Statistical Robust Beamforming for Broadcast Channels and Applications in Satellite Communication (eBook)

Preface 7
Acknowledgments 8
Introduction 9
Zusammenfassung 11
Contents 13
Acronyms 15
Nomenclature 17
List of Figures 20
List of Tables 22
1 Multi-User Downlink Communication 23
1.1 Gaussian Vector Broadcast Channel Model 27
1.2 Quality-of-Service Optimization and Rate Balancing 28
1.2.1 Quality-of-Service Optimization Problem 29
1.2.2 Rate Balancing Problem 30
1.2.3 Relation Between the Problems 31
1.3 Solutions for Perfect Transmitter Channel Knowledge 31
1.3.1 Uplink–Downlink Duality and Uplink Power Allocation 32
1.3.2 Convex Problem Reformulations 35
1.3.3 Quality-of-Service Feasibility 37
1.4 Beamformer Design with Multiple Power Constraints 39
1.4.1 QoS Optimization and Balancing with Multiple Power Constraints 40
1.5 Outline of the Chapters and the Contributions 42
1.5.1 Contributions for Ergodic Rates with Multiplicative Channel Errors 44
1.5.2 Contributions for Average MSEs with Additive Channel Errors 44
1.5.3 Contributions for Outage Rate Requirements 46
1.5.4 Applications to Satellite Communication 47
2 Models for Incomplete Channel Knowledge 50
2.1 Additive Error Models 51
2.1.1 Deterministic Channel Models 52
2.1.2 Stochastic Channel Models 53
2.1.3 Quantization Errors and Delays 54
2.2 Multiplicative Error Models 55
2.2.1 Deterministic Channel and Shadow Fading 55
2.2.2 Rank-One Channel Covariance Matrix 56
2.3 Multiplicative Approximations for Additive Fading 57
3 Precoder Design for Ergodic Rates with Multiplicative Fading 60
3.1 Closed-Form Rate Expressions 62
3.2 Lower and Upper Rate Bounds 63
3.2.1 Generalized Zero-Forcing Lower Bound 63
3.2.2 Bounds for Adaptive Beamforming 65
3.2.3 QoS and Balancing Optimization with Rate Bounds 67
3.2.3.1 Iterative Inner Rate Balancing Optimization 69
3.2.3.2 Outer Worst-Case Noise Optimization 71
3.2.3.3 Uplink–Downlink Transformation 73
3.3 Quality-of-Service Feasibility Region 73
3.4 Post-Processing Power Allocation 76
3.4.1 Post-Processing for QoS Optimization 77
3.4.2 Post-Processing for Ergodic Rate Balancing 78
3.5 Sequential Approximation Strategy 79
3.5.1 Sequential Quality-of-Service Optimization 81
3.5.2 Sequential Ergodic Rate Balancing 83
3.6 Branch and Bound Method 86
3.7 Numerical Optimization Results for Ergodic Rates 88
3.7.1 Power Minimization Results 89
3.7.2 Rate Balancing Results 91
3.7.3 Results for Sequential QoS Optimization and Balancing 93
4 Mean Square Error Transceiver Design for AdditiveFading 97
4.1 Mean Square Error Based Rate Bounds 99
4.2 Closed-Form Mean Square Error Expressions 100
4.3 Quality-of-Service Optimization 103
4.3.1 MSE Based Uplink–Downlink Dualities 106
4.3.2 Power Allocation and Worst-Case Noise Search 112
4.3.3 Primal Reconstruction of Beamformers 114
4.3.4 Expectation Evaluation for Alternating Convex Optimization 115
4.4 Quality-of-Service Feasibility Region 116
4.5 Average Mean Square Error Balancing 118
4.5.1 Alternating Convex Search for Balancing 121
4.5.2 Uplink–Downlink Dualities for Balancing 124
4.5.3 Iterative Minimum Mean Square Error Search 127
4.5.3.1 Uplink Weighted Sum-MSE Minimization 129
4.6 Ergodic Rate Balancing Approximations 130
4.7 Numerical Results for Mean Square Error Optimizations 132
4.7.1 Max–Min Mean Square Error Optimization Results 133
4.7.1.1 Per-User Mean Square Error Balancing Results 134
4.7.1.2 Per-user MSE Balancing Versus Weighted Sum-MSE Balancing 135
4.7.1.3 Complexity Comparison for MSE Balancing 137
4.7.2 Sum Mean Square Error Minimization Analysis 139
4.7.3 Approximate Rate Balancing Using MSE Optimizations 143
