About the Series Editors xi
1 Introduction 1
1.1 General remarks 1
1.2 Signals, interference, and types of parallel channels 3
2 Four-parametric model of a SISO channel 7
2.1 Multipath propagation 7
2.2 Random walk approach to modeling of scattering field 13
2.2.1 Random walk in two dimensions as a model for scattering field 13
2.2.2 Phase distribution and scattering strength 14
2.2.3 Distribution of intensity 14
2.2.4 Distribution of the random phase 17
2.3 Gaussian case 18
2.3.1 Four-parametric distribution family 18
2.3.2 Distribution of the magnitude 20
2.3.3 Distribution of the phase 27
2.3.4 Moment generating function, moments and cumulants of four-parametric distribution 29
2.3.5 Some aspects of multiple scattering propagation 29
3 Models of MIMO channels 33
3.1 General classification of MIMO channel models 33
3.2 Physical models 33
3.2.1 Deterministic models 34
3.2.2 Geometry-based stochastic models 35
3.3 Analytical models 36
3.3.1 Channel matrix model 37
3.4 Geometrical phenomenological models 47
3.4.1 Scattering from rough surfaces 48
3.5 On the role of trigonometric polynomials in analysis and simulation of MIMO channels 49
3.5.1 Measures of dependency 50
3.5.2 Non-negative trigonometric polynomials and their use in estimation of AoD and AoA distribution 51
3.5.3 Approximation of marginal PDF using non-negative polynomials 51
3.6 Canonical expansions of bivariate distributions and the structure MIMO channel covariance matrix 52
3.6.1 Canonical variables and expansion 52
3.6.2 General structure of the full covariance matrix 54
3.6.3 Relationship to other models 54
3.7 Bivariate von Mises distribution with correlated transmit and receive sides 56
3.7.1 Single cluster scenario 56
3.7.2 Multiple clusters scenario 58
3.8 Bivariate uniform distributions 58
3.8.1 Harmonic coupling 58
3.8.2 Markov-type bivariate density 61
3.9 Analytical expression for the diversity measure of an antenna array 62
3.9.1 Relation of the shape of the spatial covariance function to trigonometric moments 62
3.9.2 Approximation of the diversity measure for a large number of antennas 64
3.9.3 Examples 66
3.9.4 Leading term analysis of degrees of freedom 70
3.10 Effect of AoA/AoD dependency on the SDoF 72
3.11 Space-time covariance function 72
3.11.1 Basic equation 72
3.11.2 Approximations 73
3.12 Examples: synthetic data and uniform linear array 75
3.13 Approximation of a matrix by a Toeplitz matrix 77
3.14 Asymptotic expansions of diversity measure 78
3.15 Distributed scattering model 79
4 Modeling of wideband multiple channels 81
4.1 Standard models of channels 82
4.1.1 Cost 259/273 82
4.1.2 3gpp Scm 83
4.1.3 WINNER channel models 84
4.2 MDPSS based wideband channel simulator 84
4.2.1 Geometry of the problem 84
4.2.2 Statistical description 85
4.2.3 Multi-cluster environment 87
4.2.4 Simulation of dynamically changing environment 88
4.3 Measurement based simulator 89
4.4 Examples 91
4.4.1 Two cluster model 92
4.4.2 Environment specified by joint AoA/AoD/ToA distribution 93
4.4.3 Measurement based simulator 95
4.5 Appendix A: simulation parameters 96
5 Capacity of communication channels 99
5.1 Introduction 99
5.2 Ergodic capacity of MIMO channel 100
5.2.1 Capacity of a constant (static) MIMO channel 100
5.2.2 Alternative normalization 102
5.2.3 Capacity of a static MIMO channel under different operation modes 103
5.2.4 Ergodic capacity of a random channel 104
5.2.5 Ergodic capacity of MIMO channels 106
5.2.6 Asymptotic analysis of capacity and outage capacity 106
5.3 Effects of MIMO models and their parameters on the predicted capacity of MIMO channels 109
5.3.1 Channel estimation and effective SNR 110
