back by each of them. Furthermore, E[K] can be written as

E[K] = U Pv ,

(4.16)

with Pv given in (4.15). With B fixed, the sum feedback rate is proportional to E[K]. We derive a set of feedback thresholds such that E[K] is limited by an upper-bound, which is independent of the number of users U . By choosing a proper value for the upper-bound, we can thus satisfy any given constraint on the sum feedback rate R. The feedback thresholds γ and ² are designed for realizing the sum feedback rate constraint while retaining the multiuser diversity gain. To achieve this gain, the channel power threshold γ must select users whose channel power is at least O(log U ) so that sum capacity increases double logarithmically with U [28]. To this end, we set γ = log U − λ log log U where λ > 0. Next, we design the quantization error threshold ² such that the average sum feedback rate is bounded by a constant that is chosen to be N Nt . Thus from (4.16) and Lemma 10 Z Nt −1



E[K] ≤ U N ²

γ

100

fρ (ρ)dρ = N Nt .

(4.17)

The design of ² involves the Alzer’s bound written as the following lemma [98]. Lemma 11 (Alzer’s Inequality) The incomplete Gamma function is bounded as h iNt Z 1 − e−βγ
0, h i−1/(Nt −1) ² = U 1−ϕ (log U )ϕλ ,

where 1 ϕ = − ln γ

µ

1 Nt !

Z



Nt −1 −ρ

ρ

e

¶ dρ .

(4.21) (4.22)

(4.23)

γ

Given these thresholds, the average number of feedback users E[K] is upper-bounded as

E[K] ≤ N Nt , 101

(4.24)

where N is the cardinality of the CSI quantization codebook F. A few remarks are in order: • Given the feedback thresholds in Theorem 1, the sum feedback rate is bounded as R ≤ BN Nt where B = Bs + log2 N is the number of feedback bits per user with Bs is the number of bits for quantizing the SINR feedback3 (cf. Section 4.4). • The power and quantization thresholds in (4.21) and (4.22) are chosen jointly to ensure the capacity of each scheduled user grows with the number of users U at an optimal rate, namely log2 log2 U [28]. Detailed analysis is given in Section 4.6. • The parameter λ in (4.21) and (4.22) affects the signal-to-interference ratio (SIR) of feedback users. Its optimal value for maximizing sum capacity can be chosen via numerical methods since analytical methods seem difficult. • CSI quantization error causes interference between simultaneous users and can potentially prevent the sum capacity from increasing with the number of users as observed in [75, 76]. This motivates the design of the quantization error threshold in (4.22). This threshold ensures that the quantization error of each feedback user converges to zero with the number of users U . This result is proved shortly. By proving the following corollary of Theorem 1, we show that the quantization errors of feedback users diminish with the number of users U . Corollary 4 The quantization error threshold ² in (4.22) ensures the quantization error of a feedback user, δ, converges to zero with the number of users U

lim δ ≤ lim ² = 0.

U →∞

U →∞

Proof: See Appendix 4.9.1. 3

(4.25)

¤

Usually, Bs 0, there exists an integer U0 such that ∀ U ≥ U0 , the average number of feedback users E[K] is given as E[K] = N Nt . Proof: See Appendix 4.9.2.

(4.27)

¤

For illustration of Theorem 1 and Proposition 9, the numbers of feedback users E[K] averaged over different channel realizations and randomly generated codebooks are plotted against different numbers of users U in Fig. 4.2. First, E[K] is observed to be upper-bounded by N Nt , which agrees with Theorem 1. Second, E[K] converges to N Nt with U , which verifies Proposition 9.

4.5.2

Overflow Probability for Feedback Channel

The overflow probability is defined as the probability that the instantaneous sum feedback rate exceeds the average sum feedback rate. A small overflow probability reduces the average waiting time of feedback users and improves the system stability [99]. In this section, we show that an arbitrarily small overflow probability can be maintained by making the maximum allowable feedback rate sufficiently large relative to the average sum feedback rate. Consistent with the literature of multiuser quantized feedback [24, 28, 76, 79], we ignore the design of a multiuser feedback protocol as it is outside the scope of this chapter. 103

35

Average Number of Feedback Users

NNt = 32 30

N = 8, Nt = 4

25

20

NNt = 16

15

10

N = 6, Nt = 3 NN = 8 t

N = 4, N = 2

5

0 1 10

t

2

3

10

10

Total Number of Users

Figure 4.2: Average numbers of feedback users for OSDMA-TF We note here that this is a topic of current research. For example, the multiuser feedback channel satisfying the sum feedback rate constraint can be implemented using a contention feedback protocol as in [66, 95], which allows feedback users to compete for uplink transmission. Alternatively, to avoid transmission collisions, the multiuser feedback channel can be designed by combining the carrier sensing protocol with a multiple-access scheme such as orthogonal frequency division multiple access (OFDMA), time division multiple access (TDMA), or code division multiple access (CDMA) [100]. For simplicity and due to their equivalence, we measure the instantaneous, average and maximum allowable sum feedback rate using the instantaneous, average and maximum allowable numbers of feedback users, denoted as K, E[K] and Kmax , respectively. The overflow probability can be upper-bounded using the Chernoff bound [101] as follows. For each user, we define a Bernoulli random variable Tu indicating whether the user meets the feedback thresholds

Tu = 1{δu ≤ ² and ρn ≥ γ},

u = 1, 2, · · · , U.

(4.28)

The instantaneous number of feedback user, K, can be expressed as the sum of these 104

Bernoulli random variables, hence K =

PU

u=1 Tu .

Using the Chernoff bound for the

summation of i.i.d. Bernoulli random variables derived in [101], we can obtain an upperbound for the overflow probability. Proposition 10 The overflow probability of the feedback channel is upper-bounded as ¶ · µ Kmax Pr(K ≥ Kmax ) ≤ exp −Kmax log − E[K] µ ¶¸ U −Kmax (U −Kmax) log , U −E[K]

(4.29)

where Kmax is the maximum number of feedback users supported by the feedback channel and E[K] ≤ N Nt . The upper-bound obtained above is useful for determining the maximum data rate the feedback channel should support such that a constraint on the overflow probability is satisfied since the feedback data rate is proportional to the number of feedback users. In Fig. 4.3, the upper bound in (4.29) is compared with the actual overflow probability obtained by Monte Carlo simulation. It can be observed that both the overflow probability and its upper-bound decreases at the same slope and approximately exponentially with the difference (Kmax − E[K]).

4.6

Analysis of Sum Capacity

In this section, for a large number of users (U → ∞), we show that the sum capacity of OSDMA-TF can grow at a rate close to the optimal one, namely Nt log2 log2 N , if the sum feedback rate is sufficiently large. Before proving the main result, several useful lemmas are presented. As we know, the high sum capacity of SDMA is due to its ability of supporting up to Nt simultaneous users. The first lemma concerns the probability for the impossibility of scheduling Nt users. This probability is name probability of scheduled user shortage and denoted as Pβ . 105

0

Feedback Overflow Probability

10

Upper Bound

−1

10

−2

10

−3

10

8

13

18

23

8

13

Kmax − E[K]

(a) Nt = 2, N = 4

0

Feedback Overflow Probability

10

−1

10

Upper Bound −2

10

−3

10

−4

10

0

2

4

6

8

10

Kmax − E[K]

(b) Nt = 2, N = 2

Figure 4.3: Feedback channel overflow probability for OSDMA-TF

106

Using the index sets defined in (4.6), we can express Pβ as ( Pβ = Pr

max

1≤m≤M

Nt X

) 1{Im,n 6= ∅} < Nt

.

(4.30)

n=1

It can be upper-bounded as shown in Lemma 12. Lemma 12 The probability of scheduled user shortage is upper-bounded as Pβ < (Nt e−Nt )M .

(4.31)

Proof: See Appendix 4.9.3.

¤

As to be shown later, the probability Pβ characterizes the decrease of the asymptotic growth rate of the sum capacity caused by scheduled user shortage or equivalently the sum feedback rate constraint. From (4.5), the multi-user interference encountered by a scheduled user with channel power ρ and channel quantization error δ is P ρδ. Lemma 13 shows that the average of the multi-user interference converges to zero with the number of user U . Lemma 13 Let ρ and δ denote the channel power and quantization error of a feedback user. We have lim E[ρδ | ρ ≥ γ, δ ≤ ²] = 0,

U →∞

if λ ≥ Nt − 1,

(4.32)

where λ is the parameter of the power threshold in (4.21). Proof: See Appendix 4.9.4.

¤

The main result of this section is the following theorem. Theorem 2 For a large number of users (U → ∞), the sum capacity of OSDMA-TF grows with the number of transmit antennas Nt linearly and with the number of users double logarithmically

1 ≥ lim

U →∞

C > 1 − (Nt e−Nt )M , Nt log2 log2 U 107

if λ ≥ Nt − 1,

(4.33)

where λ is the parameter of the power threshold in (4.21). Proof: See Appendix 4.9.5.

¤

The above theorem shows the effect of a sum feedback rate constraint is to decrease the growth rate of the sum capacity with respect to that for feedback from all users, namely Nt log2 log2 U [28,96]. Nevertheless, such difference in growth rate can be made arbitrarily small by increasing the sum feedback rate, or equivalently the number of feedback bits per feedback user, as stated in the following corollary. Corollary 5 By increasing the number of feedback bits per feedback user (log2 N ) such that N = o(U ), namely lim N/U = 0, the sum capacity of OSDMA-TF grows at the U →∞

optimal rate: lim

lim

U →∞ N →∞ N =o(U )

C Nt log2 log2 U

= 1,

if λ ≥ Nt − 1.

(4.34)

Proof: The result follows from (4.33) and N = M Nt .

¤

The condition N = o(U ) is important for ensuring the number of users selecting each code vector in the codebook F with |F| = N is O(U ) so as to retain the multiuser diversity gain. Let Pβ = (Nt e−Nt )M . We can observe from Fig. 4.4 that the asymptotic growth rate for the sum capacity for OSDMA-TF converges to the optimal value, hence 1−Pβ → 1 from (4.33), very rapidly as the number of feedback bits per user (log2 N ) increases.

4.7

Numerical Results

In this section, we present numerical results for evaluating the performance of OSDMA-TF and validating the assumption AS 6.

4.7.1

Performance Comparison

The sum capacity and the sum feedback rate of OSDMA-TF are compared with the case of all-user feedback, which is equivalent to OSDMA-TF with trivial feedback thresholds 108

1.1

1

1 − Pβ

0.9

0.8

Nt = 3 Nt = 2 or 4

0.7

0.6

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Number of Feedback Bits per User

Figure 4.4: Convergence of the lower bound in (4.33), 1 − Pβ , to one with the number of feedback bits per user log2 N . γ = 0 and ² = 1. Similar comparisons are also conducted between OSDMA-TF and existing algorithms including OSDMA-LF [96], OSDMA-BS [79] and OSDMA [28]. The sum capacities of OSDMA-TF and the corresponding case of all-user feedback are plotted against the number of users U in Fig. 4.5(a) for the cases of Nt = {2, 4} transmit antennas. For these two cases, the parameter λ for the power threshold γ in (4.21) is assigned the values of 1 and 1.5 respectively, which are found numerically to be sum capacity maximizing. Each user quantizes his/her channel shape using a codebook of size N = 8 and hence each feedback user sends back log2 N = 3 bits. It can be observed from Fig. 4.5(a) that the sum feedback rate constraint for SDMA-TF incurs negligible loss in sum capacity with respect to the case of all-user feedback. Next, the number of feedback users for OSDMA-TF and all-user feedback are compared in Fig. 4.5(b). Note that the sum feedback rate R is proportional to the number of feedback users E[K]: R = 3E[K] bits. It can be observed that the number of feedback users for OSDMA-TF is upper bounded by 32 for Nt = 4 and 16 for Nt = 2 since OSDMA-TF is designed for satisfying a sum feedback constraint (cf. Section 4.5.1). In summary, OSDMA-TF achieves almost 109

identical sum capacity as the case of all user feedback but with a dramatic reduction on sum feedback rate for a large number of users U . In Fig. 4.6, the sum capacity and sum feedback rate of OSDMA-TF is compared with those of OSDMA-LF, OSDMA-BS and OSDMA for different numbers of users U , with Nt = 2 and an SNR of 5 dB. The number of feedback bits per feedback user differs for the algorithms in comparison since they use different sizes for quantization codebooks or different feedback algorithms. For OSDMA-TF, two codebook sizes N = 8 and N = 24 are considered, corresponding to 3 and 4.6 feedback bits for each feedback user, respectively4 . For OSDMA-LF, the codebook size N increases with U as: N = 5d(log2 U )Nt −1 e to avoid saturation of sum capacity due to limited feedback [96]. The codebook sizes for OSDMABS and OSDMA are both N = 2. Different from other algorithms, the CSI feedback for OSDMA-BS is performed iteratively, where each iteration penalizes the sum capacity by a factor of 0 ≤ α ≤ 1 [79]. Let I denote the number of feedback iterations. Therefore, for OSDMA-BS, the total feedback for each user is I bits and the sum capacity with feedback penalty is given as Cp = (1 − Iα)C. For fair comparison, we also apply this feedback penalty to the other algorithms5 . From Fig. 4.6(a), we can observe that OSDMA-TF (N = 24) yields the highest sum capacity and OSDMA-TF (N = 8) is outperformed only by OSDMA-LF. Moreover, the sum capacity of OSDMA-TF converges to DPC rapidly, where the DPC capacity is computed following [22] and for the same sets of scheduled users as for OSDMA-TF. From Fig. 4.6(b), the sum feedback rates for OSDMA-TF is observed to grow much more gradually with the number of users U than those for other algorithms. Asymptotically, the sum feedback rates for OSDMA-TF saturate due to sum feedback rate constraint (cf. Theorem 1) while those for other algorithms continue to increase with U . Several other observations can be made from Fig. 4.6(b). First, for U ≥ 25, OSDMA-TF with N = 8 has the smallest sum feedback rate among all algorithms but it outperforms OSDMA-BS 4 5

¯ is a function of N (cf. Section 4.5.1). The average number of feedback users K I = 1 for OSDMA-TF, OSDMA-LF and OSDMA

110

12

SNR = 5 dB N=8

Sum Capacity, C (bits/s/Hz)

11

N =4 t

Overlapping Curves for Threshold and All−User Feedback

10 9 8

N =2 t

7 6 5 4 1 10

2

3

10

10

Total Number of Users, U

(a) Sum Capacity

3

10

Number of Feedback Users

FB Bits/User = B+ 3 bits B: SINR Feedback

All Feedback

2

10

32 Th. Feedback (N = 4) t

16 1

10

Th. Feedback (N = 2) t

1

10

2

10

3

10

Total Number of Users

(b) Sum Feedback Rate

Figure 4.5: Comparison between threshold feedback (OSDMA-TF) and all-user feedback: (a) sum capacity versus total number of users and (b) average number of feedback users versus total number of users for SNR = 5 dB, the size of the shape quantization codebook N = 8, and different numbers of antennas Nt = {2, 4}.

111

7.5

SNR = 5 dB N =2 t

Capacity, C (bps/Hz)

7

DPC

α = 0.05

6.5

6

DPC OSDMA−TF OSDMA−LF OSDMA−TF OSDMA−BS OSDMA

5.5

5

4.5 10

20

30

40

50

60

(FB = 4.6 bit/user) (4.3 − 5.1 bit/user) (3 bit/user) (2 − 4 bit/user) (1 bit)

70

80

90

100

Number of Users, U

(a) Sum Capacity

500

Average Sum Feedback (bits)

450 400 350

OSDMA−LF OSDMA−BS OSDMA−SF (4.3 bits/user) OSDMA OSDMA−SF (3 bits/user) I=3

300 I=2 250

I=1

200 150 100 50 0 10

100

500

Number of Users, U

(b) Sum Feedback Rate

Figure 4.6: Comparison between OSDMA-TF, OSDMA-LF, OSDMA-BS and OSDMA: (a) sum capacity versus total number of users and (b) average sum feedback rate versus total number of users for SNR = 5 dB, the number of antennas Nt = 2 and the feedback penalty factor α = 0.05.

112

and OSDMA in terms of sum capacity. Second, for U ≥ 75, OSDMA-TF with N = 24 yields the highest sum capacity among all algorithms and also requires the smallest sum feedback rate. Third, the sum feedback rate for OSDMA-LF is the highest among all algorithms. Fourth, the drop of sum feedback rate for OSDMA-BS at U = 90 is due to the decrease of the optimal number of feedback iterations from I = 3 to I = 2, which are obtained in [79]. Last, for U ≥ 200, OSDMA-BS with I = 1 is identical to OSDMA and hence they have equal sum feedback rates.

4.7.2

Effect of SINR Quantization

Using numerical results, we will show that SINR quantization causes only very small reduction on sum capacity. Thereby, the assumption AS 6 is justified. The codebook, denoted as G, for quantizing the SINR value sent by a feedback user to the base station is constructed using a heuristic approach. Therefore, the performance of this codebook serves as a benchmark for that of an optimal codebook, which is outside the scope of our discussion. Let ηˆ1 , ηˆ2 , · · · , ηˆD ∈ R+ denote the available quantized SINR values in ascending order, hence G = [ˆ η1 , ηˆ2 , · · · , ηˆD ]. These codebook values are constrained to be equally spaced in the range [0, γ0 ] with a probability FSINR (γ0 ) close to 1, where FSINR (·) is the SINR CDF of a feedback user. Therefore, the codebook values are given as ηˆi =

i D γ0

for i = 1, · · · , D. For constructing the codebook, we set

FSINR (γ0 ) = 99% and compute γ0 using the Monte Carlo simulation. In Fig. 4.7, the sum capacities for prefect and quantized SINR feedback are compared, where the codebook G described previously is used by all users and the quantized SINR is ηˆ = arg minηˆ∈G (ˆ η − η)2 . Moreover, the number of antennas is Nt = 2, the codebook size for quantizing channel shapes is N = 8 (cf. Section 4.4.1) and the SNR is 5 dB. As observed from Fig. 4.7, compared with perfect SINR feedback, the loss of sum capacity due to SINR quantization is negligible for log2 D = 4 bits of SINR feedback from each user and smaller than 0.1 bps/Hz for 3 bits. In conclusion, quantized SINR feedback yields sum capacity close to that for perfect SINR feedback and hence the assumption AS 6 is 113

7.5

Sum Capacity, C (bps/Hz)

7

Perfect 4 bits 3 bits 2 bits 1 bit

SNR = 5 dB N =2 t

N=8

6.5

6

5.5

5

4.5

4 10

20

30

40

50

60

70

Number of Users, U

Figure 4.7: Capacity comparison between perfect and quantized SINR feedback for SNR = 5 dB, the number of antennas Nt = 2, and the size of the codebook for channel-shape quantization is N = 8. justified.

4.8

Summary

In this chapter, we have proposed a SDMA downlink algorithm with a sum feedback rate constraint, which is applied by using feedback thresholds on users’ channel power and channel quantization errors. We have derived the expressions for these thresholds and the upper bound for the corresponding feedback overflow probability. Furthermore, we have obtained the asymptotic growth rate of the sum capacity with the number of users. We showed that it can be made arbitrarily close to the optimal value by increasing the sum feedback rate. From numerical results, we have found that limiting the sum feedback rate incurs negligible loss on sum capacity. Moreover, we have demonstrated that the proposed SDMA algorithm is capable of outperforming existing algorithms despite having a much smaller sum feedback rate.

114

4.9 4.9.1

Appendix Proof of Corollary 4

From Lemma 11 and (4.23), we have ³

1 − e−βγ

´Nt

¡ ¢N < 1 − Nt e−ϕγ < 1 − e−γ t .

(4.35)

From the definition in (4.21), we observe that γ monotonically increases with U when U is large. Therefore, from (4.21) and (4.35), there exists an integer U0 such that ∀ U ≥ U0 , 1 − Nt e−βγ < 1 − Nt e−ϕγ < 1 − Nt e−γ .

(4.36)

It follows that β < ϕ < 1. Combining this with (4.22), the result in Corollary 4 follows.

4.9.2

Proof of Proposition 9

We observe from (4.16) and (4.15) that E[K] = U Pγ P² ≤ U N Pγ Pr{s ∈ Vn },

(4.37)

where 1 ≤ n ≤ N is arbitrary. We will prove the existence of an integer U0 such that the equality in the above equation holds ∀ U ≥ U0 . Following (4.22), such an integer exists such that ²≤

∆δmin , 2

∀ U ≥ U0 ,

(4.38)

where ∆δmin > 0 is the minimum distance for the codebook F as defined in (4.26). Assume that there exist two overlapping sets Va and Vb , Va ∩ Vb 6= ∅. Let s ∈ Va ∩ Vb . From the triangular inequality and the definition in (4.26), (1 − |sH fa |2 ) + (1 − |sH fb |2 ) ≥ ∆δmin .

(4.39)

On the other hand, from the definition in (4.10), (1 − |sH fa |2 ) + (1 − |sH fb |2 ) ≤ 2². 115

(4.40)

Nevertheless, (4.38) leads to the contradiction between (4.39) and (4.40). Thereby, we prove that given (4.38), Va ∩ Vb = ∅,

∀ U ≥ U0 and 1 ≤ a, b ≤ N.

(4.41)

It follows that Pr{s ∈ ∪n Vn } = N Pr{s ∈ V1 }.

(4.42)

Therefore, K = U N Pγ Pr{s ∈ V1 },

∀ U ≥ U0 .

(4.43)

Substitution of (4.21) and (4.22) into the above equation completes the proof.

4.9.3

Proof of Lemma 12

From (4.30), ( Pβ

=

Pr

M \

(N t X

))

(a)

=

=

M Y m=1 M Y

(b)

m=1 M Y

(c)

m=1 M Y



=

Pr Pr

n=1 (N t X

(4.44)

)

1{Im,n 6= ∅} < Nt

n=1 (N [t

,

1{Im,n 6= ∅} < Nt

m=1

,

) {Im,n = ∅} ,

n=1

(Nt Pr {Im,n = ∅}) , £ ¤ Nt (1 − Pγ Pr{δ ≤ ²})U ,

m=1 (d)

= ≤

£ ¤M Nt (1 − Nt /U )U , £ ¤M Nt e−Nt .