5 Outage Constrained Beamformer Design 146
5.1 Chance-Constrained Optimization 147
5.1.1 Basic Mathematical Background 147
5.1.2 Chance Constraints in Downlink Communication 150
5.2 Multiplicative Fading Example 152
5.3 Outage Probability Computation for Additive Fading 155
5.4 Power Allocation and Feasibility for Fixed Beamforming 156
5.4.1 Characteristic of the Chance Constraints 157
5.4.2 Fixed Point Framework for Power Allocation 159
5.4.3 Feasibility Detection for QoS Optimization 161
5.5 Robust Uncertainty Reformulations 162
5.5.1 Sphere Bounds for the Additive Channel Errors 162
5.5.2 Quadratic Bounds for the Orthogonal Channel Errors 164
5.5.3 QoS Optimization with Uncertainty Constraints 166
5.5.4 Balancing Optimization with Uncertainty Constraints 167
5.5.5 Beamformer Reconstruction and Direct Beamformer Optimization 168
5.6 Tractable Bounds with Concentration Inequalities 168
5.6.1 Markov's Inequality Based MSE Approximation 169
5.6.2 Bernstein-Type Inequality Bound for the SINR 169
5.6.3 Bernstein-Type Inequality Bound for the MSE 170
5.6.4 Related MSE Based Chance Constraint Approximation 173
5.7 Numerical Results for Chance-Constrained Optimization 174
5.7.1 Post-Processing Power Allocation for QoS Optimization 175
5.7.2 QoS Optimization with Channel Uncertainty Constraints 177
5.7.3 Rate Balancing with Approximated Chance Constraints 182
5.7.4 Comparison of the MSE Based Approximations 187
6 Applications in Satellite Communication 190
6.1 Satellite Channel Characteristic 191
6.1.1 Multi-Spotbeam Model 191
6.1.2 Fading Characteristics 194
6.1.3 Channel Error Model 195
6.2 Balancing Optimization for Satellite Communication 196
6.3 Ergodic Rate and Mean Square Error Optimization 197
6.4 Results for Rate and Mean Square Error Balancing 198
6.4.1 Results for Rain and Rank-One Additive Channel Fading 198
6.4.2 Results for Rain and Full-Rank Additive Channel Fading 200
6.4.3 Performance Limits with Per-Antenna Power Constraints 202
6.5 Outage Constrained Rate Optimization 205
6.5.1 Conservative Inner Optimization 206
6.5.2 Outer Optimization of Priors 208
6.6 Results for Outage Constrained Rate Balancing 209
6.6.1 Equal Fading Conditions for the Terminals 210
6.6.2 Effects for Distinct Fading Conditions 213
6.6.3 Outage Probabilities at the Terminals 214
7 Summary, Conclusions, and Open Research 217
7.1 Research Results for Robust Beamforming 217
7.1.1 Achievements and Open Problems for Multiplicative Channel Errors 217
7.1.2 Summary of Results for Average MSE Optimization 218
7.1.3 Contributions for Outage Rate Requirements 219
7.1.4 Summary and Conclusions for Satellite Communication 220
7.2 Other Research on Robust Beamforming 220
A Additional Information 222
A.1 Basic Properties of the Rate Based Optimizations 222
A.1.1 Positivity and Monotonicity for the Optimum of the QoS Problem 222
A.1.2 Positivity and Monotonicity for the Optimum of the Rate Balancing Problem 223
A.1.3 Relation Between the QoS Optimization and the Rate Balancing Problem 223
A.2 Interference Functions and Property Preserving Transforms 223
A.3 Ergodic Rate Bounds for Multiplicative Fading 227
A.3.1 Derivation of Lower and Upper Bounds on the Ergodic Rate 227
A.4 Feasible QoS Region with Ergodic Rate Bounds 229
A.5 On Uniqueness of the QoS Optimal Power Allocation 230
A.6 Duality for Second Order Cone Programs 231
A.6.1 Application to Uplink–Downlink Duality for MSE Based QoS Optimization 235
A.6.2 Reconstruction of the Primal Variables 236
A.7 Properties of the Dual Uplink MSE Optimizations 237
A.8 Some Distribution and Quantile Functions 239
References 243
Index 259