The equality (a) results from the independence of the M events in (4.44) due to the independent generations of the M sub-codebook in the codebook F. The inequality (b) is obtained by applying the union bound as well as using the equal probabilities of the events {Im,n = ∅} for m = 1, 2, · · · , M . The equality (c) follows from the definition of the 116

set Im,n in (4.6). The equality (d) is obtained from (4.8), (4.14), (4.21) and (4.22).

4.9.4

Proof of Lemma 13

Given AS 4, ρ and δ are independent, hence E[ρδ | ρ ≥ γ, δ ≤ ²] = E[ρ | ρ ≥ γ]E[δ | δ ≤ ²]. By definition,

R∞ E[ρ | ρ ≥ γ] =

ρ · ρNt −1 e−ρ dρ Γ(Nt + 1, γ) R∞ = , N −1 −ρ t Γ(Nt , γ) e dρ γ ρ

γ

where Γ(·, ·) denotes the incomplete Gamma function [98]. By expanding Γ(·, ·), we obtain an upper-bound for E[ρ | ρ ≥ γ] as Nt !e−γ

PNt

i i=0 γ /i! , P t −1 i (Nt − 1)!e−γ N i=0 γ /i! Ã ! γ Nt /Nt ! = Nt 1 + PNt −1 , i i=0 γ /i! µ ¶ γ Nt /Nt ! < Nt 1 + Nt −1 , γ /(Nt − 1)!

E[ρ | ρ ≥ γ] =

= Nt + γ.

(4.45)

Next, we obtain the expression of E[δ | δ ≤ ²] as: Z

²

E[δ | δ ≤ ²] =

δfδ (δ | δ < ²)dδ, Z ² = ²−(Nt −1) δ L dδ,

(4.46)

0

(4.47)

0

=

Nt − 1 ². Nt

(4.48)

From (4.45) and (4.48), 0 ≤ E[ρδ] < (Nt − 1)² +

Nt − 1 γ². Nt

(4.49)

From (4.22) and ϕ < 1, lim ² = 0.

U →∞

117

(4.50)

Moreover, from (4.22) and (4.21), γ² = U

ϕ−1 Nt −1

1− Nϕλ −1

(log2 U )

t

µ ¶ log2 log2 U . 1−λ log2 U

(4.51)

If λ ≥ Nt − 1, it follows that lim E[γ²] = 0,

U →∞

if λ ≥ Nt − 1.

(4.52)

Combining (4.49), (4.50) and (4.52) completes the proof.

4.9.5

Proof of Theorem 2

The lower and upper bounds of the asymptotic sum capacity given in (4.33) are proved in subsequent sections.

4.9.6

Lower Bound for Asymptotic Sum Capacity

Define m? = arg max

1≤m≤M

T? =

Nt X

Nt X

1{Im,n 6= ∅}

n=1

1{Im? ,n 6= ∅}.

(4.53)

n=1

From (4.7) C ≥ E

"N t X n=1

= E

"N t X

+E

µ ¶# log2 1 + max SINRu u∈Im? ,n

# µ ¶ log2 1 + max SINRu | T ? = Nt (1 − Pβ )

n=1 "N t X n=1

≥ E

"N t X

n=1

u∈Im? ,n

# µ ¶ ? log2 1 + max SINRu | T < Nt Pβ u∈Im? ,n

# µ ¶ log2 1 + max SINRu | T ? = Nt (1 − Pβ ) u∈Im? ,n

(4.54)

where Pβ is defined in (4.30). In the above expression, the condition T ? = Nt ensures that Im? ,n 6= ∅ ∀ 1 ≤ n ≤ Nt so that there exist Nt simultaneous users with orthogonal 118

beamforming vectors. Under this condition, let un denote an arbitrary member of the index set Im? ,n . From (4.54) C≥E

"N t X

# log2 (1 + SINRun ) | T ? = Nt (1 − Pβ ).

(4.55)

n=1

By substituting (4.5) into (4.55), "N # µ ¶ t X 1 + P ρun ? C≥E log2 | T = Nt (1 − Pβ ). 1 + P ρun δun

(4.56)

n=1

Next, the pairs of random variables {(ρu , δu ) | u ∈

SU

n=1 Im? ,n , T

?

= Nt } are

proved to be i.i.d. in Section 4.9.8. It follows that conditioned on T ? = Nt , the random variables in (4.56), namely {(ρun δun )}, are i.i.d.. To simplify notation, let (ρ, δ) denote a pair of random variables following the common distribution of {(ρun δun )}. From (4.65) in Appendix, the conditional PDF of (ρ, δ) is given as f (ρ, δ | T ? = Nt ) = f (ρ, δ | ρ ≥ γ, δ ≤ ²). Using above results, (4.56) can be simplified as · C

µ

1 + Pρ 1 + P ρδ



¸ | ρ ≥ γ, δ ≤ ²



(1 − Pβ )Nt E log2



(1 − Pβ )Nt {log2 (1 + P γ) − E [log2 (1 + P ρδ) | ρ1 ≥ γ, δ1 ≤ ²]}

(a)



(1 − Pβ )Nt {log2 (γ) + log2 (P ) − log2 (1 + P E[ρδ | ρ ≥ γ, δ ≤ ²])}

(4.57)

where the inequality (a) is obtained by applying the Jensen’s inequality. By substituting (4.21) into (4.57) ¶ · µ λ log2 log2 U + log2 (P )− C ≥ (1 − Pβ )Nt log2 log2 U + log2 1 − log2 U log2 (1 + P E[ρδ | ρ ≥ γ, δ ≤ ²])] . Therefore lim

U →∞

C Nt log2 log2 U

≥ (1 − Pβ )[1 + Π1 + Π2 − Π3 ]

119

(4.58)

where Π1 Π2 Π3

µ ¶ 1 λ log2 log2 U = lim log2 1 − U →∞ log2 log2 U log2 U log2 (P ) = lim U →∞ log2 log2 U 1 log2 (1 + P E[ρδ | ρ ≥ γ, δ ≤ ²]). = lim U →∞ log2 log2 U

(4.59)

The values of Π1 and Π2 are observed to be zeros. From Lemma 10, the value of Π3 is also equal to zero if λ ≥ Nt − 1. Therefore, we obtain from (4.58) that lim

U →∞

C ≥ (1 − Pβ ), Nt log2 log2 U

if λ ≥ Nt − 1.

(4.60)

By applying Lemma 10, we obtain the lower bound of the asymptotic sum capacity in (4.33).

4.9.7

Upper Bound for Asymptotic Sum Capacity

From (4.7) C≤E

"N t X

µ ¶# log2 1 + max SINRu , C+.

n=1

1≤u≤U

(4.61)

The upper bound C + in (4.61) is identical to that for the sum capacity of OSDMA given in [28, (8)]. As shown in [28] C+ = 1. U →∞ Nt log2 log2 U lim

(4.62)

From (4.61) and (4.62), we obtain the upper bound of the asymptotic sum capacity in (4.33). Thereby, we complete the proof of Theorem 2.

120

4.9.8

Additional Proof

In this appendix, the pairs of random variables {(ρu , δu ) | u ∈

SU

n=1 Im? ,n , T

?

= Nt } are

proved to be i.i.d.. To simplify notation, define the set ¯ ( ) U ¯ [ ¯ C := (ρu , δu ) ¯u ∈ Im? ,n , T ? = Nt ¯ n=1 ¯ ( ) U ¯ [ (a) ¯ = (ρu , δu ) ¯u ∈ Im? ,n , |Im? ,n | ≥ 1 ∀ 1 ≤ n ≤ Nt ¯ n=1

where (a) follows from the definition of T ? in (4.53). Consider an index a ∈ Im? ,k . By using Bayes’ rule, the PDF of C defined above can be written as f (C) = f ((ρa , δa ) | a ∈ Im? ,k , |Im? ,k | ≥ 1, C/{(ρa , δa )} ) × f (C/{(ρa , δa )}) (b)

= f ((ρa , δa ) | a ∈ Im? ,k , |Im? ,k | ≥ 1} ) × f (C/{(ρa , δa )})

(c)

= f ((ρa , δa ) |ρa ≥ γ, δa ≤ ² ) × f (C/{(ρa , δa )})

(4.63)

where (b) results from the fact that multiuser channels are independent, and thus (ρa , δa ) are independent of C/{(ρa , δa ). The equality (c) follows from the definition in (4.6). By repeatedly applying Bayes’ rule using (4.63), f (C) =

N Y

Y

f ((ρa , δa ) |ρa ≥ γ, δa ≤ ² ) .

(4.64)

n=1 a∈Im? ,n

Given that multiuser channels are identically distributed (cf. Assumption 4), (ρu , δu ) ∀ u follow the same distribution. Let (ρ, δ) denote a pair of random variables having the same distribution as (ρu , δu ) for an arbitrary index u. Thus, (4.64) can be re-written as f (C) = [f ((ρ, δ) |ρ ≥ γ, δ ≤ ² )]|C| . The desired result follows from the above equation.

121

(4.65)

Chapter 5

SDMA with Limited Feedback: Uplink Throughput Scaling The preceding two chapters focus on downlink SDMA with limited feedback. In this chapter, we consider uplink SDMA with limited feedback. For uplink SDMA, CSI is usually acquired by channel estimation based on uplink pilot symbols. Nevertheless, this method is sensitive to multiuser interference, and inflexible due to fixed pilot-tone locations in a broad-band system. Moreover, channel estimation does not allow CSI protection by coding. These drawbacks motivate the use of limited feedback for uplink SDMA in this chapter. In this chapter, two types of multiuser beamforming, namely zero-forcing and orthogonal beamforming, are considered. The throughput scaling laws for uplink SDMA are derived for different regimes of the SNRs. Moreover, useful guidelines for uplink scheduling for achieving optimal uplink throughput are provided.

5.1

Prior Work

The prior work on throughput scaling laws of SDMA with limited feedback target the downlink [24, 28, 96]. The existing analytical approach is to use the extreme value theory [24, 28], but this approach is not directly applicable for uplink SDMA as explained below. The key to this approach is the derivation of the probability density function (PDF) of the signal-to-interference-noise ratio (SINR). This SINR PDF allows the application of extreme value theory for analyzing the throughput scaling law. The above approach is 122

feasible for downlink SDMA because the SINR of a scheduled user depends only on this user’s CSI [24, 28]. In contrast, for uplink SDMA, this SINR is a function of the CSI of all scheduled users. Such a discrepancy is due to the difference between the downlink and uplink. To be specific, both the signal and interference received by a user (the base station) propagate through the same channel (different channels) in the downlink (uplink). Consequently, the derivation of the SINR PDF for uplink SDMA is complicated because of its dependence of the specific scheduling algorithm. This motivates us to seek new tools for analyzing the throughput scaling laws for uplink SDMA. Two beamforming and scheduling methods, zero-forcing beamforming [24,102] and orthogonal beamforming[16, 28, 96], are being discussed for enabling downlink SDMA with limited feedback in the 3GPP-LTE standard [15, 16]. Due to the uplink-downlink difference mentioned above, the scaling laws for downlink SDMA in [24, 28, 96] can not be directly extended to the uplink counterpart. Furthermore, the scaling law for orthogonal beamforming in the interference-limited regime remains unknown even for downlink SDMA. This motivates us to consider both orthogonal and zero-forcing beamforming in the analysis of uplink SDMA. Furthermore, the throughput scaling analysis covers high SNR (interference limited), normal SNR and low SNR (noise limited) regimes.

5.2

Contributions

To discuss the contributions of this chapter, the system model is summarized as follows. The uplink SDMA system model includes a base station with multi-antennas and users with single-antennas. The multiuser channels are assumed to follow the i.i.d. Rayleigh distribution. The CSI feedback of each user consists of a quantized channel-direction vector and two real scalars, namely the quantization error and the channel power, which can be assumed perfect since they require much less feedback than the vector. Moreover, both orthogonal [28, 96] and zero-forcing beamforming [24, 76], are considered for beamforming at the base station. The main contributions of this chapter are the asymptotic throughput scaling laws 123

for uplink SDMA with limited feedback in different SNR regimes and for both orthogonal and zero-forcing beamforming. The derivation of the throughput scaling laws makes use of new analytical tools including the Vapnik-Chervonenkis theorem [81] and the bins-andballs model [103] for analyzing multiuser limited feedback. Our results are summarized as follows. 1. In the high SNR regime and for orthogonal beamforming, an upper and a lower bound are derived for the throughput scaling factor. These bounds show that the throughput scales logarithmically with both the number of users U and the quantization codebook size N . Furthermore, the linear scaling factor is smaller than the number of antennas Nt , indicating the loss in the spatial multiplexing gain. 2. In the high SNR regime and for zero-forcing beamforming, the exact throughput scaling factor is derived, which provides the same observations as for orthogonal beamforming. To be specific, the throughput scales logarithmically with both U and N . The linear factor of the asymptotic throughput is smaller than Nt . 3. In the normal SNR regime, for both orthogonal and zero-forcing beamforming, the throughput is shown to scale double logarithmically with U and linearly with Nt . 4. The same results are obtained for the lower SNR regime. The analysis of the throughput scaling laws provides the following guidelines for designing uplink SDMA with limited feedback. In the high SNR regime, the scheduling algorithm should select users with minimum quantization errors. Thus, feedback of channel power for scheduling is unnecessary. In the lower SNR regime, the scheduled users should be those with maximum channel power. Consequently, scheduling requires no feedback of quantization errors. In the normal SNR regime, the scheduling criterion should include both channel power and quantization errors. This implies that the feedback of both types of CSI is needed.

124

Downlink Control Channel

User 1

Scheduled User Indices

Uplink Channel

Scheduled User Indices

User 2

Beamforming & Scheduling User U

RF Data Streams Beamforming Vectors

1 2 SDMA Nt

Finite-Rate Feedback Channels Base Station

Figure 5.1: Uplink SDMA system with limited feedback

5.3

System Description

The uplink SDMA system considered in this chapter is illustrated in Fig. 5.1. In this system, U backlogged users each with a single antenna attempt to communicate with a base station with Nt antennas. For each time slot, up to Nt users are scheduled for uplink SDMA transmission. Users learn the scheduling decisions from the indices of scheduled users broadcast by a base station. The base station separates the data packets of scheduled users by receive beamforming. The base station requires the CSI feedback from all users for scheduling and beamforming. Each user sends back CSI using limited feedback as elaborated later. Two approaches for scheduling and beamforming based on limited feedback are analyzed in this chapter, namely orthogonal beamforming [28,96] and zero-forcing beamforming [24, 76], which are discussed respectively in Section 5.4.3 and Section 5.4.3. Assuming the presence of channel reciprocity (hence a time-division multiplexing (TDD) system), each user estimates the downlink channel, equivalently the uplink channel, using pilot symbols periodically broadcast by the base station. For simplicity, we make the following assumption. Assumption 7 Each user has perfect CSI of the corresponding uplink channel. This assumption simplifies analysis by allowing omission of channel estimation errors. Consider a system with a large number of users. Even by exploiting channel reciprocity, 125

the base station can acquire the CSI of only the scheduled uplink users, which is a small subset of users. Nevertheless, the base station requires the CSI of all users for scheduling and beamforming, which motivates the CSI feedback from all users. Each user relies on a finite-rate feedback channel for CSI feedback, thus limited feedback is used for efficiently quantizing CSI for satisfying the finite-rate constraint. The uplink channel of each user is modeled as a frequency-flat block-fading vector channel. By blocking fading, channel realizations for different time slots are independent. Consequently, the uplink channel of the uth user can be represented by a random vector hu . To simplify our analysis, we make the following assumption. Assumption 8 The vector channel of each user, hu where u = 1, 2, · · · , U , is an i.i.d vector with complex Gaussian coefficients CN (0, 1). This assumption is commonly made in the literature of multi-user diversity [28, 76, 79, 84]. For analysis, the channel vector hu is decomposed into channel shape and channel power, defined as su = hu /khu k and ρu = khu k2 , respectively. Based on the above model, the vector of multi-antenna observations at the base station, denoted as y, can be written as y=

X u∈A

p P ρu su xu + ν

(5.1)

where A is the index set of scheduled users, xu is the data symbol of the uth user, and ν is the AWGN vector. Furthermore, the recovered data symbol for the scheduled uth user after beamforming is given as x ˆu = vu† y =

X p P ρu vu† su xu +

m∈A/{u}

p P ρm vu† sm xm + νu ,

(5.2)

where vu is the beamforming vector used for retrieve the data symbol of the uth user.

5.4

Limited Feedback, Scheduling and Beamforming

This section presents the analytical framework for limited feedback, scheduling and beamforming for uplink SDMA. SINR and throughput are important quantities for scheduling 126

at the base station. Their exact values are unknown to the base station because of imperfect CSI feedback. The approximated SINR and throughput, named expected SINR and expected throughput, are discussed in Section 5.4.1 and Section 5.4.2, respectively. These new quantities are computable at the base station using limited feedback. Based on limited feedback, the beamforming vectors of scheduled users are computed at the base station to satisfy the following constraint vu ⊥ ˆsu0

∀ u, u0 ∈ A and u 6= u0

(5.3)

where vu is the beamforming vector, ˆ su the quantized channel-shape and A the index set of scheduled users. This constraint has been also used for downlink SDMA with limited feedback [28, 76, 79, 96]. For perfect feedback (su = ˆ su ), the above constraint ensures no interference between scheduled users. In Section 5.4.3, two beamforming approaches for satisfying (5.3), namely orthogonal beamforming and zero-forcing beamforming, are introduced. In addition, the compatible scheduling methods are also described.

5.4.1

Expected SINR

In this section, the expected SINRs of scheduled users are defined, which are computable using limited feedback. Given the index set of scheduled users A and corresponding beamforming vectors {vu }, as in [24, 76], the SINR is obtained from (5.2) as SINRu =

γρ |v† s |2 Pu u u 1 + γ m∈A ρm ²m βm,u

(5.4)

m6=u

where the signal-to-noise ratio (SNR) γ = P/σν2 , and su and ρu are respectively the channel shape and power of the uth user, ²u = sin2 (∠(su , ˆ su )) is the quantization error of the channel shape. Moreover, βm,u is a Beta random variable that is independent of ²m and has the cumulative density function (CDF) Pr(βm,u ≤ β0 ) = β0Nt −1 . The direct feedback of SINRs in (5.4) by users is infeasible as computation of SINRs requires multi-user CSI and such information is unavailable to individual users. Note that the SINR feedback is feasible for downlink SDMA since the SINR depends only on single127

user CSI [28] or approximately so [24]. Therefore, we require that the expected SINR is computable at the base station using individual users’ CSI feedback. The expected SINR is defined as follows, which is computable from the feedback of channel power {ρu } and channel-shape quantization errors {²u } by users. In addition, the feedback of quantized channel shapes allows the base station to compute beamforming vectors {vu } that satisfy the constraint in (5.3). As feedback of a scalar requires potentially much fewer bits than that of a vector, the following assumption is made throughout this chapter unless specified otherwise. Assumption 9 The feedback of channel power {ρu } and channel-shape quantization errors {²u } from all users are perfect. Depending on the operational SNR regime, either of these two types of scalar feedback can be avoided as shall be discussed later. Given Assumption 9, limited feedback in this chapter focuses on quantization and feedback of channel shapes. Under Assumption 9, the expected SINR for the uth user, denoted as Ψu , is defined as Ψu =

5.4.2

1+γ

γρ P u

m∈A ρm ²m m6=u

.

(5.5)

Expected Throughput

In this section, the expected throughput that approximates the exact one is defined as follows R=E

hX u∈A

i log (1 + Ψu )

(5.6)

where Ψu is defined in (5.5), A is the index set of scheduled users. This quantity is estimated by the base station using limited feedback and for a given set of scheduled users. Next, the expected throughput is shown to converge to the actual one when the number of users is large. Therefore, the expected throughput can replace the actual one in the asymptotic analysis of throughput scaling, which significantly simplifies our analysis. To obtain the desired result, a useful lemma from [76] is provided below. 128

Lemma 14 Let ²(N ) be the minimum of N i.i.d. Beta random variables. The following inequalities hold E [− log(²(N ))] ≤

log N + 1 Nt − 1

E [²(N )] < (N )

− N 1−1 t

.

(5.7) (5.8)

Let ϕu denote the angle between the beamforming vector and quantized channel shape of the uth scheduled user, hence ϕu = ∠(vu , ˆ su ). Using this lemma, the following result on the difference between the expected and the exact throughput is proved. Proposition 11 If ϕu ≤ ϕ0 , ²u ≤ θ0 and (ϕ0 + θ0 ) < π2 , then ½ ¾ Nt |R − C| ≤ max 2 log cos(ϕ0 + θ0 ), [log(Nt − 1) + 1] Nt − 1 where C is the exact throughput given as hX C=E

u∈A

i log (1 + SINRu ) .

(5.9)

(5.10)

The proof is given in Appendix 5.11.1. As shown in subsequent sections, the expected throughput R increases continuously with the number of users U . Consequently, from Proposition 11, the expected throughput R has the same asymptotic scaling factor as the exact throughput in (5.10).

5.4.3

Beamforming Methods

The orthogonal and zero-forcing beamforming methods are commonly used in the literature of downlink SDMA with limited feedback [24, 28, 76, 96, 102]. These methods are adopted in this chapter for uplink SDMA as elaborated in Section 5.4.3 and Section 5.4.3, respectively. The main difference between orthogonal and zero-forcing beamforming lies in their use of the quantizer codebook. For orthogonal beamforming, the codebook of unitary vectors provides potential beamforming vectors. In other words, quantized CSI of scheduled users directly provides their beamforming vectors. For zero-forcing beamforming, the 129

codebook is used in the traditional way as in vector quantization. Beamforming vectors are computed from quantized CSI using the zero-forcing method. Orthogonal Beamforming In this section, orthogonal beamforming for downlink SDMA with limited feedback is discussed. The orthogonal beamforming method is characterized by the following constraint [28, 96] (orthogonal beamforming)

  ˆ su ⊥ ˆ su0

∀ u, u0 ∈ A and u 6= u0

  vu = ˆ su

∀u∈A

(5.11)

The above constraint can be implemented using the following joint design of limited feedback, beamforming and scheduling (see e.g. [96]). First, the channel shape of each user is quantized using a codebook that is comprised of multiple orthonormal vector sets. Let F denote the codebook, N = |F| the codebook size and M := N/Nt the number of (m)

orthonormal sets in F. Moreover, let vn

denote the nth member of the mth orthonormal

(m)

set in F. Thus, F = {vn , 1 ≤ n ≤ Nt , 1 ≤ m ≤ M }. As in [96], the M orthonormal vector sets of F are generated randomly and independently using a method such as that in [85]. Consider the quantization of su , the channel shape of the uth user. Following [10], the quantizer function is given as ˆ su = arg maxv∈F |v† su |2

(5.12)

where ˆ su represents the quantized channel shape. The quantization error is given as ²u = |ˆs† s|2 . The quantized channel shapes {ˆsu } as well as channel power {ρu } and quantization error {²u } are sent back from the users to the base station. The base station constrains the quantized channel shapes of scheduled users to belong to the same orthonormal set in the codebook F. Furthermore, the quantized channel shapes of scheduled users are applied as beamforming vectors. Thereby, the orthogonal beamforming constraint in (5.11) is satisfied. Under this constraint and for the criterion of maximizing throughput, the expected throughput defined in (5.6) can be 130

written as

" Ror = E max1≤m≤M max u

XNt (m)

n ∈In n=1,··· ,Nt

n=1

# log (1 + Ψun )

(5.13) (m)

where Ψun is the scheduling metric defined in (5.5). The user index set In , which groups users with identical quantized channel shapes, is defined as n o In(m) = 1 ≤ u ≤ U | ˆsu = vn(m)

1 ≤ m ≤ M, 1 ≤ n ≤ Nt .

(5.14)

Zero-Forcing Beamforming In this section, the zero-forcing beamforming method for SDMA with limited feedback [24, 76] is introduced, which satisfies the following constraint.    ∠(ˆ su , ˆ su0 ) ≥ ϕ0 ∀ u, u0 ∈ A and u 6= u0 (zero-forcing beamforming)   vu ⊥ ˆ su0 ∀ u, u0 ∈ A and u 6= u0

(5.15)

The constant ϕ0 , which is usually large, ensures the quantized channel shapes of scheduled users are well separated in angles [24]. The second condition of the above constraint is satisfied by computing beamforming vectors {vu , u ∈ A} from {ˆsu , u ∈ A} using the zero-forcing method [24,76]. Following [24,76], the channel shape of each user is quantized using the random vector quantization method, where the codebook F consists of N i.i.d. isotropic unitary vectors. To derive an expression of the expected throughput for the criterion of maximizing throughput, define all subsets of users whose quantized channel shapes satisfy the first condition of the beamforming constraint in (5.15) as follows © ª {B} = B ⊂ U | |B| ≤ Nt , ∠(ˆ su , ˆ su0 ) ≥ ϕ0 ∀ u, u0 ∈ B and u 6= u0 .

(5.16)

In terms of the above subsets, the expected throughput can be written as h X Rzf = E maxA⊂{B}

u∈A

where the expected SINR Ψu is given in (5.5). 131

i log (1 + Ψu )

(5.17)

U balls

1

2

N

Area of small bin = p

N+1

Area of big bin = 1-Np

Figure 5.2: The bins and balls model for multiuser feedback of quantized channel shapes

5.5

Background: Analytical Tools

In this section, a bins and balls model for multi-user feedback of quantized channel shapes is introduced. This model provides a useful tool for analyzing throughput scaling law for orthogonal beamforming in Section 5.6.1. In this model as illustrated in Fig. 5.2, U balls are thrown into N + 1 bins: N small bins and one big one, whose total volume is equal to one. Some useful results are derived using the bins and balls model. Let the probability that a ball falls into a specific bin be equal to p for each small bin and q for the big bin, hence q = 1 − N p. The first question to ask is how many small bins are nonempty. The answer to this question is provided in the following lemma, obtained Using the Chebychev’s inequality [103]. Lemma 15 Denote p˜ = 1 − (1 − p)U . The number of nonempty small bins W satisfies ³ ´ p Pr W ≥ N p˜ − log N (N p˜ − N p˜2 ) ≥ 1 −

1 . log N

(5.18)

Next, consider clusters of Nt neighboring small bins. In Section 5.6.1, each cluster is related to an orthonormal vector set in the quantizer codebook for orthogonal beamforming. Each cluster is said to be nonempty if it contains no empty bins. Then, the second question to ask is how many clusters are nonempty. The answer is provided in the 132

following corollary of Lemma 15. Corollary 6 Denote the number of non-empty clusters of small bins as Q. Then Q satisfies ¶ µ q 1 Pr Q ≥ M p˜Nt − log M (M p˜Nt − M p˜2Nt ) ≥ 1 − (5.19) log M where M is the total number of clusters.

5.6

Throughput Scaling: High SNR

In this section, the throughput scaling law of uplink SDMA in the high SNR regime (γ À 1) is analyzed. The expected SINR in (5.5) for this regime is simplified as Ψ(α) u = P

ρu m∈A ρm ²m

(5.20)

m6=u

where the superscript (α) is added to indicate the high SNR regime. Using the analytical tools discussed in Section 5.5, the throughput scaling laws are derived in Section 5.6.1 and 5.6.2 for orthogonal and zero-forcing beamforming, respectively.

5.6.1

Throughput Scaling for Orthogonal Beamforming

In this section, we analyze the throughput scaling laws for orthogonal beamforming in the high SNR regime. Two cases are considered. First, both the number of users U and the quantization codebook size N are large. For this case, we derive an upper and a lower bounds for the throughput scaling factor as functions of U and N . Second, U is large but N is fixed. For this case, the exact throughput scaling factor in terms of U is obtained. U → ∞ and N → ∞ To derive the throughput scaling law for U → ∞ and N → ∞, the following approach is adopted. First, we derive an upper-bound for the throughput scaling factor of the expected throughput, which is defined in (5.6). To avoid confusion, the expected throughput is (α)

denoted as Ror where the superscript specifies the high SNR regime and the subscript indicates orthogonal beamforming. Second, an achievable lower-bound is obtained by 133

constructing a sub-optimal scheduling algorithm. Last, the throughput scaling law for (α)

Ror is shown to hold for the exact throughput. (α)

An upper-bound for scaling factor of Ror is derived as follows. To avoid considering any specific scheduling algorithm in the derivation, the following assumption is made. Assumption 10 The channel power of a scheduled user is lower bounded as ρu ≥

1 log U + c

∀ u ∈ A.

(5.21)

This assumption is justifiable under the current design criterion of maximizing throughput. Under this criterion, as U grows, the channel power of scheduled users increases but the lower bound in (5.21) converges to zero. Since ρu ≥ 0 and we are interested in the case of U → ∞, Assumption 10 is justified. Using this assumption, an upper bound for the (α)

scaling factor of Ror is derived and shown in the following lemma. Lemma 16 In the high SNR regime and for the case of U → ∞ and N → ∞, the scaling (α) factor of the expected throughput Ror in (5.6) is upper bounded as (α)

lim

U →∞ B→∞

Ror ≤ 1. Nt Nt −1 (log U + log N )

(5.22)

The proof is given in Appendix 5.11.2. (α)

Next, an achievable lower bound for the scaling factor of Ror is obtained. The (α)

direct derivation of a scheduling algorithm for maximizing the scaling factor of Ror in (5.6) is very difficult if not impossible. To overcome this difficulty, we argue that it is unnecessary to consider channel power in scheduling. In the sequel, we prove that the scheduling neglecting channel power leads to a reasonable lower bound of the optimum throughput scaling factor for orthogonal beamforming. The reason for the above argument is that scheduling users with largest channel power can at most increase the scaling factor by only O(log log U ) since the largest power scales as log U [28]. Such an increment is negligible because the expected scaling factor is O(log U ) as shown in Lemma 16. Thus, to achieve the optimum throughput scaling, using minimum quantization errors {²u } as the scheduling criterion suffices. In the high SNR regime that is interference limited, 134

such a criterion minimizes interference caused by quantization errors. The use of only (α)

quantization errors as the scheduling criterion leads to the following lower bound for Ror . Let χ21 , χ22 , · · · , χ2Nt denote a sequence of chi-squared random variables representing the channel power of scheduled users. From (5.6) and (5.20)    2 XNt χ    Ror ≥ E max1≤m≤M max u ∈I (m) log 1 + PNt n  2 k n=1 k k=1 χk ²uk k=1,··· ,N t



k6=n





XNt   ≥ E max1≤m≤M log 1 + PNt n=1 

2 k=1 χk k6=n



  ≥ Nt E max1≤m≤M log 1 + 



  = Nt E log 1 +

χ2n minu∈I (m) ²u

 

k

χ2n max1≤n≤Nt minu∈I (m) ²u n 

χ2n   P N 2 ? ² k=1 χk

 PNt

2 k=1 χk k6=n

 

(5.23)

k6=n

where ²? = min

max

min ²u .

(5.24)

1≤m≤M 1≤n≤Nt u∈I (m) n

A scheduling algorithm directly follows from the throughput lower bound in (5.23). Define à ! m? = arg min

max

min ²u

(5.25)

k

Then the scheduled user set A is given as ( A=

.

1≤n≤Nt u∈I (m)

1≤m≤M

)

arg min? ²u , 1 ≤ n ≤ Nt (m )

.

(5.26)

u∈In

Using this scheduled algorithm, an achievable lower bound of the throughput scaling factor is obtained and shown in the following lemma. Lemma 17 In the high SNR regime and for the case of U → ∞ and N → ∞, the scaling

135

(α)

factor of the expected throughput Ror in (5.6) is lower bounded as (α)

lim

U →∞ N →∞

Ror ≥ 1. Nt 1 Nt −1 log U + Nt −1 log N

(5.27)

The proof is given in Appendix 5.11.3. The proof procedure involves using the bins-andballs model and Lemma 6. Proposition 11 implies the identical throughput scaling factors for the expected (α)

(α)

throughput Ror and the exact one, denoted as Cor , because their difference is no more than a constant. By combining Proposition 11, Lemma 17, and Lemma 16, the main result of this section is obtained and summarized in the following theorem. Theorem 3 In the high SNR regime and for the case of U → ∞ and N → ∞, the scaling law of the throughput for orthogonal beamforming is given as (α)

lim

U →∞ N →∞

Cor ≤ 1, Nt Nt Nt −1 log U + Nt −1 log N

(α)

lim

U →∞ N →∞

Cor ≥ 1. Nt 1 Nt −1 log U + Nt −1 log N

(5.28)

A few remarks are in order. • The bounds in (5.28) agree on that the throughput scaling factor with respect to U is

Nt Nt −1

log U .

• The lower and the upper bound in (5.28) differ by Nt times in the throughput scaling factor with respective to N . The smaller scaling factor in the constructive lower bound is due to the use of a suboptimal scheduling algorithm. The design of a scheduling algorithm for achieving the upper-bound for the scaling factor in (5.28) is a topic for future investigation. • No feedback of channel power is required for achieving the lower bound for the throughput scaling factor in (5.28), because scheduling is independent of channel power.

136

U → ∞ and N fixed In this section, the throughput scaling law for orthogonal beamforming is analyzed for the high SNR regime and the case where the codebook size N is fixed and the number of users U → ∞. The upper-bound of the throughput scaling factor is shown in the following lemma. The proof can be easily modified from that for Lemma 16 by substituting limU →∞ log N/ log U = 0. Lemma 18 In the high SNR regime and with N fixed, the throughput scaling factor for orthogonal beamforming is upper bounded as (α)

Ror lim Nt ≤ 1. U →∞ Nt −1 log U

(5.29)

Next, the equality in (5.29) is shown to hold using the following scheduling al(m)

gorithm. First, among users belonging to the index set In , the one with the smallest quantization error is selected. Second, among the selected users corresponding to the (m)

index sets {In }, an arbitrary set of users with orthogonal quantized channel shapes are scheduled and these orthogonal vectors are applies as their beamforming vectors. Using this scheduling algorithm, the index set of scheduled users can be written as n o A = arg minu∈I (m) ²u , 1 ≤ n ≤ Nt . Based on the above scheduling algorithm and from n

(5.6), the expected throughput is bounded as     (α) Ror ≥ Nt E log 1 + PN

2 k=1 χk k6=n

 χ2n minu∈I (m) ²u

  .

k

Using the above throughput lower-bound and Lemma 18, the following lemma is proved. Lemma 19 The upper-bound of the throughput scaling factor in (5.29) is achievable (α)

Ror lim Nt =1 U →∞ Nt −1 log U

(5.30)

The proof is given in Appendix 5.11.4. This proof makes use of the theory of uniform convergence in the weak law of large numbers in Lemma 5 in Section 3.5.1. 137

By combining Lemma 19 and Proposition 11, the main result of this section is obtained and summarized in the following theorem. Theorem 4 In the high SNR regime (γ À 1) and with a fixed codebook size N , the throughput scaling law for orthogonal beamforming is (α)

Cor lim = 1. U →∞ Nt log U Nt −1

(5.31)

Two remarks are given. • The current throughput scaling factor is identical to the first terms of the bounds in (5.28) corresponding to the case of N → ∞. • For Nt ≥ 3, the linear scaling factor in (5.31), namely Nt /(Nt − 1), is smaller than Nt , which is the number of available spatial degrees of freedoms. This indicates the loss in multiplexing gain for Nt ≥ 3.

5.6.2

Throughput Scaling for Zero-Forcing Beamforming

In this section, the scaling law for zero-forcing beamforming in the high SNR regime is analyzed. Two cases are considered: 1) U → ∞ and N → ∞ and 2) U → ∞ and N is fixed, which are jointly analyzed due their similarity in analysis. Denote the expected and α and the exact throughput for zero-forcing beamforming in the high SNR regime as Rzf α. Czf

The upper-bounds of the throughput scaling factor for orthogonal beamforming in Lemma 16 and Lemma 18 can be shown to hold for zero-forcing beamforming by trivial modifications of the proofs. Thus, (α)

(α)

lim

Rzf

U →∞ Nt (log U N →∞ Nt −1

+ log N )

≤ 1,

lim

U →∞

Rzf Nt Nt −1

log U

≤ 1.

(5.32)

The above upper-bounds for the throughput scaling factor of zero forcing beamforming can be achieved using the following scheduling algorithm. Consider an arbitrary basis of CNt , denoted as {q1 , q2 , · · · , qNt }. Using this basis, we define the following index 138

sets

n o Jk = 1 ≤ u ≤ U | 1 − |q†nˆsu |2 ≤ τo

where τo = sin2

¡π 4



ϕo ¢ 2

=

1+sin(ϕo ) 2

1 ≤ k ≤ Nt

(5.33)

and ˆsu is the quantized channel shape. The purpose

of these index sets is to select users who satisfy the zero-forcing beamforming constraint in (5.15). Among the users in each of the index sets {Jk }, the one with the smallest quantization error is scheduled. In other words, the index set of the scheduled users is ¾ ½ A = arg min ²u , 1 ≤ k ≤ Nt .

(5.34)

u∈Jk

The beamforming vectors of the scheduled users are computed from their quantized channel shapes using the zero-forcing method. From the above scheduling algorithm results in the following throughput lower bound  

 χ2n

  (α) Rzf ≥ Nt E log 1 + PN

2 k=1 χk minu∈Jk ²u

  .

k6=n

Using the above throughput lower bound, we prove the following theorem. Theorem 5 In the high SNR regime, the throughput scaling law for zero-forcing beamforming is given as follows. 1. For U → ∞, N → ∞ (α)

Czf

lim

U →∞ Nt N →∞ Nt −1

log U +

Nt Nt −1

log N

= 1.

(5.35)

2. For U → ∞, N fixed (α)

lim

Czf

U →∞ Nt N →∞ Nt −1

log U

= 1.

(5.36)

The proof is given in Appendix 5.11.5. The proof uses the uniform convergence in the weak law of large numbers. As before, Proposition 11 is applied to equate the scaling laws between the expected and the exact throughput. A few remarks are in order. • For U → ∞, N → ∞, the throughput scaling factor for zero-forcing beamforming 139

upper bounds that for orthogonal beamforming (cf. (5.28)). Note that this does not implies the former is larger since the achievability of the same scaling factor for orthogonal beamforming is unknown. • The same scaling laws as in (5) have been also proved for downlink SDMA with limited feedback [24]. They are derived using a different approach based on the extreme value theory, though. This similarity demonstrates uplink-downlink duality. • As for orthogonal beamforming, the scheduling algorithm, which achieves the above scaling laws for zero-forcing beamforming, requires no feedback of channel power.

5.7

Throughput Scaling: Normal SNR

In this section, the throughput scaling law for uplink SDMA in the normal SNR regime is analyzed. In this regime, neither the noise nor the interference dominates, thus the SINR and scheduling metric are given respectively in (5.4) and (5.5). The throughput scaling law for orthogonal beamforming and zero-forcing beamforming are analyzed separately in Section 5.7.1 and Section 5.7.2.

5.7.1

Orthogonal Beamforming

In this section, the throughput scaling factor for orthogonal beamforming is obtained by deriving an upper-bound and an achievable lower bound of this factor. The upper-bound of the scaling factor is given in the following lemma. This upperbound also holds for the low SNR regime and the zero-forcing beamforming. Lemma 20 For both the normal and low SNR regimes, the throughput scaling factors for both orthogonal and zero-forcing beamforming are upper-bounded as Ror/zf ≤ 1. U →∞ Nt log log U lim

(5.37)

The proof is similar to that for Lemma 16 and hence omitted. In the proof, the upperbound of the throughput scaling factor in (5.37) is derived by omitting interference. This 140

implies that reducing interference by increasing the codebook size N has no effect on this upper bound. Thus it is unnecessary to consider the case of N → ∞ in the analysis for the normal SNR regime. The scheduling algorithm for achieving the equality in (5.37) is provided as follows. Define the user index sets ( T´n(m) =

Ã

1 ≤ u ≤ U | su ∈ Bn(m)

1

!)

(log U )Nt −1

(m) (m) and a scalar Uβ := exp (−dmin /4). Then T´n ⊂ In

(5.38)

(m) ∀ U ≥ Uβ . From each set T´n ,

the user with the maximum channel power is selected. Next, among the selected users, up to Nt users are scheduled using the criterion of maximizing throughput. Using this scheduling algorithm and from (5.13), a lower-bound of the throughput is obtained as    γ maxu∈T (m) ρu XNt    n Ror ≥ E max1≤m≤M log 1 +  U ≥ Uβ P Nt 1 n=1 1 + γ k=1 maxu0 ∈T (m) ρu0 log U k k6=u    γ maxu∈T (m) ρu XNt   n ≥ E log 1 + (5.39)  U ≥ Uβ . P Nt 1 n=1 1 + γ k=1 maxu0 ∈T (m) ρu0 log U k6=u

k

Using the above lower bound, we prove the following theorem. Theorem 6 In the normal SNR regime, the scaling law for orthogonal beamforming is Cor = 1. U →∞ Nt log log U lim

(5.40)

The proof is given in Appendix 5.11.6. Again, the proof relies on the uniform convergence in the weak law of large numbers. A few remarks are in order. • The throughput in the normal SNR regime scales as log log U but that in the high SNR regime increases as log U . Therefore, the throughput scaling rate is much higher in the high SNR regime than in the normal SNR regime. • The scaling law in Theorem 6 shows the full multiplexing gain. 141

• Besides quantized channel shapes, feedback of both channel power and quantization errors from users are required.

5.7.2

Zero-Forcing Beamforming

This section focuses on the throughput scaling law for zero-forcing beamforming in the normal SNR regime. A scheduling algorithm for achieving the scaling upper-bound in Lemma 20 is constructed as follows. Define the index sets, {Tn }N n=1 , similar to (5.38) but based on the RVQ codebook for zero-forcing beamforming (cf. Section 5.4.3). Next, define a new index set ¡ ¢ Lk = Jk ∩ ∪N n=1 Tn

1 ≤ k ≤ Nt

(5.41)

where Jk is given in (5.33). From users in each of the sets {Lk }, the one with the maximum channel power is scheduled. Thus, the index set of scheduled users is given as A = {maxu∈Lk ρu , 1 ≤ k ≤ Nt } .

(5.42)

Using the above scheduling algorithm, we obtain the following theorem by proving the achievability of the throughput-scaling upper-bound in Lemma 20. Theorem 7 In the normal SNR regime, the scaling law for zero-forcing beamforming is lim

U →∞

Czf = 1. Nt log log U

The proof is given in Appendix 5.11.7.

(5.43)

The proof involves repeated applications of

Lemma 5, which show the uniform convergence of the numbers of users in the index sets {Tn } and Jn defined (5.33), respectively. Comparing Theorem 7 and Theorem 6, the same scaling law holds for both orthogonal and zero-forcing beamforming in the normal SNR regime. Furthermore,this scaling law is identical to that for downlink SDMA with limited feedback [24, 28, 96].

142

5.8

Throughput Scaling: Low SNR

In this section, the analysis of the throughput scaling law for uplink SDMA focuses on the lower SNR regime where channel noise is dominant. In this regime, the expected SINR in (5.5), denoted as Ψ(β) , reduces to γρu . The following analysis is presented in Section 5.8.1 and Section 5.8.2, which correspond respectively to orthogonal and zeroforcing beamforming.

5.8.1

Orthogonal Beamforming

In the lower SNR regime, the throughput scaling law for orthogonal beamforming is obtained by achieving the upper-bound for the throughput scaling factor in Lemma 20 using (β)

a specific scheduling algorithm. Denote the expected and exact throughput as Ror and (β)

Cor , respectively. A suitable scheduling algorithm can be modified from that in Section 5.7.1 by replacing the index sets in (5.38) with the following ones n ³ ´o Tˇn(m) = 1 ≤ u ≤ U | su ∈ Bn(m) (dmin /4)Nt −1

1 ≤ m ≤ M, 1 ≤ n ≤ Nt .

(5.44)

0

(m) (m ) Note that Tˇn ∩ Tˇn0 = ∅ for all (m, n) 6= (m0 , n0 ). The modified scheduling algorithm

leads to the following throughput lower bound h ³ ´i (β) Ror ≥ Nt E log 1 + γ maxu∈Tˇ (m) ρu . n

(5.45)

Using the above throughput lower bound, the throughput scaling law is obtained and summarized in the following theorem. Theorem 8 In the low SNR regime, the scaling law of uplink SDMA with orthogonal beamforming is given as (β) Cor lim = 1. (5.46) U →∞ Nt log log U The proof is similar to that for Theorem 6. Specifically, the proof uses the result of the extreme value theory in (3.33) and Lemma 5 of the uniform convergence in the weak law 143

of large numbers. The details of the proof are omitted. Comparing Theorem 6 and Theorem 8, the scaling laws in the normal and the low SNR regimes are identical. The intuition is that the interference power decreases contemptuously with U . Thus, for a large U , both the low and normal SNR regimes become noise limited, resulting in the same throughput scaling laws.

5.8.2

Zero-Forcing Beamforming

As in the last section, the derivation of the throughput scaling law for zero-forcing beamforming in the low SNR regime relies on the use of a specific scheduling for achieving the scaling upper-bound in Lemma 20. This scheduling algorithm is simplified from that in Section 5.7.2 as follows. For the current algorithm, the scheduled users are selected from the index sets {Jk } in (5.33) rather than {Lk } as in Section 5.7.2. Consequently, the index set of scheduled users is A = {maxu∈Lk ρu , 1 ≤ k ≤ Nt } .

(5.47)

Using the above scheduling algorithm, we prove the following theorem. Theorem 9 In the low SNR regime, the scaling law for zero-forcing beamforming is Czf = 1. U →∞ Nt log log U lim

(5.48)

The proof is a simplified version of that for Theorem 9 due the similarity in scheduling algorithms. Unlike the previous proof, the current proof requires only one-time application of Lemma 5. Similar remarks for Theorem 8 are also applicable here.

5.9

Numerical Results

In this section, based on simulation, orthogonal and zero-forcing beamforming are compared in terms of uplink SDMA throughput for an increasing number of users U . Such a comparison is to evaluate the throughput difference between orthogonal and zero-forcing beamforming in the practical regime of U . Note that the throughput scaling laws derived 144

in previous sections indicate the same slopes for the throughput vs. U curves for both beamforming methods in the asymptotic regime of U . Furthermore, uplink SDMA with limited feedback is compared with uplink channel-aware random access proposed in [1], which requires no CSI feedback. Orthogonal and zero-forcing beamforming are compared for both the high and the low SNR regimes. For simulation, the scheduling criterion is minimum quantization error in the high SNR regime and maximum channel power in the low SNR regime. These criteria are shown to achieve optimum throughput scaling in Section 5.6 and Section 5.8. For zero-forcing beamforming, the scheduling algorithms are modified from that proposed in [24] by using the above criterions in greedy-search scheduling. For orthogonal beamforming, the scheduling algorithms are identical to those proposed in Section 5.6.1 and Section 5.8.1. The throughput of orthogonal and zero-forcing beamforming are compared in Fig. 5.3 for an increasing number of users U . For this comparison, the number of antennas is Nt = 2, the quantizer codebook size is N = 8, and the SNRs are {−5, 0} dB for the low SNR regime and {20, 30} dB for the high SNR regime. Several observations are made from Fig. 5.3. First, as shown Fig. 5.3(a) for the high SNR regime, orthogonal beamforming provides higher (smaller) throughput than zero-forcing beamforming if the number of users is large (small). The crossing point between the curves for orthogonal and zero-forcing beamforming is at U = 20 for SNR = 20 dB and at U = 28 for SNR = 30 dB. Second, from Fig. 5.3(b) for the low SNR regime, orthogonal beamforming always achieves higher throughput than zero-forcing beamforming. Note that for U → ∞, the curves for orthogonal and zero-forcing beamforming have identical slops according the throughput scaling laws. In Fig. 5.4, the throughput of uplink SDMA is compared with that of SDMA with random scheduling and uplink random access [1], both of which require no CSI feedback. For SDMA with random scheduling, a random set of users is scheduled and their beamformers are columns of a random orthonormal basis. Note that with a single scheduled users, SDMA with random scheduling reduces to TDMA. For uplink random access, trans145

16 Orthogonal (30 dB) 15 14

Throughput (b/s/Hz)

Orthogonal (20 dB) 13 ZF (30 dB)

12 11 10 9

ZF (20 dB) 8 7 6

0

20

40

60

80

100

120

140 150

120

140 150

Number of users

(a) High SNR 4.5 Orthogonal (0 dB)

Throughput (b/s/Hz)

4

ZF (0 dB)

3.5

3 Orthogonal (−5 dB) 2.5

2 ZF (−5 dB) 1.5

1

0

20

40

60

80

100

Number of users

(b) Low SNR

Figure 5.3: Throughput comparisons between orthogonal and zero-forcing beamforming for uplink SDMA in (a) the high SNR regime and (b) the low SNR regime. The number of antennas at the base station is Nt = 2 and the quantizer codebook size is N = 8. The plotted values in brackets specify the SNR values in dB.

146

mitting users are selected distributively using a channel power threshold, which increases with the total number of users [1]. For fair comparison, the uplink random access design originally proposed in [1] for SISO channels is modified to allow transmit beamforming at each user who has Nt antennas. For uplink SDMA with limited feedback, the scheduling algorithms used in the previous comparison for the low SNR regime are applied. The simulation parameters are SNR = 5 dB, Nt = 2 and N = 8. Several observations are made from Fig. 5.4. First, the throughput for uplink SDMA is much higher than that of SDMA with random scheduling and uplink random access. The throughput gains of uplink SDMA result from scheduling at the base station and the support of Nt simultaneous users. Second, the throughput of SDMA with random scheduling and uplink random access is insensitive to changes on the number of users U for the following reasons. Without giving preference to users with large channel power, random scheduling is incapable of exploiting multiuser diversity. Next, uplink random access achieves the throughput scaling of log log U but such a function grows extremely slowly with U . In summary, uplink SDMA outperforms SDMA with random scheduling and uplink random access in [1] by a large margin at the expense of finite-rate feedback from each user. Note that it is possible to schedule feedback users so as to constraint the total feedback overhead for uplink SDMA by following an approach similar to those proposed in [86, 102].

5.10

Summary

In this chapter, the scaling law of uplink SDMA with limited feedback is analyzed for different SNR regimes and both orthogonal and zero-forcing beamforming. In the high SNR regime and for orthogonal beamforming, for an increasing quantizer codebook size, the throughput scales logarithmically with both the number of users and the codebook size; for a fixed codebook size, the throughput scales logarithmically only with the codebook size. For both cases, the linear scaling factor is smaller than the number of antennas, indicating the loss in spatial multiplexing gain. Similar results are obtained for zero-forcing beamforming. In the normal SNR regime, for both orthogonal zero-forcing beamforming, 147

SDMA (Orthogonal)

6

Throughput (b/s/Hz)

5

SDMA (ZF) 4

3

Random scheduling (# scheduled users = 2) Random scheduling (# scheduled users = 1) Random access (# scheduled users = 1)

2

1 0

50

100

150

Number of Users

Figure 5.4: Throughput comparisons between uplink SDMA with limited feedback, SDMA with random scheduling and uplink random access in [1]. The number of antennas at the base station is Nt = 2; the quantizer codebook size is N = 8; the SNR = 5dB. the throughput is found to scale double logarithmically with the number of users and linearly with the number of antennas. The same results are obtained for the low SNR regime. Simulation results suggest that orthogonal and zero-forcing beamforming achieve different uplink throughput in non-asymptotic regimes even though they may follow the same throughput scaling laws asymptotically. For a small SNR or a large SNR coupled with many users, orthogonal beamforming outperforms zero-forcing beamforming. The reverse is true for a large SNR and a small number of users.

148

5.11

Appendix

5.11.1

Proof of Proposition 11

Using the triangular inequality, |∠(vu , ˆ su ) − ∠(su , ˆ su )| ≤ ∠(vu , su ) ≤ ∠(vu , ˆ su ) + ∠(su , ˆ su ). By definitions of ϕu and θu , the above expression can be rewritten as |ϕu − θu | ≤ ∠(vu , su ) ≤ ϕu + θu . From the given condition ϕu + θu ≤ ϕo + θo
1− 157

¶¾

1 2 (log U )Nt −1

1 ≤ n ≤ Nt .

(5.76)

∀ U > U1

(5.77)

n where U1 = max

3 2

h i h io (log U )Nt −1 log 10c (log U )Nt −1 , 24 (log U )Nt −1 log 2 (log U )Nt −1 .

There exists U0 such that Ln ∩ L0n = ∅ for all n 6= n0 . ³ ´ Nt t Next, define Jn = |Jn ∩ ∪N L n=1 n | and L = | ∪n=1 Ln |. Again, by applying Lemma 5 ¡ ¢ log L Pr Jn ≥ τoNt −1 L − log L > 1 − ∀ L ≥ L1 (5.78) L n i io h h 3L 4L L L ˜ = U/(log U )Nt −1 . where L1 = max log , . Denote U log 10c log 2 L log L log L log L  R≥

XNt n=1

XNt n=1



  E log 1 +

 ≥

 1+γ

PN

γ maxu∈Jn ρu

2−1/(Nt −1) k=1 maxu0 ∈Jk ρu0 log U k6=n



  E log 1 +

 ˜) ˜  Pr(L ≥ U |L≥U

 1+γ

PN

γ maxu∈Jn ρu

2−1/(Nt −1) k=1 maxu0 ∈Jk ρu0 log U k6=n



 N −1 ˜ ˜ − log U  | Jn ≥ τo t U 

˜ − log U ˜ |L≥U ˜ ) Pr(L ≥ U ˜) Pr(Jn ≥ τoNt −1 U !# " Ã XNt γ log U U →∞ ≥ E log 1 + −1/(N −1) n=1 1 + γ log U 2 log Ut The desired result following from the last inequality and Proposition 11.

158



(5.79)

Chapter 6

Spatial Interference Cancelation for MIMO Ad Hoc Networks In a mobile ad hoc network (MANET), the mutual interference between nodes limits throughput for peer-to-peer communication over the network. In this chapter, the degrees of freedom created by employing antenna arrays at receivers are exploited for canceling interference from the strongest interferers. Thereby the number of successful communication links per unit area, related to network transmission capacity [104], increases significantly even with imperfect CSI.

6.1

Prior Work

For a multi-hop ad hoc network, the notion of transport capacity was introduced in [87], and bounds were derived for a large number of nodes. This work started a series of studies on network transport capacity (see e.g. [105–109]). Prior results on transport capacity typically focus on scaling laws of network throughput with an asymptotically large number of nodes. These results suggest that the throughput per node diminishes with the total number of nodes in the network unless they collaborate in transmission. Such asymptotic results may differ significantly from the actual throughput of finite-size networks, and thus have limited practical applications. This motivates an alternative definition of network capacity that allows more accurate characterization of network throughput. For single-hop ad hoc networks, the transmission capacity metric introduced in 159

[110] is defined as the maximum number of successful communication links per unit area under signal-to-interference-noise ratio (SINR) and outage constraints. Through suitably modeling network nodes as a Poisson point process on a 2-D plane, this notion of network capacity enjoys more accurate analysis and easier computability compared with transport capacity [104, 110]. Transmission capacity has been used to make tractable analysis of opportunistic transmission [110], distributed scheduling [111], coverage [112], network irregularity [113], bandwidth partitioning [114], successive interference cancelation (SIC) [115] and multi-antenna transmission [116] in an ad hoc network. Interference directly limits the throughput of MANETs. The optimal approach for reducing interference in MANETs is known as interference alignment, which achieves the number of degrees of freedom equal to half of the number of interference links [117]. Nevertheless, this approach appears daunting because it requires nodes to employ jointly designed precoders and obtain perfect channel sate information (CSI) of interference channels. One simpler method for reducing interference in MANETs is to create an interfererfree area – a guard zone – around each receiving node through carrier sensing [111]. By optimizing the guard-zone size, this method leads to a significant gain in network throughput with respect to purely random access. An alternative approach is to employ physical layer techniques of the multiuser detection family for suppressing interference [27]. Nevertheless, these algorithms appear to be very sensitive to residual interference resulting from imperfect interference cancelation, and the near-far problem. This chapter considers zero-forcing beamforming for interference cancelation in multi-antenna MANETs. Furthermore, this chapter assumes single data streams for communication links in the networks. Therefore, stream control [118] is unnecessary, and the spatial degrees of freedoms created by multi-antennas are dedicated for interference cancelation. Recently, beamforming or directional antennas [6] have been integrated with the medium access control (MAC) protocols for MANETs to achieve higher network spatial reuse or energy efficiency [119–131]. In addition, these multi-antenna techniques can also improve the efficiency of routing protocols for MANETs [132–136]. Directional an160

tennas suppress interference by spatial filtering, which works only in an environment of sparse scattering [126–130]. In contrast, beamforming is suitable for both sparse and rich scattering, and is hence adopted in this chapter as well as in [121–123, 131] for spatial interference cancelation. Most prior work focuses on designing MAC protocols and relies on simulations for investigating network throughput [119–130]. The capacity of MANETs with beamforming or direction antennas are analyzed in [116, 137, 138]. In [137, 138], the use of directional antennas are shown to increase the linear scaling factor of network transport capacity. In [116], the transmission capacity for multi-antenna MANETs is analyzed, where interference is treated as noise and suppressed by averaging through beamforming. In view of prior work, there still lacks of theoretic characterization of the relationship between the transmission capacity of MANETs and spatial interference cancelation. Furthermore, the important issue of how CSI inaccuracy affects the throughput of MANETs has not been analyzed [116, 119–130, 137, 138]. These issues are addressed in this chapter.

6.2

Contributions

The contributions of this chapter are summarized as follows. This chapter targets a MANET with single-stream data links and perfect synchronization between nodes. First, zero-forcing beamforming is applied for canceling interference at receivers and thereby increasing network transmission capacity1 . The number of canceled interferers depends on the number of antennas at each receiver. Moreover, transmit beamforming vectors are randomly selected to avoid iterative receive beamforming and potential network instability. Second, assuming low mobility, an opportunistic CSI estimation algorithm is proposed, which provides the CSI required for spatial interference cancelation. By opportunistically estimating CSI in the absence of strong interferers, this algorithm provides a trade-off between transmission power for training signals and the duration of CSI estimation. Third, based on the Poisson assumption on transmitting-node locations and the spatially i.i.d. 1

The zero-forcing method is used for analytical simplicity, and the extension to minimum-mean-squarederror (MMSE) beamforming is straightforward

161

Rayleigh fading channel model, bounds on the signal-to-interference ratio (SIR) outage probability are derived for the cases of perfect and imperfect CSI. These results are useful for computing network transmission capacity. The derived bounds are found to be reasonably tight especially for the case of imperfect CSI. Third, the scaling laws for transmission capacity are derived for asymptotically small target outage probability. Specifically, with interference cancelation, the asymptotic transmission capacity grows following a power law for both perfect and imperfect CSI. The base of the power law is the target outage probability, and the exponent is the inverse of the antenna-array size if it is smaller than the pass-loss exponent. Otherwise, the exponent is bounded between the inverse of the antenna-array size and that of the pass-loss exponent. Simulation results are also presented and provide the following observations. First, employment of a few (2-4) antennas per node is sufficient to harvest most of the capacity gains promised by spatial interference cancelation. In particular, compared with the case of single antennas, a capacity gain of more than an order of magnitude can be achieved by using only one additional antenna at each node, even if CSI is imperfect. This demonstrates the effectiveness of spatial interference cancelation for practical applications. Second, residual interference caused by imperfect CSI significantly reduces network transmission capacity, which emphasizes the importance of accurate CSI estimation. Third, for effective interference cancelation, the overhead for CSI acquisition including pilot transmission power and estimation delay is larger for a more stringent SIR outage constraint (or smaller node density). Finally, the derived asymptotic scaling laws of transmission capacity are found to accurately characterize transmission capacity even for non-asymptotic target outage probability (up to 0.1).

6.3 6.3.1

Network and Channel Models Network Model

In this chapter, the locations of potential transmitting nodes in the mobile ad hoc network, including both active and inactive transmitters, are modeled as a Poisson point process 162

following the common approach in the literature [104, 110, 115, 116, 139]. Specifically, the positions of the potential transmitters form a homogeneous Poisson point process with the density denoted by λo . The Poisson assumption allows the use of tools from stochastic geometry for analyzing the network capacity [140]. To access wireless channels, potential transmitting nodes follow a random access protocol. They transmit independently and with fixed probability denoted by Pt . Let Tn denote the coordinate of the ith transmitting node on the 2-D plane. Given the random access protocol, the set Φ = {Tn } is also a homogeneous Poisson point process but with the smaller density λ = Pt λo [141]. Each transmitting node is associated with a receiving node located at a fixed distance denoted as d. As shown in [110], randomness of the distances has no significant effect on the analysis of transmission capacity and is thus omitted for simplicity. Consider a typical receiving node located at the origin, denoted as R0 , and hence |T0 | = d. This location constraint of T0 does not compromise the generality since the transmitting node process Φ is translation invariant. Furthermore, according to Slivnyak’s theorem [140], other transmitting nodes, namely Φ/{T0 }, remain as a homogeneous Poisson point process with the same node density λ . The ad hoc network is assumed to be interference limited and thus noise is neglected for simplicity. Consequently, the reliability of data packets received by the node R0 is determined by the SIR. Moreover, we assume that each data link in the network has a single stream, and communications between nodes are perfectly synchronized. Let S denote the random channel power for the link from T0 to R0 , and the function I(Tn ) gives the power of interference from the transmitting node Tn to R0 . Thus, assuming uniform data transmission power for all transmitting nodes, denoted as PD , the SIR at R0 is given as SIR0 = P

S Tn ∈Φ/{T0 } I(Tn )

.

(6.1)

To simplify notation, I(Tn ) is denoted as In in the sequel. Since the SIR0 is independent of PD , PD = 1 is assumed for simplicity. The correct decoding of received data packets 163

requires the SIR to exceed a threshold θ, which is identical for all receiving nodes. In other words, the rate of information sent from a transmitter to a receiver is log2 (1 + θ) assuming Gaussian signaling. To support this information rate with high probability, the outage probability that SIR0 is below θ must be smaller than or equal to a given threshold 0 < ² < 1, i.e. Pout (λ) = Pr(SIR0 ≤ θ) ≤ ²

(6.2)

where Pout (λ) denotes the SIR outage probability as a function of λ. Given an outage constraint ², Pout determines the transmission capacity, which is defined as [104] C(²) = (1 − ²)λ²

(6.3)

where Pout (λ² ) = ². Note that this equality maximizes transmission capacity under the outage constraint (6.2) since Pout (λ² ) increases monotonously with λ² .

6.3.2

Channel Model

The channel model is characterized by narrow-band and flat fading. Each node in the network is equipped with L antennas. Consequently, there exists a L × L multipleinput-multiple-output (MIMO) channel between every pair of nodes. Each MIMO channel consists of path-loss and spatially i.i.d. small fading components, corresponding to rich scattering. Specifically, the channel from a node Tn to the typical receiving node R0 is −α/2

Hn = dn

−α/2

Gn . The factor dn

represents path-loss, where dn = |Tn | is the Euclidean

distance and α > 2 is the path-loss exponent. The other factor of Hn , Gn , models spatially i.i.d. Rayleigh fading, and hence G is a L × L matrix of i.i.d. CN (0, 1) components. The path-loss and spatially i.i.d. Rayleigh fading channels are commonly assumed in the literature of ad hoc networks (see e.g. [104, 139]) and MIMO communication (see e.g. [142, 143]), respectively. The above channel model simplifies analysis in this chapter. Finally, given single-stream data links, beamforming is applied at each transmitter and receiver. Let fn and v0 denote the beamforming vectors at Tn and R0 , respectively. Then the effective channel power for the data link from T0 to R0 is S = |v0† H0 f0 |2 , and that for 164

Symbol C Pout λ rn ² W Hn (Gn ) 2 σR

L fn , vn

PD /PP 2) η (σR

Table 6.1: Summary of Notation Description Symbol Description network transmission capacity α path loss exponent (α > 2, default = 4) SIR outage probability Φ point process modeling all transmitting nodes density of transmitting nodes and Tn /Rn nth transmitting/receiving Φ node distance between Tn and the oriIn power of interference from Tn to R0 gin SIR outage constraint Γ/γ upper/lower incomplete Gamma function fading gain of the effective chanrmin distance threshold for channel nel from T0 to R0 estimation MIMO channel from Tn to R0 G power of the primary interfer(fading component) ence variance of residual interference IΠ (g) secondary interference power due to imperfect CSI conditioned on G = g number of antennas at each node d (dn ) distance between T0 (Tn ) and (default = 4) R0 (default = 5 m) transmit and receive beamformθ required SIR for successful ers of the node Tn and Rn , respeccommunication (default = 3 or tively 4.8 dB) transmission power of data/pilot T0 /R0 typical node pair with R0 losymbols cated at the origin normalized power (variance) of δ constant equal to α2 residual interference from secondary interferers

the interference link from Tn to R0 is Sn = |v0† Hn fn |2 .

6.3.3

Notation

The notation commonly used in this chapter is summarized in Table 6.1. Unless stated otherwise, the default values for variables as provided in Table 6.1 are used for generating numerical results.

165

6.4

Spatial Interference Cancelation: Algorithm and Model

The algorithm of zero-forcing beamforming for spatial interference cancelation is described in Section 6.4.1. The resultant effective network and channel models are proposed for two cases, namely perfect and imperfect CSI, in Section 6.4.1 and 6.4.2, respectively. Furthermore, the algorithm for opportunistic CSI estimation and the model of the resultant residual interference are presented in Section 6.4.2.

6.4.1

Perfect CSI

Assume perfect CSI, synchronization between nodes, and single-stream data links. Under these assumptions, spatial interference cancelation uses zero-forcing beamforming by following the procedure described in Section 6.4.1. To analyze the effect of interference cancelation on network transmission capacity, the effective network and channel models are presented in Section 6.4.1. Without loss of generality, the discussion focuses on the typical pair of nodes T0 and R0 (cf. Section 6.3.1). Spatial Interference Cancelation and Opportunistic Transmission The idea of spatial interference cancelation is to apply zero-forcing beamforming at R0 for canceling interference from strong interferers. Thereby the network can support a higher density of active node pairs without violating their QoS requirements. The details of the interference cancelation algorithm are provided below. Let fn and v0 denote the transmit beamformer at Tn and the receive beamformer at R0 , respectively. From the perspective of R0 , the interference channel from Tn (n 6= 0) appears as an effective channel vector hn = Hn fn , where Hn denotes the actual MIMO channel. To facilitate our discussion, the indices of the transmitting nodes interfering with R0 are sorted according to their effective interference channel norms, namely kh1 k ≥ kh2 k ≥ · · · ≥ khL k · · · . The crux of the interference cancelation algorithm is to constrain the beamforming vector v0 of R0 to be in the null space of the matrix [h1 , h2 , · · · , hL−1 ].

166

Thereby, the interference from L − 1 strongest interferers to R0 is nulled2 : |v0† h1 | = |v0† h2 | · · · = |v0† hL−1 | = 0. Note that perfect CSI estimation of h1 , h2 , · · · , hL−1 by R0 is required to completely cancel the interference from L − 1 strongest interferers. CSI estimation at each receiver uses pilot symbols broadcast by transmitters. The assumption of perfect CSI is made in this section, and the issue of CSI inaccuracy is addressed in Section 6.4.2. An arbitrary transmit beamformer is applied at T0 , represented by f0 . Note that an attempt to perform maximum ratio transmission [144] causes iterative updating of beamforming vectors at all nodes and potential network instability. Remarks on other MIMO transmission techniques are given later. By such beamforming, multiple transmit antennas contribute no diversity gain, but they are needed for interference cancelation when the transmitter becomes a receiver. To avoid deep fading due to the lack of diversity gain, opportunistic transmission is applied. Consequently, transmission at each transmitter is turned on only if the channel gain S is above a threshold denoted by β, where S = |v0† H0 f0 |2 . It follows that the randomaccess probability for each potential transmitter is Pt = Pr(S ≥ β) (cf. Section 6.3.1). This algorithm is also used in [110] for single-antenna ad hoc networks and similar concepts exist in optimal power control for fading channels [145, 146]. The threshold β should be sufficiently small so as not to cause long awaiting-time for transmission. Remarks: Two alternative MIMO transmission techniques are either impractical or in conflict with spatial interference cancelation. First, as mentioned earlier, the optimal approach–interference alignment–requires each transmitter to have perfect CSI of its channels to all receivers [117]. Acquiring such CSI can result in excessive overhead and eliminate the capacity gain due to interference alignment. Second, space-time block coding [142, 147, 148] reduces the number of spatial degrees of freedom for spatial interference cancelation. The reason is that space-time block coding transmits multiple data 2 With probability one, the matrix [h1 , h2 , · · · , hL−1 ] has full rank. Thus, with L antennas, R0 can cancel at most L − 1 interferers.

167

symbols over different antennas rather than one symbol as for transmit beamforming. In view of the drawbacks of existing transmission techniques, designing efficient transmit beamforming algorithms for increasing network capacity is an important topic for future work. Effective Channel and Network Models The following effective channel and network models result from the application of the interference cancelation algorithm in the preceding section. With perfect interference cancelation, R0 receives interference only from the nodes {Tn | n ≥ L}3 . Let rn and In denote respectively the distance between Tn and the origin, and the interference power from Tn to R0 . Using this notation and based on the channel model in Section 6.3.2, for n ≥ L, In = PD |v0† Hn fn |2 = rn−α |v0† Gn fn |2 . Because both fn and v0 are independent of Gn 4 and Gn is an i.i.d. complex Gaussian matrix, the random variable ρn = |v0† Gn fn |2 follows the exponential distribution with unit variance. Therefore, for n ≥ L, the effective interference power from Tn to Rn is In = rn−α ρn . The effective power of the data link from T0 to R0 is given by S = |v0† H0 f0 |2 = d−α |v0† G0 f0 |2 with S ≥ β due to opportunistic transmission. Recall that d is the distance between T0 and R0 , β is the transmission threshold, and G0 is the fading component of the MIMO channel H0 . Because the beamformers v0 and f0 are independent of G0 as discussed in the preceding section, the random variable W = |v0† G0 f0 |2 has the exponential distribution with the following probability density function fW (w) = exp(−w)/Pt ,

w ≥ βdα

(6.4)

where Pt = exp(−βdα ). Last, the effective channel and network models resulting from interference cancelation is illustrated in Fig. 6.1(a). Note that the density of transmitting nodes is λ = Pt λo . 3

Note that the indices of the nodes {Tn } are sorted according to the power of their interference to R0 Note that the interference canceling beamformer v0 is a function of {Gn | 1 ≤ n ≤ L − 1} but independent of {Gn | n ≥ L}. Moreover, the transmit beamformer fn is dedicated for enhancing the corresponding data link and is hence independent of any interference channel. 4

168

T yp e s o f n o de T ra n sm ittin g n o d e R e ce ivin g n o d e P rim a ry in te rfe re r S e co n d a ry in te rfe re r Cancele d in te rfe re r

In te rfe re n ce cancelation zo n e

In te rfe re n ce cancelation zo n e

(a) Perfect CSI

(b) Im perfect CSI

Figure 6.1: Effective channel and network models resulting from interference cancelation with (a) perfect CSI or (b) imperfect CSI for antenna arrays of three elements. The distance in the figures is proportional to the effective channel power. The data and interference links are plotted by using solid and dashed lines, respectively. Based on the above models, the SIR at R0 is given as d−α W (Perfect CSI) SIR0 = P∞ . −α n=L rn ρn

6.4.2

(6.5)

Imperfect CSI

In this section, the assumption of perfect CSI in the preceding section is relaxed, and the effect of imperfect CSI on interference cancelation is characterized. In Section 6.4.2, the algorithm for opportunistic CSI estimation is proposed. Based on this algorithm, the residual interference resulting from imperfect CSI is characterized in Section 6.4.2. In Section 6.4.2, the effective channel and network model in the presence of residual interference is summarized. Again, the discussion in following subsections focuses on the perspective of the typical receiving node R0 . Opportunistic CSI Estimation As discussed in Section 6.4, interference cancelation at R0 requires the estimation of CSI on the effective channels vectors corresponding to L − 1 strongest interferers of R0 . The 169

CSI estimation is facilitated by the transmission of pilot symbols from these interferers. In the CSI estimation phase, exceedingly large transmission power for pilot symbols may be required to cope with strong interferers. This motivates opportunistic CSI estimation in the absence of strong interferers. The algorithm for opportunistic CSI estimation by R0 is summarized as follows. For interference cancelation in the data reception phase, R0 must sequentially estimate the CSI of the channel vectors {hn | 1 ≤ n ≤ L−1} corresponding to the interfering nodes. To reduce CSI inaccuracy, R0 estimates a specific channel vector opportunistically whenever no interferers are present within the distance of rmin from R0 . The threshold rmin is chosen such that interference is sufficiently attenuated by channel path loss. Here, we assume that R0 tracks the positions of nearby interferers. Furthermore, the feasibility of the above opportunistic CSI estimation method relies on the following important assumption. Assumption 11 The channel coherence time is much longer than the transmission duration of each transmitting node. This assumption ensures that the delay due to opportunism of CSI estimation does not cause the CSI to be outdated. This assumption is valid for a network with bursty traffic and relatively low mobility. Furthermore, the above assumption leads to fixed variance for the residual interference due to the CSI error as shown in the next section, and thereby simplifies analysis5 . Last, Assumption 11 results in small instantaneous transmission power for CSI training signals since their energy can be widely spread in time. Based on the above opportunistic CSI estimation, the model of an estimated channel vector is constructed as follows. Consider the scenario that R0 estimates the effective ˜ k corresponding to the transmitter Tk , where the fading comchannel vector hk = rk−α h ˜ k is an i.i.d. CN (0, 1) vector (cf. Section 6.4.1). The nodes interfering with this ponent h 5

Assumption 1 is invalid for a network with lasting traffic flows or high mobility. For this case, the relaxation of Assumption 1 complicates analysis by causing the variance of the residual interference to be a random variable. Accordingly, the residual interference can be treated in the same way as other interference components, which also have random variance (cf. Lemma 25).

170

CSI estimation process are grouped into a point process Υ = {Tn | Tn ∈ Φ/{T0 , Tk }, rn ≥ rmin }

(6.6)

where rmin is the distance threshold for opportunistic CSI estimation as discussed earlier. The received training signal at R0 is given as q Z=

˜ k uT + PP rk−α h

p

PD Q

(6.7)

where PP and PD are transmission power for pilot and data symbols. The M × 1 vector u represents a pilot sequence with unit energy, namely that |um |2 =

1 M

and kuk2 = 1.

The pilot symbols in u are scattered over multiple transmission durations of Tk and hence encounters different realizations of the interferer process Υ in (6.6), over which the CSI, ˜ k , remains constant (cf. Assumption 11). The L × M matrix Q represents namely rk and h the interference from interfering nodes in Υ in (6.6) with the mth column given as qm =

X p ˜ n xn rn−α h

(6.8)

Tn ∈Υm

where Υm is a realization of Υ in the mth pilot symbol duration, and xn = CN (0, 1) is the complex Gaussian signal transmitted by the interferer Tn . By using least-squares CSI ˆ n , is given as6 estimation, the estimated channel vector, denoted as h ˆk = h

Zu∗ q rk−α PP

˜ k + IΥ = h where Z is in (6.7), and the channel estimation error vector IΥ is given by s M rkα PD X X p −α IΥ = rn hn xn u∗m . PP

(6.9)

(6.10)

m=1 Tn ∈Υm

Last, upon the completion of CSI estimation, R0 computes the receive beamformer ˆ n | 1 ≤ n ≤ L − 1}. Subsesuch that it is orthogonal to the estimated channel vectors {h 6

Alternatively, MMSE CSI estimation can be used, which, however, complicates notation.

171

quently, R0 performs interference cancelation as described in Section 6.4. Nevertheless, the imperfect CSI causes incomplete interference cancelation and hence residual interference from the nodes {Tn | 1 ≤ n ≤ L − 1}, which is characterized in the next section. Residual Interference In this section, we analyze the distribution of residual interference from the nodes {Tn | 1 ≤ n ≤ L − 1} after interference cancelation with imperfect CSI. The analysis is based on the model of estimated CSI constructed in the preceding section. The distribution of the CSI estimation error IΥ in (6.10) is derived and shown in the following lemma. Lemma 23 For a long sequence of pilot symbols M → ∞, IΥ converges in distribution to an i.i.d. vector of CN (0, σk2 ) components where σk2 =

rkα PD 2πλ × α−2 . α − 2 rmin PP

Proof: See Appendix 6.9.1.

(6.11) ¤

From the proof, the exact condition for the above convergence of IΥ is that the pilot sequence is sufficiently long and widely scattered in time to attain the ergodicity of the interferer process Υ defined in (6.6). This is feasible given Assumption 11. Therefore, the CSI error vector IΥ is assumed hereafter to have the distribution as specified in Lemma 23. A few remarks on the variance of CSI error are in order. First, from (6.11), σk2 can be decreased by increasing either the ratio of data and pilot transmission power PP /PD , or the distance threshold rmin for opportunistic CSI estimation. The latter is preferred because larger pilot transmission power leads to additional interference between nodes. Nevertheless, increasing rmin requires a larger difference in time scale between channel and traffic to minimize the impact of the resultant longer estimation delay on CSI accuracy. Second, σk2 decreases with rk , the distance of the node targeted in CSI estimation process. In other words, stronger interference channel leads to smaller CSI estimation error. Last, the variance of CSI error is proportional to the node density λ. Next, based on the above results, the residual interference power from L − 1 strongest interferers of R0 is readily obtained as follows. By using (6.9) and Lemma 23, 172

the residual interference from the nodes {Tn | 1 ≤ n ≤ L − 1} to R0 is written as Ir =

L−1 X

v0† hn =

n=1

L−1 X

³ ´ ˆ n − σn pn rn−α/2 v0† h

(6.12)

n=1

where v0 is the receive beamformer at R0 , σn is given in (6.11), pn is an i.i.d. CN (0, 1) vector, and pn and pm for n 6= m are independent. Following the interference cancelation ˆ n for all 1 ≤ n ≤ L − 1. Thus, from (6.12) algorithm, v0 ⊥ h Ir =

L−1 X

(a)

−rn−α/2 σn v0† pn =

√ L − 1rn−α/2 σn ν

(6.13)

n=1

where ν is a CN (0, 1) random variable, and the equality (a) follows from the independence between v0 and pn . From (6.11) and (6.13), the main result of this section is obtained and summarized in the following theorem. Theorem 10 The aggregate residual interference from the nodes {Tn | 1 ≤ n ≤ L − 1} to 2 ) random variable where the variance σ 2 is given as R0 is a CN (0, σR R 2 σR =

2π(L − 1)λ PD × α−2 . α−2 PP rmin

(6.14)

2 is written as σ 2 = ηλ where η is defined as the normalized For convenience, the variance σR R

residual interference power. In the subsequent analysis, η is used as a parameter for controlling the amount of residual interference. For a given value of η, the combination of the parameters PD , PP , and rmin should be optimized based on the network configuration, which is outside the scope of this chapter. Last, it is interesting to observe that the residual 2 is independent of the interference channels h , h , · · · , h interference power σR 1 2 L−1 .

Effective Channel and Network Models As illustrated in Fig. 6.1(b), the effective channel and network models for interference cancelation with imperfect CSI is identical to that for the case of perfect CSI except for the additional residual interference from the nodes {Tn | 1 ≤ n ≤ L − 1}. For the present case, the SIR in (6.5) is modified as (Imperfect CSI) SIR0 = 173

2 + σR

d−α W P∞

. −α n=L rn ρn

(6.15)

It is worth mentioning that the present model of residual interference is more accurate than that in [115] for SIC with imperfect CSI. In [115], the residual interference P −α where the parameter 0 ≤ z ≤ 1 controls the degree of power is modeled as z L n=1 rn CSI accuracy. The overhead and algorithm for CSI estimation considered in the present model are not accounted for in [115].

6.5

Outage Probability and Transmission Capacity: Perfect CSI

This section focuses on the analysis of the outage probability for a SIR constraint and the related network transmission capacity for the case of prefect CSI. In Section 6.5.1, some useful auxiliary results are introduced. Using these results, bounds on the outage probability are obtained in Section 6.5.2. Based on these bounds, the scaling law of transmission capacity is derived for the regime of small outage probability in Section 6.5.3.

6.5.1

Auxiliary Results

To facilitate analysis, the interference nodes of R0 after perfect interference cancelation, namely {Tn | n ≥ L}, are separated into the strongest interferer TL and others {Tn | n ≥ L + 1}, referred to respectively as the primary and the secondary interferers. For convenience, denote the random interference power from TL as G = IL . There are two reasons for the above separation of the interferers. First, considering the strongest interferer TL alone yields a lower bound for the outage probability to be derived in the next section. Second, the separation of interferers provides a useful result that conditioning on G, the secondary interferers {Tn | n ≥ L + 1} form a Poisson point process as shown shortly. The result stated above is obtained by using the Marking Theorem [141]. To apply this theorem, a marked point process is defined for the secondary interferers, where the mark of the node Tn is the corresponding interference power In . Specifically, conditioning on the interference power of the primary interferer G = g, the desired marked point process 174

is Π(g) = {(Tn , In } | Tn ∈ Φ/{T0 }, 0 ≤ In < g}.

(6.16)

where Φ is the homogeneous Poisson point process modeling all active transmitters (cf. Section 6.3.1). Note that conditioning on G = g, the marks for different nodes are independent. Given this condition, the result in the following lemma directly follows from the Marking Theorem. Lemma 24 The process Π is a Poisson point process with the average number of nodes given by Z∞ Zg µ(g) = 2πλ rp(r, dI)dr. (6.17) 0

0

This result is useful for analyzing the aggregate interference from the secondary interferers to R0 . Conditioned on G = g, this interference is written as X

IΠ (g) =

In .

(6.18)

(Tn ,In )∈Π? (g)

Next, the distribution of the interference power from both the primary and the secondary interferers are characterized in the following lemma. Lemma 25 After perfect interference cancelation, the interference at R0 has the following properties. (a) The interference power of the primary interferer G has the following probability density function ³ ´ δcL λL g −δL−1 exp −c1 λg −δ (6.19) fG (g) = 1 Γ(L) where δ =

2 α

and c1 = πPt Γ(δ + 1).

(b) Conditioned on G = g, the mean and variance of the aggregate interference power from the secondary interferers are given by E[IΠ (g)] = Var(IΠ (g)) = Proof: See Appendix 6.9.2.

2πλΓ(δ + 1) 1−δ g α−2 πΓ(δ + 1)λ 2−δ g . α−1

(6.20) (6.21) ¤

175

Note that IΠ (g) is known as a shot noise process [141]. The probability density function of IΠ (g) is difficult to derive. In general, the distribution function of a shot noise process has no closed-form expression except for some simple cases [110, 139].

6.5.2

Bounds on Outage Probability

In this section, bounds on the outage probability are obtained for interference cancelation with perfect CSI. From (6.5) and the separation of interferers in the preceding section, the outage probability Pout can be written as Pout (λ) = E [Pr(SIR0 ≤ θ)] £ ¤ = E Pr(IΠ (g) ≥ wθ−1 d−α − g | g, w) | G = g, W = w

(6.22)

where W is the fading component of the data link power for R0 (cf. Section 6.4.1), G and IΠ (g) are respectively the interference power of the primary and secondary interferers. The direct analysis of the exact outage probability by using (6.22) is infeasible due to the difficulty in deriving the distribution function of IΠ (g). Therefore, we resolve to obtaining bounds for Pout , following the approaches in [104, 110, 115]. The expression of the outage probability in (6.22) can be rewritten as £ ¤ Pout (λ) =E Pr(IΠ (G) ≥ W θ−1 d−α − G | G, W ) | W G−1 > θdα × Pr(W G

−1

α

−1

≥ θd ) + Pr(W G

(6.23)

α

≤ θd ).

Thus, a lower bound of Pout is given as Pout (λ) ≥ Pr(W G−1 ≤ θdα ).

(6.24)

This lower bound considers only the primary interference, and hence is tight if the primary interferer TL is the dominant source of interference. Next, an upper bound of the outage probability can be derived by applying the following Chebyshev’s inequality on (6.23) ½ Pr(IΠ (g) ≥ a) ≤ min

¾ Var(IΠ (g)) ,1 . {a − E [IΠ (g)]}2

176

(6.25)

Based on (6.23), (6.24) and (6.25), bounds on the outage probability are derived as shown in the following proposition. Proposition 12 For perfect CSI, the bounds on the outage probability are given as follows. 1. The lower bound is L Pout (λ) =

h ³ ´i 1 E γ L, c2 λW −δ Γ(L)

(6.26)

where c2 = πΓ(1 + δ)θδ d2 , and W has the probability density function in (6.4). 2. The upper bound is £ ¤ U L L Pout (λ) = Pout (λ) + 1 − Pout (λ) Pα (λ)

(6.27)

where ½ Pα (λ) = E and c3 =

2πΓ(δ+1) α−2

·

¸¯ ¾ ¯W c3 λG2−δ α ¯ min ,1 ¯ >d θ (d−α θ−1 W − G − c4 λG2−δ )2 G

and c4 =

(6.28)

πΓ(1+δ) α−1 .

Proof: Seep Appendix 6.9.3.

¤

The above bounds on Pout do not provide simple closed-form expressions in terms of the node density λ. The difficulty in deriving such closed-form expression is mainly due to the existence of multiple random variables, namely W , G and IΠ (G), which jointly determine the outage probability. The tightness of the above bounds on Pout is evaluated using simulation in Section 6.7.

6.5.3

Asymptotic Transmission Capacity

In this section, the scaling law for transmission capacity is derived for small target outage probability (² → 0) and perfect CSI. This scaling law also accurately characterizes transmission capacity in the non-asymptotic outage regime (up to 0.1) as shown by simulations in Section 6.7. Small target outage probability results in a network of sparse transmitting nodes (i.e. λ → 0). For such a sparse network, the useful relationship between the outage probability and node density is derived and shown in the following lemma. 177

Lemma 26 For perfect CSI and λ → 0, the outage probability scales with λ as follows. 1. For L ≤ α, κ1 ≤ lim

λ→0

where κ1 =

Γ(1−δL,βdα )[πΓ(δ+1)θδ d2 ]L Pt Γ(L+1)

2. For L > α, κ1 ≤ lim

λ→0

where κ3 =

Pout (λ) ≤ κ1 (1 + κ2 ) λL ³ ´ δL α and κ2 = 2L α−2 − 2−δ .

Pout (λ) , λL

lim

λ→0

(6.29)

Pout (λ) ≤ κ3 λα

(6.30)

8(c1 d2 θδ )α Γ(−1,βdα )Γ(L−α+1) . (α−2)Pt Γ(L)

Proof: See Appendix 6.9.4.

¤

Using Lemma 26 and the definition of transmission capacity in (6.3), the main result of this section is obtained and summarized in the following theorem. Theorem 11 For perfect CSI and small target outage probability ² → 0, the transmission capacity scales as 1. For L ≤ α, lim

²→0

C(²) 1 −L

[κ1 (1 + κ2 )]

²

1 L

≥ 1,

C(²)

lim

²→0

−1

1

≤1

(6.31)

κ1 L ² L

where κ1 and κ2 are specified in Lemma 26. 2. For L > α, lim

C(²)

1 ²→0 − α 1 κ3 ² α

≥ 1,

lim

²→0

C(²) −1

1

≤1

(6.32)

κ1 L ² L

where κ3 is given in Lemma 26. The above theorem shows that as the target outage probability decreases, transmission capacity grows following the power law a²t where a and t are constants. For L > α, only bounds of the exponent t are known. The derivation of the exact exponent requires a more accurate upper bound on outage probability than that based on Chebyshev’s inequality in (6.25). This may require analyzing the distribution function of the secondary interference power (cf. Section 6.4.1), which, however, has no closed-form expression for the present case [149]. For L ≤ α, the exponent of the transmission capacity power law a²t is shown in 1

−1

Theorem 11 to be t = 1/L, and α is bounded as [κ1 (1 + κ2 )]− L ≤ α ≤ κ1 L . This power 178

law indicates that the size of antenna array L determines the sensitivity of transmission capacity to the change on the outage constraint. To facilitate our discussion, rewrite the 1 scaling law in Theorem 11 as C(²) ∼ = α² L where “∼ =” represents asymptotic equivalence

for ² → 0. Moreover, consider two sets of values (C1 , ²1 ) and (C2 , ²2 ), and define the C1 logarithmic ratios ∆C = log C and ∆² = log ²²12 . Using this notation, the above scaling 2

law can be simplified as ∆C ∼ 1 = . ∆² L The above quantity

∆C ∆²

(6.33)

represents the sensitivity of transmission capacity towards the

change of the outage constraint. Its value decreases inversely with the size of antenna array. Specifically, computed using (6.33), a hundred-time decrease on ² reduces network transmission capacity by {10, 3.2, 1.8} times for L = {2, 4, 8}, respectively. For the extreme case of L = ∞, transmission capacity is independent of the outage constraint since

∆C ∆²

= 0.

Last, from simulation results in Section 6.7, the capacity scaling law in Theorem 11 is observed to also hold in the outage regime of practical interest (² ≤ 0.1).

6.6

Outage Probability and Transmission Capacity: Imperfect CSI

In this section, the SIR outage probability and transmission capacity are analyzed for the case of imperfect CSI discussed in Sections 6.6.1 and 6.6.2, respectively. Furthermore, in Section 6.6.3, we analyze the power ratio between residual interference due to imperfect CSI and other types of interference. Such analysis provides insights into the required amount of overhead for CSI estimation.

6.6.1

Bounds on Outage Probability

In this section, bounds on outage probability are obtained for the case of imperfect CSI. The procedure of deriving the outage probability bounds is similar to that in Sec-

179

tion 6.5. From (6.15), the SIR outage probability is given as µ Pout (λ) = Pr

W d−α 2 + G + I (G) ≤ θ σR Π

¶ (6.34)

2 is the power of the residual interference given in (6.14), and other variables are where σR

identical to those in (6.22). To obtain its bounds, the outage probability in (6.34) can be decomposed as µ Pout (λ) = Pr | µ Pr |

¯ ¶ µ ¶ µ ¶ ¯ W W W α α ¯ W α α 2 ≤ d θ + Pr 2 ≤ d θ ¯ σ 2 > d θ Pr σ 2 > d θ σR G + σR R R {z } | {z } p1 (λ)

p2 (λ)

¯ ¶ µ ¶ (6.35) ¯ W W W α ¯ α α 2 + G + I (G) ≤ d θ ¯ G + σ 2 > d θ Pr G + σ 2 > d θ . σR Π R {z R } p3 (λ)

Roughly speaking, the three terms in the above equation p1 , p2 and p3 correspond to outage caused by one, two and three types of interference, respectively. It follows from (6.35) that a lower bound of the outage probability is Pout (λ) ≥ p1 (λ) + p2 (λ).

(6.36)

This lower bound is tight if the summation of the residue and primary interference dominates over the secondary one (cf. Section 6.4.2). As for the case of perfect CSI, the upper bound of the outage probability is obtained by applying Chebyshev’s inequality on the term p3 (λ) in (6.35). Based on the above procedure, bounds on the outage probability are derived as shown in the following proposition. Proposition 13 For the case of imperfect CSI, the bounds on the SIR outage probability are given as follows. 1. The lower bound is

L P˜out (λ) = p1 (λ) + p2 (λ).

(6.37)

where p1 (λ) = [1 − exp(βdα − ηdα θλ)]+ and p2 (λ) =

i ¡ ¢−δ ´¯¯ 1 − p1 (λ) h ³ E γ L, πΓ(1 + δ)λ W d−α θ−1 − ηλ ¯ W > max(dα θηλ, βdα ) . Γ(L)

180

2. The upper bound is h i U L L P˜out (λ) = P˜out (λ) + 1 − P˜out (λ) pˆ3 (λ)

(6.38)

where ½ pˆ3 (λ) = E and c3 =

2πΓ(δ+1) α−2

¸¯ ¾ ¯ W c3 λG2−δ α ¯ min 2 − c λG2−δ )2 , 1 ¯ G + σ 2 > d θ (W d−α θ−1 − G − σR 4 R ·

and c4 =

πΓ(1+δ) α−1 .

The proof is similar to that for Proposition 12 and is thus omitted. The tightness of the above bounds is evaluated by using simulation in Section 6.7. In general, the presence of residual interference tightens the bounds on outage probability. The reason is that the residual interference reduces the significance of the interference from secondary interferers, which is the source of the difference between the outage probability and its bounds.

6.6.2

Asymptotic Transmission Capacity

In this section, for the regime of small target outage probability (² → 0), the scaling law of transmission capacity for imperfect CSI is show to be identical to that for perfect CSI. The main result of this section is summarized in the following theorem. Theorem 12 For imperfect CSI and small target outage probability ² → 0, transmission capacity follows the same scaling law as given in Theorem 11 for perfect CSI. Proof: See Appendix 6.9.5.

¤

Based on Theorem 12, the discussion on the result in Theorems 11 is also applicable for the present case of imperfect CSI. Theorem 12 implies that CSI inaccuracy has no effect on transmission capacity for asymptotically small target outage probability. Nevertheless, CSI inaccuracy does decrease transmission capacity for non-asymptotic target outage probability, as shown by simulation results in Section 6.7.

6.6.3

Power Ratio between Residual and Other Interference

In this section, we derive the average power ratio between the residual and other interference. This result allows the computation of the required value of the normalized residual 181

interference power η (cf. Section 6.4.2). The value of η in turn provides estimation of the required overhead for CSI estimation such as pilot transmission power and training duration. The main result of this section is given in the following proposition. Proposition 14 The average power ratio between the residual interference due to imperfect CSI and other interference components is given as V =

2 α σR = c6 ηλ1− 2 E [G + IΠ (G)]

(6.39)

2 /λ and σ 2 is given in (6.14) and the constant where η = σR R

c6 =

(L − 1)Γ(L) h ¡ ¢ ¡ δ [πΓ(δ + 1)] Γ L − 1δ + 1−δ Γ L− 1 δ

1 δ

¢i . +1

(6.40)

The detailed proof is omitted as it is similar to that for the second property of Lemma 25. In practice, small values of V are preferred for ensuring the effectiveness of interference cancelation. From (6.39), decreasing V reduces η linearly, leading to more CSI estimation overhead including larger pilot transmission power or longer estimation delay (cf. Section 6.4.2). In Fig. 6.2, the normalized residual interference power η is plotted for an increasing node density and V = {0.1, 1}. As observed from Fig. 6.2, the required value of η for a given power ratio V decreases with the node density7 , implying larger CSI estimation overhead. The reason is that the power of other types of interference is smaller as the node density decreases. Correspondingly, smaller η and hence more accurate CSI estimation is required to prevent residual interference from becoming a dominant interference source.

6.7

Simulation and Discussion

In this section, the bounds on outage probability and the network transmission capacity are evaluated by using Monte Carlo simulation in Section 6.7.1 and 6.7.3, respectively. 7

A smaller node density corresponds to a more small target outage probability.

182

−5 V =1

Normalized Residual Interference Power (dB)

−10

V = 0.1

−15

−20

−25

−30

−35

−40 −3 10

−2

10 NodeDensity

10

−1

Figure 6.2: Normalized residual interference power for different transmitting node densities. The number of antennas per node is L = 4 and the path loss exponent is α = 4.

6.7.1

Bounds on Outage Probability

In this section, the bounds on outage probability in Proposition 12 and 13 are compared with values computed using Monte Carlo simulation. The procedure for simulating a MANET follows that in [150]. For perfect CSI, the bounds on outage probability from Proposition 12 and simulated values are compared in Fig. 6.3. The path loss exponent is α = 4 and the number of antenna per node is L = {2, 4}. Two observations can be made from Fig. 6.3. First, the bounds for L = 2 are tighter than those for L = 4. Second, the bounds and the simulated values of the outage probability converge as the transmitting node density λ decreases. These two observations can be explained by the dominance of the primary interference over the secondary one as L increases or λ decreases, where the secondary interference causes the looseness of the bounds on outage probability. Last, the outage probability is approximately proportional to λL . For the case of imperfect CSI, the SIR outage probability obtained by simulation as well as the bounds in Proposition 13 are plotted in Fig. 6.4 for different transmitting node 183

0

10

Simulation Lower bound Upper bound −1

Outage probability

10

−2

10

−3

10

−4

10

−4

10

−3

−2

10 10 Contending node density

−1

10

(a) Perfect CSI, L = 2 0

10

Simulation Lower bound Upper bound −1

Outage probability

10

−2

10

−3

10

−4

10

−2

10 Contending node Density

−1

10

(b) Perfect CSI, L = 4

Figure 6.3: Outage probability for different transmitting node densities and perfect CSI. The size of the antenna array is (a) L = 2 and (b) L = 4

184

densities. The number of antennas per node is L = {2, 4}, and the normalized power of residual interference is η = {−10, −15} dB. Two observations are made. First, for imperfect CSI, the bounds on outage probability are tighter than those for perfect CSI. This is particularly obvious by considering L = 4 and comparing Fig. 6.4(b) and Fi.g. 6.3(b). The reason is that for relative large values of L and η, the residual interference due to imperfect CSI is the dominant interference source, reducing the looseness of bounds contributed by the secondary interferers. Second, larger residual interference power, correspondingly to less accurate CSI, causes higher outage probability. Nevertheless, the effect of CSI inaccuracy diminishes as the transmitting node density decreases, reflected in the convergence of curves for different levels of residual interference power. Last, by comparing Fig. 6.4 and 6.3, the effect of CSI inaccuracy is negligible for (η = −14 dB, L = 2) and (η = −20 dB, L = 4). This indicates interference cancelation with larger antenna arrays is more sensitive to residual interference because the primary and secondary interference is weaker.

6.7.2

Scaling Laws of Transmission Capacity

In this section, from simulation results, the asymptotic capacity scaling laws derived in Sections 6.5.3 and 6.6.2 are observed to accurately characterize transmission capacity in the non-asymptotic regime of the outage constraint. In Fig. 6.5, asymptotic bounds on transmission capacity in Theorem 11 are compared with the exact values obtained by simulation for perfect CSI and the range of target outage probability ² ∈ [10−5 , 0.1]. As observed from Fig. 6.5, the asymptotic upper bound on transmission capacity is very tight for L = {2, 3} even in the non-asymptotic range e.g. ² ∈ [0.01, 0.1]. The tightness of this bound is due to the dominance of primary interference for interference cancelation with small sizes of antenna array. For L = 4, both the asymptotic lower and upper bounds are tight. The tightness of asymptotic bounds implies the slopes of the transmission capacity vs. target outage probability curves are approximately equal to

1 L.

In other words, the reduction of transmission capacity by an

order of magnitude corresponds to the decrease of the constrained outage probability by 185

10

0

Simulation Lower bound Upper bound

Outage probability

10

10

10

10

η=-8 dB

−1

−2

η=-14 dB

−3

−4

10

−5

10

−4

−3

10 Transmitting node density

10

−2

(a) Imperfect CSI, L = 2 10

0

η=-12 dB

Outage probability

10

10

−1

−2

η=-20 dB

10

10

−3

−4

Simulation Lower bound Upper bound 10

−3

−2

10 Transmitting node density

(b) Imperfect CSI, L = 4

Figure 6.4: Outage probability for different node densities and imperfect CSI. The size of the antenna array is (a) L = 2 and (b) L = 4

186

10

−2

Simulation Asymptotic Upper Bound Asymptotic Lower Bound

Transmission capacity

L=4 10

−3

L=3

10

−4

L=2

10

−5

10

−6

10

−5

−4

−3

10 10 Target outage probability

10

−2

10

−1

Figure 6.5: Comparison between asymptotic bounds on transmission capacity and the exact values obtained by simulation perfect CSI and the size of antenna array L = {2, 3, 4} four-order of magnitude for L = 4, or two-order for L = 2. The above results demonstrate the usefulness of the derived scaling laws for estimating transmission capacity. Fig. 6.6 compares asymptotic bounds on transmission capacity in Theorem 11, identical for both perfect and imperfect CSI, with the exact values obtained by simulation for imperfect CSI for the range of target outage probability ² ∈ [10−5 , 0.1]. The size of antenna array is L = {2, 4}, and the normalized power of residual interference is η = {−10, −15, −20} dB. Two observations can be made from Fig. 6.6. First, regardless of the residual interference power, transmission capacity converges either the upper or the lower asymptotic bound as the target outage probability decreases. This observation agrees with Theorem 12. Moreover, faster convergence is observed for a smaller size of antenna array (L = 2) or smaller residual interference power e.g. η = −20 dB. Second, CSI inaccuracy reduces transmission capacity. This effect of CSI inaccuracy is observed to be more significant for interference cancelation with a larger size of antenna array (L = 4 for this comparison). Despite the negative effect of CSI inaccuracy, spatial interference cancelation still yields large gains on transmission capacity as shown in the next section. 187

10

−2

Transmission capacity

Asymptotic Upper Bound Asymptotic Lower Bound η = −20 dB η = −15 dB η = −10 dB 10

10

10

−3

−4

−5

10

−6

10

−5

−4

−3

10 10 Target outage probability

10

−2

10

−1

(a) Imperfect CSI, L = 2

10

−2

Transmission capacity

Asymptotic Upper Bound Asymptotic Lower Bound η = −20 dB η = −15 dB η = −10 dB

10

−3

10

−6

10

−5

−4

−3

10 10 Target outage probability

10

−2

10

−1

(b) Imperfect CSI, L = 4

Figure 6.6: Comparison between asymptotic bounds on transmission capacity and the exact values obtained by simulation for imperfect CSI, (a) the size of antenna array L = 2 and the normalized power of residual interference η = {−10, −15, −20} dB, and (b) L = 4 and η = {−10, −15, −20} dB.

188

Transmission Capacity

10

10

10

−2

−3

−4

ε = 0.1 ε =1e−2 ε =1e−3 10

−5

1

2

3

4 5 6 7 Number of Antennas per Node

8

9

10

Figure 6.7: Transmission capacity by simulation for different node densities and perfect CSI. The size of the antenna array is L = 4 and the outage constraint is ² = {10−1 , 10−2 , 10−3 }.

6.7.3

Transmission Capacity vs. Size of Antenna Array

In Fig. 6.7, the transmission capacity is plotted for an increasing number of antennas per node assuming perfect CSI. Furthermore, different outage constraints, namely ² = {10−1 , 10−2 , 10−3 }, are considered. From Fig. 6.7, the following observations are made. First, the use of multiple antennas for interference cancelation leads to the increase in transmission capacity by an order of magnitude or more with respect to the case of single-antenna per node. This capacity gain is especially large for a small number of antennas and small target outage probability. For example, for ² = 10−1 , the use of three antennas per node provides transmission capacity seven times of that for the singleantenna case. The capacity gain by using additional antennas diminishes rapidly as the number of antennas per node increases. Second, the outage constraint affects transmission capacity significantly for a small number of antennas per node. Nevertheless, transmission capacity becomes insensitive to the change on the outage constraint as the number of antennas increases. 189

The effect of imperfect CSI on transmission capacity is shown in Fig. 6.8, where transmission capacity is plotted for different numbers of antennas per node L. The size of the antenna array is L = 4 and the outage constraint is ² = 10−2 . The residual interference due to imperfect CSI has the normalized power of η = {−10, −15, −20} dB. Several observations can be made from Fig. 6.8. First, the power of residual interference impacts transmission capacity for L ≥ 2. As an example, for L = 9, the increase of η from −20 to −10 dB reduces transmission capacity by six times. This suggests the importance of accurate CSI for interference cancelation. Second, even for a relatively small amount of CSI estimation overhead (i.e. η = −10 dB), a capacity gain of more than an order of magnitude can be achieved using interference cancelation. This supports the practical applications of interference cancelation. Third, most capacity gains are contributed by the cancelation of the strongest interferer to each receiving node. The cancelation of other interferers has a much less significant effect on the network capacity Third, most capacity gains are contributed by the cancelation of the strongest interferer to each receiving node. The cancelation of more interferers has a much less significant effect on the network capacity since it becomes limited by residual interference.

6.8

Summary

In this chapter, the spatial interference cancelation algorithm have been applied and the resultant network capacity gains have been characterized. To enable interference cancelation, an opportunistic CSI estimation algorithm have been designed, which trades estimation delay for smaller pilot transmission power. To investigate network capacity for interference cancelation, the SIR outage probability has been analyzed, which is useful for computing network transmission capacity. For asymptotically small outage probability, the scaling laws of transmission capacity have been derived, which follow the power law for both perfect and imperfect CSI. These scaling laws also accurately predict transmission capacity for the non-asymptotic outage regime. Through simulation, interference cancelation has been observed to provide significant network capacity gains even for just a few 190

Transmission capacity

10

10

10

−2

−3

η = −10 dB η = −15 dB η = −20 dB

−4

1

2

3

4 5 6 7 Number of antennas per node

8

9

10

Figure 6.8: Transmission capacity by simulation for different node densities and imperfect CSI. The size of the antenna array is L = 4 and the outage constraint is ² = 10−2 . (2-3) antennas per node. As also observed, network capacity is sensitive to the inaccuracy of CSI, indicating the importance of accuracy CSI for interference cancelation.

191

6.9 6.9.1

Appendix Proof of Lemma 23

For M pilot symbols scattered in time, the lth component of IΥ in (6.10), denoted as Il (M ), can be written as s Il (M ) =

M X X rkα PD × PP

s

m=1 Tn ∈Υm

rn−α Hn,l Xn M

(6.41)

where Hn,l = CN (0, 1) is the lth component of the channel vector hn , and Xn = CN (0, 1) is the data symbol xn rotated by a random phase µ∗m M . Because symbols {Xn } are independent Gaussian random variables, Il (M ) can be re-written as Il (M ) =

q ˜l σl2 (M )X

(6.42)

˜ l = CN (0, 1), and σ 2 (M ) is given as where X l σl2 (M ) =

M rα PD 1 X X −α rn ρn,l × k . M PP

(6.43)

m=1 Tn ∈Υm

where ρn,l = |Hn,l |2 is an random variable following the exponential distribution with unit variance. For M → ∞, the variance of different components of Il (M ) converges to an identical value. The asymptotic value of σl2 (M ) is " σl2

= lim

M →∞

σl2 (M )

=E

X

# rn−α ρn,l

Tn ∈Υ

×

rkα PD PP

(6.44)

where the expectation is taken over the interferer process Υ. From the definition, Υ is a Poisson point process with the node density λ. Thus, by applying Campbell’s theorem [141] on (6.44) Z σl2



= 2πλ rmin

=

r1−α E[ρn,l ]dr ×

2−α 2πλrmin rα PD × k . α−2 PP

192

rkα PD PP (6.45)

Because σl2 is independent of the index l, the components of IΥ for M → ∞ follow the same distribution. Next, for M → ∞, the components of IΥ are shown to be asymptotically independent. For detection, the time scale of interest is a finite number of symbol durations. Let ES denote the expectation corresponding to a time duration sufficiently long for a random data symbol to achieve ergodicity. The covariance of two components, Ia (M ) and Ib (M ) where a 6= b, for M → ∞ is · ES

"

¸ lim

M →∞

Ia (M )Ib∗ (M )

=

M M X p X rα PD X X p −α ∗ lim k rn Hn,a Xn rz−α Hz,b Xz∗ M →∞ M PP

ES

m=1 Tn ∈Υm

(a)

=

=

n=1 Tz ∈Υn

M £ ¤ rkα PD X X −α ∗ rn Hn,a Hn,b ES |Xn |2 M →∞ M PP m=1 Tn ∈Υm " # X rkα PD ∗ E rn−α Hn,a Hn,b . M PP

lim

(6.46)

Tn ∈Υ

It is important to note that the equality (a) holds because based on Assumption 11 channels remain constant in the ergodic duration for the expectation ES . From (6.46) " # · ¸ X rkα PD (b) ∗ −α ∗ ES lim Ia (M )Ib (M ) = E rn Hn,a Hn,b = 0. M →∞ PP Tn ∈Υ

∗ are independent. This completes the proof. The equality (b) holds because Hn,a and Hn,b

6.9.2

#

Proof of Lemma 25

Define a marked Poisson point process as M(g) = {(Tn , In } | Tn ∈ Φ/{T0 }, In ≥ g}. Given that In = rn−α ρn , the node density of M(g) follows from the Marking Theorem as Z

∞ Z (t/g)1/α

µ(M(g)) = λ 0

= πλg

0

Z −δ

2πrfρ (t)drdt ∞

tδ e−t dt,

0

= πΓ(δ + 1)λg −δ .

193

δ :=

2 α (6.47)

The cumulative density function of G is the probability that the number of nodes in the subset M(g) is no more than L − 1. Thus, Pr(G ≤ g) =

L−1 X k=0

¡ ¢k ³ ´ c1 λg −δ exp −c1 λg −δ . k!

(6.48)

The probability density function of G is obtained by differentiating the above function. Using Campbell’s theorem[140], the expressions for Var(IΠ (g)) and E[IΠ (g)] are obtained as Z E[IΠ | g] = 2πλ =

0

Z = 2πλ

0

6.9.3

ρ g

r1−α ρe−ρ drdρ

2πλΓ(δ + 1) 1−δ g . α−2

Var(IΠ | g) = 2πλ

=

³ ´1/α

0

Z

=

∞Z ∞

∞Z ∞ ³ ´1/α ρ g

∞Z ∞ ³ ´1/α ρ g

(6.49)

¡ ¢2 r r−α ρ e−ρ dρdr r1−2α ρ2 e−ρ drdρ

Z πλ 2−δ ∞ δ −ρ g ρ e dρ α−1 0 πΓ(δ + 1)λ 2−δ g . α−1

(6.50)

Proof of Proposition 12

L . Thus, P L = The lower bound of the outage probability in (6.24) is denoted as Pout out ¡ ¢ ˆ = G−1 and Q = W G−1 . It Pr W G−1 ≤ θdα . Define two new random variables G

ˆ is follows from (6.19) that the probability density function of G fGˆ (g) =

³ ´ L δL−1 δcL 1λ g exp −c1 λg δ . Γ(L)

(6.51)

The probability density function of the product random variable Q can be written as [151] Z fQ (q) =

0



fW (w) ³ q ´ fGˆ dw w w

194

(6.52)

where fW (w) and fGˆ are the probability density functions of the random variables W and ˆ respectively. Thus, G, Z L Pout =

θdα

0

Z



= 0

fQ (q)dq Z α ³q´ fW (w) θd fGˆ dq dw. w w 0 | {z }

(6.53)

Π(w)

By substituting(6.51), the term Π(w) as defined above is simplified as Π(w) = = =

1 Γ(L)

Z 0

θdα

L δcL 1λ

Z

³ q ´δL−1 w

³ ´ exp −c1 λw−δ q δ dq

³ ´ L 1−δL δcL 1λ w q δL−1 exp −c1 λw−δ q δ dq Γ(L) 0 w δ 2 γ(L, c1 θ d λw−δ ). Γ(L) θdα

(6.54) (6.55)

Substituting (6.55) into (6.53) gives the desired lower bound for the outage probability. The upper bound of the outage probability is obtained by combining Lemma 25, (6.23), and (6.25).

6.9.4

Proof of Lemma 26

Lower Bound on Outage Probability Consider the lower bound of the outage probability given in Proposition 12 and obtained in the preceding section. By expanding the integrand in (6.54) around λ = 0 using Taylor’s series Π(w) = =

L 1−δL Z θdα δcL 1λ w q δL−1 dq + O(λL+1 ) Γ(L) 0 £ ¤L πΓ(δ + 1)θδ d2 λL w1−δL + O(λL+1 ) Γ(L + 1)

(6.56)

By substituting (6.56) into (6.53) L Pout (λ) = κλL + O(λL+1 )

where κ is given in Lemma 26. 195

(6.57)

Upper Bound on Outage Probability The asymptotic expansion of the upper bound on the outage probability in Proposition 12 is obtained as follows. To simplify notation, define B = d−α θ−1 W . The term in (6.28) can be written as Pα (λ) = E[Λ(B)] where the function Λ(b) is defined below ½ Λ(b) = E

·

¸¯ ¾ ¯ c3 λG2−δ ¯ , 1 ¯ G < b Pr(G < b). min (b − G − c4 λG2−δ )2

It can be expanded as8 ¸¯ ¾ µ ¶ ¯ c3 λG2−δ b b ¯ ,1 ¯G ≤ E min Pr G ≤ + (b − G − c4 λG2−δ )2 2 2 ½ · ¸¯ ¾ µ ¶ ¯b c3 λG2−δ ¯ < G < b Pr b < G < b , 1 E min ¯2 (b − G − c4 λG2−δ )2 2 ½ · ¾ µ ¶ µ ¶ ¸¯ 2−δ ¯ c3 λG b b b E min Pr G ≤ + Pr α 1  1 Γ(L)  cL−1 λL−1 (b/2)−δ(L−α)   [1 + O(λ)], L ≤ α.  Λ1 (b) ≤ 1 Γ(L) 8

(6.60)

Expansion by using an alternative value other than b/2 may tighten the asymptotic upper bound on the outage probability but has no effect on the exponent of its scaling law.

196

Again, by using (6.19), Λ2 (b) = = ≤ = =

L Z b δcL 1λ g −δL−1 exp(−c1 λg −δ )dg Γ(L) b/2 Z c1 λ(b/2)−δ 1 g L−1 exp(−g)dg Γ(L) c1 λb−δ ³ ´ ³ ´i [c1 λ(b/2)−δ ](L−1) h exp −c1 λb−δ − exp −c1 λ(b/2)−δ Γ(L) o L−1 L−1 c1 λ (b/2)−δ(L−1) n c1 λb−δ (2δ − 1) + O(λ2 ) Γ(L) L −δL 2δL (1 − 2−δ )cL 1λ b + O(λL+1 ). Γ(L)

By combining (6.61), (6.59) and (6.58),  ³ ´ −2 ]c cα−1 Γ L − α + 1, c λ (b/2)−δ  4E[B  3 1 1    λα + O(λα+1 ), L > α  Γ(L) ´ E[Λ(B)] ≤ ³  α −δ 2δL cL E[B −δL ]  − 2  1 α−2   λL + O(λL+1 ), L≤α  Γ(L)

(6.61)

(6.62)

where by using (6.4) d2L θδL Γ(1 − δL, βdα ) Pt d2α θ2 Γ(−1, βdα ) . Pt

E[B −δL ] = E[B −2 ] =

(6.63)

The desired upper bound is obtained by combining (6.27), (6.57) and (6.62).

6.9.5

Proof of Theorem 12

It is sufficient to prove that Lemma 26 also holds for the case of imperfect CSI. Define ˜ = W − dα θηλ. For λ < βθ−1 η −1 , the variables in Proposition 13 a random variable W reduce to p1 (λ) = 0 p2 (λ) =

h i 1 ˜ −δ λ E γ(L, c2 W Γ(L)

197

( pˆ3 = E

) ¸¯ ˜ ¯W c3 λG2−δ min , 1 ¯¯ > dα θ . ˜ − G − c4 λG2−δ )2 G (d−α θ−1 W ·

(6.64)

By substituting above expressions into the bounds on the outage probability in Proposition 13, they are observed to have identical forms as their counterparts for perfect CSI ˜ is replaced with W . Therefore, the as given in Proposition 12 if the random variable W desired result follows by combining the proof procedure in Appendix 6.9.4 for perfect CSI ˜ and we have E[B ˜ −δL ] = E[B ˜ −δL ][1+O(λ)] ˜ = d−α θ−1 W and the following results. Define B ˜ −2 ] = E[B −2 ][1 + O(λ)], where B is defined in Appendix 6.9.4. and E[B

198

Chapter 7

Conclusion 7.1

Summary

In this dissertation, we have thoroughly studied the impact of imperfect CSI on the performance of MIMO techniques in point-to-point links, SDMA systems and mobile ad hoc networks. We have discovered that the negative effects of CSI inaccuracy on MIMO communication can be contained by carefully selecting system parameters, and jointly designing CSI quantization algorithms with other techniques such as SDMA, multiuser diversity, and spatial interference cancelation. As a result, despite CSI inaccuracy, several key MIMO techniques, namely beamforming, SDMA and spatial interference cancelation, have been shown to increase throughput significantly in cellular systems or ad hoc networks. Existing results have shown that very low-rate quantized CSI feedback enables close-to-optimal transmit beamforming. Prior work assumes simple blocking fading channel where channel temporal correlation is neglected. In this dissertation, we have considered transmit beamforming over a temporally correlated channel and investigated a set of issues related to CSI feedback. Specifically, based on a periodic feedback protocol, we have derived the CSI feedback rate as a function of channel coherence time. This result is useful for allocating bandwidth for a CSI feedback channel. We have shown analytically that the capacity gain contributed by transmit beamforming decreases with feedback delay at least at an exponential rate, which is a function of channel coherence time. Furthermore, 199

we have proposed an algorithm for compressing CSI feedback in time. This algorithm contributes a compression ratio up to 25%. Finally, we have demonstrated that different system parameters, including the feedback bit rate, feedback delay, and vehicular speed, can be jointly designed under a constraint on throughput. For downlink SDMA, CSI quantization, scheduling and beamforming have been joint designed. The proposed SDMA design has several features. Fist, this design enables a base station in cellular system to broadcast multiple spatial data streams to different subscriber units. Second, this design requires only finite-rate CSI feedback from each subscriber unit. Third, the proposed SDMA design exploits multiuser diversity for increasing downlink throughput. In particular, throughput increases at an optimal rate as the number of subscribers grows. Building on the above SDMA design, a distributed algorithm is proposed for multiuser CSI feedback in MIMO downlink. The proposed algorithm requires each user to apply thresholds on CSI for turning CSI feedback on or off. The thresholds have been designed to constrain the sum feedback rate without requiring coordination on multiuser CSI feedback. The proposed multiuser feedback algorithm is particularly useful for a SDMA system where users share a common CSI feedback channel. We have shown that the proposed feedback algorithm does not compromise the throughput of MIMO downlink despite the sum feedback constraint. The use of limited feedback for uplink SDMA has been proposed. The throughput scaling laws for uplink SDMA with limited feedback have been derived for different SNR regimes. In particular, throughput has been proved to scale with the number of users at a much faster rate at high SNRs than that at normal to low SNRs. Nevertheless, uplink SDMA has been shown to lose spatial multiplexing gain at the high SNRs. Thus, SDMA has no significant advantages over time-division multiple access (TDMA) at high SNRs due to multiuser interference. We have analyzed the capacity of a mobile ad hoc network with spatial interference cancelation. We have characterized the effect of CSI inaccuracy by deriving the 200

distribution of the resultant residual interference. This derivation has built on a proposed opportunistic algorithm for CSI estimation. For a small target outage probability, the scaling laws of network transmission capacity have been shown to scale identically for both perfect and imperfect CSI. Furthermore, we have demonstrated that spatial interference cancelation contributes significant gains on network capacity even in the presence of CSI inaccuracy.

7.2

Future Work

There are several directions for future research: Base station coordination: Base station coordination is a new breakthrough in MIMO technologies. In cellular networks, such coordination exploits the high-speed connections between base stations and clusters them to form gigantic MIMO systems. The base station coordination reduces interference between users in different cells and substantially increases the capacity of cellular networks. One key challenge in implementing this technology is the requirement of CSI feedback from each subscriber unit to multiple base stations. Consequently, the amount of CSI overhead in a cellular network potentially increases linearly with the number of coordinated base stations, reducing significantly the throughput gain. Designing suitable CSI feedback techniques for reducing CSI overhead are crucial for implementing base station coordination in practical cellular systems. MIMO cognitive ratio with imperfect CSI: Motivated by the shortage of radio spectrum, extensive research is being carried out on spectrum sensing cognitive radio for improving the efficiency of spectrum utilization. A cognitive radio device monitors the interference levels in different spectrum slots, and intelligently accesses wireless channels without interrupting the communications of other devices. Given spatial degrees of freedom, this intelligent device can exploit communication opportunities not only in spectrum but also in space. This advanced transmission technology, called MIMO cognitive radio, relies on accurate and real-time CSI of MIMO broadband channels, placing a stringent requirement on CSI acquisition. This dissertation research provides a foundation for de201

veloping efficient CSI acquisition algorithms for MIMO cognitive radio and investigating the sensitivity of such systems to CSI inaccuracy. Applications of CSI in mobile ad hoc networks: For mobile ad hoc networks, this dissertation focuses on estimating CSI of interference channels and using it for spatial interference cancelation. There also exist other types of CSI such as geographic locations of nodes, queue lengths, and the quality of received data. CSI applications include routing, scheduling, re-transmission, and traffic control. The many possible combinations of the types and applications of CSI cause uncertainty and controversy on the optimal designs of ad hoc networks. Furthermore, few results exist on efficient ways of acquiring different types of CSI and the effects of their inaccuracy on network capacity.

202

Bibliography [1] X. Qin and R. A. Berry, “Distributed approaches for exploiting multiuser diversity in wireless networks,” IEEE Trans. on Info. Theory, vol. 52, pp. 392–413, Feb. 2006. [2] “IEEE 802.16e amendment: Physical and medium access control layers for combined fixed and mobile operation in licensed bands,” IEEE Standard 802.16, 2005. [3] “IEEE 802.16 task group m (tgm),” Available: http: // www. ieee802. org/ 16/ tgm/ , 2008. [4] “Working group for IEEE 802.11 wlan standards,” Available:

http: // www.

ieee802. org/ 11/ , 2008. [5] “3GPP TR 25.814: Physical layer aspects for evolved universal terrestrial radio access (release 7),” June 2006. [6] A. Goldsmith, Wireless Communications. Cambridge University Press, 2005. [7] T. L. Marzetta and B. M. Hochwald, “Fast transfer of channel state information in wireless systems,” IEEE Trans. on Signal Processing, vol. 54, pp. 1268–78, Apr. 2006. [8] D. J. Love, R. W. Heath Jr., W. Santipach, and M. L. Honig, “What is the value of limited feedback for MIMO channels?,” IEEE Comm. Mag., vol. 42, pp. 54–59, Oct. 2004. [9] D. J. Love and R. W. Heath Jr., “Feedback techniques for MIMO channels,” in 203

MIMO Antenna Technology for Wireless Communications, (Boca Raton, FL), CRC Press Inc, 2006. [10] D. J. Love, R. W. Heath Jr., and T. Strohmer, “Grassmannian beamforming for multiple-input multiple-output wireless systems,” IEEE Trans. on Info. Theory, vol. 49, pp. 2735–47, Oct. 2003. [11] K. K. Mukkavilli, A. Sabharwal, E. Erkip, and B. Aazhang, “On beamforming with finite rate feedback in multiple antenna systems,” IEEE Trans. on Info. Theory, vol. 49, pp. 2562–79, Oct. 2003. [12] J. Choi, B. Mondal, and R. W. Heath, “Interpolation based unitary precoding for spatial multiplexing MIMO-OFDM with limited feedback,” IEEE Trans. on Sig. Proc., vol. 54, pp. 4730–40, Dec. 2006. [13] B. Mondal and R. W. Heath, Jr., “Adaptive feedback for mimo beamforming systems,” in Proc. of IEEE Workshop on Signal Processing Advances in Wireless Comm., July 2004. [14] D. Gesbert, M. Kountouris, R. W. Heath, C.-B. Chae, and T. Salzer, “Shifting the MIMO paradigm,” IEEE Signal Proc. Magazine, vol. 24, pp. 36–46, Sept. 2007. [15] Motorola, Inc, “Downlink MIMO summary,” in 3GPP TSG RAN WG1 # 49bis/R1-072693, June 2006. [16] Samsung Electronics, “Downlink MIMO for EUTRA,” in 3GPP TSG RAN WG1 # 44/R1-060335, Feb. 2006. [17] Freescale Semiconductor, “Details of zero-forcing MU-MIMO for DL EUTRA,” in 3GPP TSG RAN WG1 # 48-bis/R1-071510, Mar. 2007. [18] M. Costa, “Writing on dirty paper,” IEEE Trans. on Info. Theory, vol. 29, no. 3, pp. 439 – 441, 1983. 204

[19] G. Caire and S. Shamai, “On the achievable throughput of a multiantenna Gaussian broadcast channel,” IEEE Trans. on Info. Theory, vol. 49, no. 7, pp. 1691–1706, 2003. [20] S. Vishwanath, N. Jindal, and A. Goldsmith, “Duality, achievable rates, and sumrate capacity of Gaussian MIMO broadcast channels,” IEEE Trans. on Info. Theory, vol. 49, pp. 2658–68, Oct. 2003. [21] W. Yu and J. M. Cioffi, “Sum capacity of Gaussian vector broadcast channels,” IEEE Trans. on Info. Theory, vol. 50, no. 9, pp. 1875–1892, 2004. [22] P. Viswanath and D. Tse, “Sum capacity of the vector Gaussian broadcast channel and uplink-downlink duality,” IEEE Trans. on Info. Theory, vol. 49, pp. 1912–21, Aug. 2003. [23] W. Yu, D. P. Varodayan, and J. M. Cioffi, “Trellis and convolutional precoding for transmitter-based interference pre-subtraction,” IEEE Trans. on Communications, vol. 53, pp. 1220–30, July 2005. [24] T. Yoo, N. Jindal, and A. Goldsmith, “Multi-antenna broadcast channels with limited feedback and user selection,” IEEE Journal on Sel. Areas in Communications, vol. 25, pp. 1478–91, July 2007. [25] N. Jindal, “Antenna combining for the MIMO downlink channel,” to appear in IEEE Trans. on Wireless Communications. [26] R. Knopp and P. Humblet, “Information capacity and power control in single-cell multiuser communications,” in Proc., IEEE Intl. Conf. on Communications, vol. 1, pp. 331–5, 1995. [27] S. Verdu, Multiuser Detection. Cambridge, UK: Cambridge, 1998. [28] M. Sharif and B. Hassibi, “On the capacity of MIMO broadcast channels with partial side information,” IEEE Trans. on Info. Theory, vol. 51, pp. 506–522, Feb. 2005. 205

[29] A. Gersho and R. M. Gray, Vector Quantization and Signal Compression. Kluwer Academic Press, 1992. [30] J. C. Roh and B. D. Rao, “Efficient feedback methods for MIMO channels based on parameterization,” IEEE Trans. on Wireless Communications, vol. 6, pp. 282–292, Jan. 2007. [31] P. Xia, S. Zhou, and G. B. Giannakis, “Achieving the Welch bound with difference sets,” IEEE Trans. on Info. Theory, vol. 51, pp. 1900–07, May 2005. [32] D. J. Love and R. W. Heath Jr., “Limited feedback unitary precoding for orthogonal space-time block codes,” IEEE Trans. on Sig. Proc., vol. 53, pp. 64–73, Jan. 2005. [33] D. J. Love and R. W. Heath Jr., “Limited feedback unitary precoding for spatial multiplexing systems,” IEEE Trans. on Info. Theory, vol. 51, pp. 1967–76, Aug. 2005. [34] B. C. Banister and J. R. Zeidler, “Feedback assisted transmission subspace tracking for MIMO systems,” IEEE Journal on Sel. Areas in Communications, vol. 21, no. 3, pp. 452–63, 2003. [35] C. Simon and G. Leus, “Feedback reduction for spatial multiplexing with linear precoding,” in Proc., IEEE Int. Conf. Acoust., Speech and Sig. Proc., vol. 3, Apr. 2007. [36] K. Huang, B. Mondal, R. W. Heath, Jr., and J. G. Andrews, “Markov models for multi-antenna limited feedback systems,” in Proc., IEEE Int. Conf. Acoust., Speech and Sig. Proc., pp. IV–9–IV–12, May 2006. [37] R. Gallager, Information Theory and Reliable Communication. Wiley, 1968. [38] Y. Isukapalli and B. D. Rao, “Finite rate feedback for spatially and temporally correlated MISO channels in the presence of estimation errors and feedback delay,” in Proc., IEEE Globecom, pp. 2791–2795, Nov. 2007. 206

[39] S. H. Ting, K. Sakaguchi, and K. Araki, “A Markov-Kronecker model for analysis of closed-loop MIMO systems,” IEEE Commun. Lett., vol. 10, pp. 617–619, Aug. 2006. [40] E. Au, S. Jin, M. R. McKay, W. Mow, X. Gao, and I. B. Collings, “Analytical performance of MIMO-SVD systems in Ricean fading channels with channel estimation error and feedback delay,” to appear in IEEE Trans. on Communications, Mar. 2008. [41] H. T. Nguyen, J. B. Andersen, and G. F. Pedersen, “Capacity and performance of MIMO systems under the impact of feedback delay,” in Proc., IEEE PIMRC, vol. 1, pp. 53–57, Sept. 2004. [42] J. Du, Y. Li, D. Gu, A. F. Molisch, and J. Zhang, “Estimation of performance loss due to delay in channel feedback in MIMO systems,” in Proc., IEEE Veh. Technology Conf., vol. 3, pp. 1619–22, Sept. 2004. [43] P.-H. Kuo, P. J. Smith, L. M. Garth, and M. Shafi, “Instantaneous signal and selfinterference power of MIMO eigenmode transmission with feedback time delay,” in Proc., IEEE Intl. Conf. on Communications, vol. 9, pp. 4143–48, June 2006. [44] K. Kobayashi, T. Ohtsuki, and T. Kaneko, “MIMO systems in the presence of feedback delay,” in Proc., IEEE Intl. Conf. on Communications, vol. 9, pp. 4102– 06, June 2006. [45] A. Duel-Hallen, “Fading channel prediction for mobile radio adaptive transmission systems,” Proceedings of the IEEE, vol. 95, pp. 2299–2313, Dec. 2007. [46] R. C. Daniels, K. Mandke, K. Truong, S. Nettles, and J. R. W. Heath, “Throughput and delay measurements of limited feedback beamforming for indoor wireless networks,” Preprint: http://users.ece.utexas.edu/∼ rdaniels/, 2008.

207

[47] H. Wang and N. Moayeri, “Finite-state Markov channel – a useful model for radio communication channels,” IEEE Trans. on Veh. Technology, vol. 44, pp. 163–71, Feb. 1995. [48] C. Pimentel, T. H. Falk, and L. Lisboa, “Finite-state Markov modeling of correlated Rician-fading channels,” IEEE Trans. on Veh. Technology, vol. 53, pp. 1491–1501, Sept. 2004. [49] C. C. Tan and N. C. Beaulieu, “On first-order Markov modeling for the Rayleigh fading channel,” IEEE Trans. on Communications, vol. 48, pp. 2032–40, Dec. 2000. [50] Q. Zhang and S. A. Kassam, “Finite-state Markov model for Rayleigh fading channels,” IEEE Trans. on Communications, vol. 47, pp. 1688–92, Nov. 1999. [51] F. Babich, O. E. Kelly, and G. Lombardi, “Generalized Markov modeling for flat fading,” IEEE Trans. on Communications, vol. 48, pp. 547–551, Apr. 2000. [52] F. Babich, G. Lombardi, and E. Valentinuzzi, “Variable order markov modelling for LEO mobile satellite channels,” Electronics Letters, vol. 35, pp. 621–623, Apr. 1999. [53] A. Chen and R. R. Rao, “On tractable wireless channel models,” in Proc., IEEE PIMRC, vol. 2, pp. 825–830, Sept. 1998. [54] A. Saadani, P. Gelpi, and P. Tortelier, “A variable-order Markov chain based model for Rayleigh fading and RAKE receiver,” IEEE Signal Processing Letters, vol. 11, pp. 356–358, Mar. 2004. [55] Y. L. Guan and L. F. Turner, “Generalised FSMC model for radio channels with correlated fading,” in IEE Proc. Comm., vol. 146, pp. 133–137, Apr. 1999. [56] H. Wang and P. Chang, “On verifying the first-order Markovian assumption for a Rayleigh fading channel model,” IEEE Trans. on Veh. Technology, vol. 45, pp. 353– 357, May 1996. 208

[57] P. Sadeghi and P. Rapajic, “Capacity analysis for finite-state markov mapping of flat-fading channels,” IEEE Trans. on Communications, vol. 53, pp. 833–840, May 2005. [58] H. Bischl and E. Lutz, “Packet error rate in the non-interleaved Rayleigh channel,” IEEE Trans. on Communications, vol. 43, no. 234, pp. 1375–82, 1995. [59] M. R. Hueda and C. E. Rodriguez, “A new information theoretic test of the Markov property of block errors in fading channels,” IEEE Trans. on Veh. Technology, vol. 54, pp. 425–434, Mar. 2005. [60] L. Galluccio, F. Licandro, G. Morabito, and G. Schembra, “An analytical framework for the design of intelligent algorithms for adaptive-rate MPEG video encoding in next-generation time-varying wireless networks,” IEEE Journal on Sel. Areas in Communications, vol. 23, pp. 369–384, Feb. 2005. [61] W. Turin, “MAP symbol decoding in channels with error bursts,” IEEE Trans. on Info. Theory, vol. 47, pp. 1832–38, July 2001. [62] J. Razavilar, K. J. R. Liu, and S. I. Marcus, “Jointly optimized bit-rate or delay control policy for wireless packet networks with fading channels,” IEEE Trans. on Communications, vol. 50, pp. 484–494, Mar. 2002. [63] R. Buche and H. J. Kushner, “Control of mobile communications with time-varying channels in heavy traffic,” IEEE Trans. on Automatic Control, vol. 47, pp. 992–1003, June 2002. [64] J. Yun and M. Kavehrad, “Markov error structure for throughput analysis of adaptive modulation systems combined with ARQ over correlated fading channels,” IEEE Trans. on Veh. Technology, vol. 54, pp. 235–245, Jan. 2005. [65] L. Li and A. J. Goldsmith, “Low-complexity maximum-likelihood detection of coded 209

signals sent over finite-state Markov channels,” IEEE Trans. on Communications, vol. 50, pp. 524–531, Apr. 2002. [66] T. Tang and R. W. Heath, Jr., “Opportunistic feedback for downlink multiuser diversity,” IEEE Commun. Lett., vol. 9, pp. 948–950, Oct. 2005. [67] R. Gallager, Discrete Stochastic Processes. Springer, 1995. [68] P. Bremaud, Markov Chains. Springer, 1999. [69] N. Merhav and M. Feder, “Universal prediction,” IEEE Trans. on Info. Theory, vol. 44, pp. 2124–47, Oct. 1998. [70] L. H. Ozarow, S. Shamai, and A. D. Wyner, “Information theoretic considerations for cellular mobile radio,” IEEE Trans. on Veh. Technology, vol. 43, pp. 359–378, May 1994. [71] A. Paulraj, R. Nabar, and D. Gore, Introduction to Space-Time Wireless Communications. Cambridge, UK: Cambridge University Press, 2003. [72] J. A. Fill, “Eigenvalue bounds on convergence to stationarity for non-reversible markov chain, with an application to the exclusion process,” The annals of applied probability, vol. 1, pp. 62–87, Jan. 1991. [73] K. Huang, B. Mondal, R. W. Heath, Jr., and J. G. Andrews, “Multi-antenna limited feedback for temporally correlated channel: Feedback compression,” in Proc., IEEE Globecom, Nov. 2006. [74] J. G. Andrews, A. Ghosh, and R. Muhamed, Fundamentals of WiMAX: Understanding Broadband Wireless Networking. Prentice Hall, 2007. [75] P. Ding, D. J. Love, and M. D. Zoltowski, “On the sum rate of channel subspace feedback for multi-antenna broadcast channels,” in Proc., IEEE Globecom, vol. 5, pp. 2699–2703, Nov. 2005. 210

[76] N. Jindal, “MIMO broadcast channels with finite-rate feedback,” IEEE Trans. on Info. Theory, vol. 52, pp. 5045–60, Nov. 2006. [77] H. Yin and H. Liu, “Performance of space-division multiple-access (SDMA) with scheduling,” IEEE Trans. on Wireless Communications, vol. 1, no. 4, pp. 611–618, 2002. [78] M. Schubert and H. Boche, “Solution of the multiuser downlink beamforming problem with individual SINR constraints,” IEEE Trans. on Veh. Technol., vol. 53, pp. 18–28, Jan. 2004. [79] W. Choi, A. Forenza, J. G. Andrews, and R. W. Heath Jr., “Opportunistic space division multiple access with beam selection,” IEEE Trans. on Communications, vol. 55, pp. 2371–80, Dec. 2007. [80] C. K. Au-Yeung and D. J. Love, “On the performance of random vector quantization limited feedback beamforming in a MISO system,” IEEE Trans. on Wireless Communications, vol. 6, pp. 458–462, Feb. 2007. [81] V. N. Vapnik and A. Chervonenkis, “On the uniform convergence of relative frequencies of events to their probabilities,” Theory of Probab. and its Applic., vol. 16, pp. 264–280, Feb. 1971. [82] K.-B. Huang, R. W. Heath Jr., and J. G. Andrews, “Uplink SDMA with limited feedback: Throughput scaling,” EURASIP Journal on Advances in Sig. Processing, vol. 2008, Article ID 479357, 2008. doi:10.1155/2008/479357. [83] C. Swannack, E. Uysal-Biyikoglu, and G. W. Wornell, “MIMO broadcast scheduling with limited channel state information,” in Proc., Allerton Conf. on Comm., Control, and Computing, Sept. 2005. [84] T. Yoo and A. Goldsmith, “On the optimality of multiantenna broadcast scheduling 211

using zero-forcing beamforming,” IEEE Journal on Sel. Areas in Communications, vol. 24, pp. 528–541, Mar. 2006. [85] K. Zyczkowski and M. Kus, “Random unitary matrices,” J. Phys., vol. A27, pp. 4235–45, June 1994. [86] K.-B. Huang, R. W. Heath Jr., and J. G. Andrews, “SDMA with a sum feedback rate constraint,” IEEE Trans. on Signal Processing, vol. 55, pp. 3879–91, July 2007. [87] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE Trans. on Info. Theory, vol. 46, pp. 388–404, Mar. 2000. [88] N. Jindal, W. Rhee, S. Vishwanath, S. A. Jafar, and A. Goldsmith, “Sum power iterative water-filling for multi-antenna gaussian broadcast channels,” IEEE Trans. on Info. Theory, vol. 51, pp. 1570–80, Apr. 2005. [89] D. A. Brannan, M. F. Esplen, and J. J. Gray, Geometry. Cambridge University Press, 1999. [90] B. Rosenfeld, Geometry of Lie Groups. Springer, 1997. [91] D. Gesbert and M.-S. Alouini, “How much feedback is multi-user diversity really worth?,” in Proc., IEEE Intl. Conf. on Communications, vol. 1, pp. 234–238, June 2004. [92] V. Hassel, M.-S. Alouini, D. Gesbert, and G. Oien, “Exploiting multiuser diversity using multiple feedback thresholds,” in Proc., IEEE Veh. Technology Conf., vol. 2, pp. 1302–1306, 2005. [93] S. Sanayei and A. Nosratinia, “Exploiting multiuser diversity with only 1-bit feedback,” in Proc., IEEE Wireless Communications and Networking Conf., vol. 2, pp. 978–983, 2005.

212

[94] S. Sanayei and A. Nosratinia, “Opportunistic beamforming with limited feedback,” in Proc., IEEE Asilomar, pp. 648–652, Nov. 2005. [95] T. Tang, R. W. Heath Jr., S. Cho, and S. Yun, “Opportunistic feedback for multiuser MIMO systems with linear receivers,” accepted to IEEE Trans. on Communications, Sept. 2006. [96] K.-B. Huang, J. G. Andrews, and R. W. Heath Jr., “Orthogonal beamforming for SDMA downlink with limited feedback,” IEEE Int. Conf. Acoust., Speech and Sig. Proc. 2007, Apr. 2007. [97] L. Yang, M.-S. Alouini, and D. Gesbert, “Further results on selective multiuser diversity,” in 7th ACM/IEEE Intl. Symp. on Model., Analysis and Sim. of Wireless and Mobile Sys., pp. 25–30, Oct. 2004. [98] H. Alzer, “On some inequalities for the incomplete Gamma function,” Mathematics of Computation, vol. 66, pp. 771–778, Apr. 2005. [99] D. Bertsekas and R. Gallager, Data networks. Prentice Hall, 1992. [100] T. S. Rappaport, Wireless Communications: Principles and Practice. Prentice Hall, 2001. [101] S. Janson, T. Luczak, and A. Rucinski, Random Graphs. John Wiley, 2000. [102] C. Swannack, G. W. Wornell, and E. Uysal-Biyikoglu, “MIMO broadcast scheduling with quantized channel state information,” in Proc., IEEE Intl. Symposium on Info. Theory, pp. 1788–92, July 2006. [103] M. Mitzenmacher and E. Upfal, Probability and Computing. Cambridge University Press, 2005. [104] S. P. Weber, X. Yang, J. G. Andrews, and G. de Veciana, “Transmission capacity 213

of wireless ad hoc networks with outage constraints,” IEEE Trans. on Info. Theory, vol. 51, pp. 4091–02, Dec. 2005. [105] M. Grossglauser and D. N. C. Tse, “Mobility increases the capacity of ad hoc wireless networks,” IEEE Trans. on Networking, vol. 10, pp. 477–486, Aug. 2002. [106] M. Franceschetti, O. Dousse, D. N. C. Tse, and P. Thiran, “Closing the gap in the capacity of wireless networks via percolation theory,” IEEE Trans. on Info. Theory, vol. 53, pp. 1009–18, Mar. 2007. [107] A. Jovicic, P. Viswanath, and S. R. Kulkarni, “Upper bounds to transport capacity of wireless networks,” IEEE Trans. on Info. Theory, vol. 50, pp. 2555–65, Nov. 2004. [108] F. Xue, L.-L. Xie, and P. R. Kumar, “The transport capacity of wireless networks over fading channels,” IEEE Trans. on Info. Theory, vol. 51, pp. 834–847, Mar. 2005. [109] A. Ozgur, O. Leveque, and D. N. C. Tse, “Hierarchical cooperation achieves optimal capacity scaling in ad hoc networks,” IEEE Trans. on Info. Theory, vol. 53, pp. 3549– 3572, Oct. 2007. [110] S. Weber, J. G. Andrews, and N. Jindal, “The effect of fading, channel inversion, and threshold scheduling on ad hoc networks,” IEEE Trans. on Info. Theory, vol. 53, pp. 4127–4149, Nov. 2007. [111] A. Hasan and J. G. Andrews, “The guard zone in wireless ad hoc networks,” IEEE Trans. on Wireless Communications, vol. 6, pp. 897–906, Mar. 2007. [112] R. K. Ganti and M. Haenggi, “Regularity, interference, and capacity of large ad hoc networks,” in Proc., IEEE Asilomar, Oct. 2006. [113] J. Venkataraman, M. Haenggi, and O. Collins, “Shot noise models for the dual problems of cooperative coverage and outage in random networks,” in Proc., Allerton Conf. on Comm., Control, and Computing, Sept. 2006. 214

[114] N. Jindal, J. Andrews, and S. Weber, “Bandwidth partitioning in decentralized wireless networks,” submitted to IEEE Trans. on Wireless Communications, Nov. 2007. [115] S. Weber, J. G. Andrews, X. Yang, and G. de Veciana, “Transmission capacity of wireless ad hoc networks with successive interference cancellation,” IEEE Trans. on Info. Theory, vol. 53, pp. 2799–2814, Aug. 2007. [116] A. M. Hunter, J. G. Andrews, and S. P. Weber, “Capacity scaling of ad hoc networks with spatial diversity,” submitted to IEEE Trans. on Wireless Communications, Sept. 2007. [117] V. R. Cadambe and S. A. Jafar, “Interference alignment and the degrees of freedom for the k user interference channel,” Preprint: http://arXiv.org:0707.0323, July 2007. [118] K. Sundaresan, R. Sivakumar, M. A. Ingram, and T.-Y. Chang, “Medium access control in ad hoc networks with MIMO links: optimization considerations and algorithms,” IEEE Trans. on Mobile Computing, vol. 3, no. 4, pp. 350–365, 2004. [119] S. Kumar, V. S. Raghavan, and J. Deng, “Medium access control protocols for ad hoc wireless networks: A survey,” Ad Hoc Networks, vol. 4, pp. 326–358, May 2006. [120] M. Zorzi, J. Zeidler, A. Anderson, B. Rao, J. Proakis, A. L. Swindlehurst, M. Jensen, and S. Krishnamurthy, “Cross-layer issues in MAC protocol design for MIMO ad hoc networks,” IEEE Wireless Communications Magazine, vol. 13, pp. 62–76, Aug. 2006. [121] B. Hamdaoui and K. G. Shin, “Characterization and analysis of multi-hop wireless mimo network throughput,” in Proc., ACM Intl. Symposium on Mobile Ad Hoc Networking and Computing, pp. 120–129, 2007.

215

[122] J. C. Mundarath, P. Ramanathan, and B. D. V. Veen, “A cross layer scheme for adaptive antenna array based wireless ad hoc networks in multipath environments,” Wireless Networks, vol. 13, no. 5, pp. 597–615, 2007. [123] J.-S. Park, A. Nandan, M. Gerla, and H. Lee, “SPACE-MAC: enabling spatial reuse using MIMO channel-aware MAC,” in Proc., IEEE Intl. Conf. on Communications, vol. 5, pp. 3642–3646, May 2005. [124] M. Z. Siam, M. Krunz, A. Muqattash, and S. Cui, “Adaptive multi-antenna power control in wireless networks,” in Proc., Intl. Conf. on Wireless Comm. and Mobile Computing, pp. 875–880, 2006. [125] Z. Huang, Z. Zhang, and B. Ryu, “Power control for directional antenna-based mobile ad hoc networks,” in Proc., Intl. Conf. on Wireless Comm. and Mobile Computing, pp. 917–922, 2006. [126] R. Ramanathan, J. Redi, C. Santivanez, D. Wiggins, and S. Polit, “Ad hoc networking with directional antennas: a complete system solution,” IEEE Journal on Selected Areas in Communications, vol. 23, pp. 496–506, Mar. 2005. [127] A. Deopura and A. Ganz, “Provisioning link layer proportional service differentiation in wireless networks with smart antennas,” Wireless Networks, vol. 13, no. 3, 2007. [128] K. Sundaresan and R. Sivakumar, “A unified mac layer framework for ad-hoc networks with smart antennas,” in Proc., ACM Intl. Symposium on Mobile Ad Hoc Networking and Computing, pp. 244–255, 2004. [129] H. Singh and S. Singh, “Smart-aloha for multi-hop wireless networks,” Mobile Networks and Applications, vol. 10, no. 5, pp. 651–662, 2005. [130] R. Ramanathan, “On the performance of ad hoc networks with beamforming antennas,” in Proc., ACM Intl. Symposium on Mobile Ad Hoc Networking and Computing, pp. 95–105, 2001. 216

[131] A. Singh, P. Ramanathan, and B. Van Veen, “Spatial reuse through adaptive interference cancellation in multi-antenna wireless networks,” in Proc., IEEE Globecom, vol. 5, Nov. 2005. [132] T. Joshi, H. Gossain, C. C., and D. P. Agrawal, “Route recovery mechanisms for ad hoc networks equipped with switched single beam antennas,” in Proc., Annual Symposium on Simulation, 2005. [133] Y. Wu, L. Zhang, Y. Wu, and Z. Niu, “Interest dissemination with directional antennas for wireless sensor networks with mobile sinks,” in Proc., Intl. Conf. on Embedded networked sensor systems, pp. 99–111, 2006. [134] C. M. Yago and P. M. Ruiz, “Energy-efficient multicast with directional antennae and localized tree reconfiguration,” in Proc., ACM Intl. Symposium on Model., Analysis & Simulation of Wireless & Mobile Systems, pp. 151–154, 2006. [135] K. Sundaresan, W. Wang, and S. Eidenbenz, “Algorithmic aspects of communication in ad-hoc networks with smart antennas,” in Proc., ACM Intl. Symposium on Mobile Ad Hoc Networking and Computing, pp. 298–309, 2006. [136] R. Vilzmann, C. Bettstetter, D. M., and C. Hartmann, “Hop distances and flooding in wireless multihop networks with randomized beamforming,” in Proc., ACM Intl. Symposium on Model., Analysis & Simulation of Wireless & Mobile Systems, pp. 20– 27, 2005. [137] J. Zhang and S. C. Liew, “Capacity improvement of wireless ad hoc networks with directional antennae,” SIGMOBILE Mobile Computing Comm. Review, vol. 10, no. 4, 2006. [138] S. Yi, Y. Pei, S. Kalyanaraman, and B. Azimi-Sadjadi, “How is the capacity of ad hoc networks improved with directional antennas?,” Wireless Networks, vol. 13, no. 5, pp. 635–648, 2007. 217

[139] F. Baccelli, B. Blaszczyszyn, and P. Muhlethaler, “An Aloha protocol for multihop mobile wireless networks,” IEEE Trans. on Info. Theory, vol. 52, pp. 421–36, Feb. 2006. [140] D. Stoyan, W. S. Kendall, and J. Mecke, Stochastic Gemoetry and its Applications. Wiley, 2nd ed., 1995. [141] J. F. C. Kingman, Poisson processes. Oxford University Press, 1993. [142] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. on Info. Theory, vol. 45, pp. 1456–1467, Jul. 1999. [143] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” European Trans. on Telecomm., vol. 10, no. 6, pp. 585–595, 1999. [144] T. K. Y. Lo, “Maximum ratio transmission,” IEEE Trans. on Communications, vol. 47, pp. 1458–1461, Oct. 1999. [145] A. J. Goldsmith and P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. on Info. Theory, vol. 43, pp. 1986–92, Nov. 1997. [146] G. Caire, G. Taricco, and E. Biglieri, “Optimum power control over fading channels,” IEEE Trans. on Info. Theory, vol. 45, pp. 1468–89, July 1999. [147] S. M. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE Jour. Select. Areas in Commun., vol. 16, pp. 1451 – 1458, Oct. 1998. [148] G. Jongren and M. Skoglund, “Quantized feedback information in orthogonal spacetime block coding,” IEEE Trans. on Info. Theory, vol. 50, pp. 2473–86, Oct. 2004. [149] S. B. Lowen and M. C. Teich, “Power-law shot noise,” IEEE Trans. on Info. Theory, vol. 36, pp. 1302–18, Nov. 1990. 218

[150] S. Weber and M. Kam, “Computational complexity of outage probability simulations in mobile ad-hoc networks,” in Proc., Conf. on Information Sciences and Systems, Mar. 2005. [151] V. K. Rohatgi, An Introduction to Probability Theory. John Wiley & Sons, 1976.

219

Vita Kaibin Huang is a Ph.D. candidate in the Department of Electrical and Computer Engineering at The University of Texas at Austin. He received the B.Eng. (1st Class Honors) and M.Eng. from the National University of Singapore in 1998 and 2000, respectively. From 2000-2004, he was an associate scientist at the Institute for Infocomm Research in Singapore, developing software defined radio systems. During the summers of 2005 and 2006, he was a summer research intern with the advanced technology group at the Freescale Semiconductor. At Freescale, he performed research on physical layer system design for the IEEE 802.16e and 3GPP-LTE standards. He has received the Motorola Partnerships in Research Grant, the University Continuing Fellowship at UT Austin, and the Best Student Paper (Communication Systems) award at IEEE Globecom 2006. He visited Northwestern university in the summer of 2007. His research interests focus on limited feedback techniques for multiuser wireless networks.

Permanent Address: BLK 403, #12-20, Pandan Gardens, Singapore 600403

This dissertation was typeset with LATEX 2ε 1 by the author.

1 A LT

EX 2ε is an extension of LATEX. LATEX is a collection of macros for TEX. TEX is a trademark of the American Mathematical Society. The macros used in formatting this dissertation were written by Dinesh Das, Department of Computer Sciences, The University of Texas at Austin, and extended by Bert Kay, James A. Bednar, and Ayman El-Khashab.

220