5.3.2 Achievable rates in Rayleigh channels with partial CSI 113
5.3.3 Examples 116
5.4 Time evolution of capacity 119
5.4.1 Time evolution of capacity in SISO channels 119
5.4.2 SISO channel capacity evolution 120
5.5 Sparse MIMO channel model 122
5.6 Statistical properties of capacity 124
5.6.1 Some mathematical expressions 124
5.7 Time-varying statistics 125
5.7.1 Unordered eigenvalues 125
5.7.2 Single cluster capacity LCR and AFD 126
5.7.3 Approximation of multi-cluster capacity LCR and AFD 126
5.7.4 Statistical simulation results 129
6 Estimation and prediction of communication channels 131
6.1 General remarks on estimation of time-varying channels 131
6.2 Velocity estimation 131
6.2.1 Velocity estimation based on the covariance function approximation 131
6.2.2 Estimation based on reflection coefficients 132
6.3 K-factor estimation 133
6.3.1 Moment matching estimation 133
6.3.2 I/Q based methods 134
6.4 Estimation of four-parametric distributions 135
6.5 Estimation of narrowband MIMO channels 138
6.5.1 Superimposed pilot estimation scheme 138
6.5.2 LS estimation 140
6.5.3 Scaled least-square (SLS) estimation 142
6.5.4 Minimum MSE 144
6.5.5 Relaxed MMSE estimators 146
6.6 Using frames for channel state estimation 148
6.6.1 Properties of the spectrum of a mobile channel 149
6.6.2 Frames based on DPSS 150
6.6.3 Discrete prolate spheroidal sequences 150
6.6.4 Numerical simulation 154
7 Effects of prediction and estimation errors on performance of communication systems 157
7.1 Kolmogorov–Szego-Krein formula 160
7.2 Prediction error for different antennas and scattering characteristics 162
7.2.1 SISO channel 162
7.2.2 SIMO channel 165
7.2.3 MISO channel 167
7.2.4 MIMO channel 170
7.3 Summary of infinite horizon prediction results 174
7.4 Eigenstructure of two cluster correlation matrix 175
7.5 Preliminary comments on finite horizon prediction 176
7.6 SISO channel prediction 178
7.6.1 Wiener filter 178
7.6.2 Single pilot prediction in a two cluster environment 179
7.6.3 Single cluster prediction with multiple past samples 181
7.6.4 Two cluster prediction with multiple past samples 182
7.6.5 Role of oversampling 187
7.7 What is the narrowband signal for a rectangular array? 188
7.8 Prediction using the UIU model 190
7.8.1 Separable covariance matrix 191
7.8.2 1 × 2 unseparable example 192
7.8.3 Large number of antennas: no noise 193
7.8.4 Large number of antennas: estimation in noise 194
7.8.5 Effects of the number of antennas, scattering geometry, and observation time on the quality of prediction 195
7.9 Numerical simulations 198
7.9.1 SISO channel single cluster 198
7.9.2 Two cluster prediction 198
7.10 Wiener estimator 199
7.11 Approximation of the Wiener filter 201
7.11.1 Zero order approximation 202
7.11.2 Perturbation solution 202
7.12 Element-wise prediction of separable process 203
7.13 Effect of prediction and estimation errors on capacity calculations 204
7.14 Channel estimation and effective SNR 205
7.14.1 System model 205
7.14.2 Estimation error 205
7.14.3 Effective SNR 207
7.15 Achievable rates in Rayleigh channels with partial CSI 208
7.15.1 No CSI at the transmitter 208
7.15.2 Partial CSI at the transmitter 209
7.15.3 Optimization of the frame length 211
7.16 Examples 211
7.16.1 P(0, 0) Estimation 211
7.16.2 Effect of non-uniform scattering 213
7.17 Conclusions 214
7.18 Appendix A: Szego summation formula 215
7.19 Appendix B: matrix inversion lemma 216
8 Coding, modulation, and signaling over multiple channels 219
8.1 Signal constellations and their characteristics 219
8.2 Performance of OSTBC in generalized Gaussian channels and hardening effect 224
8.2.1 Introduction 224
8.2.2 Channel representation 225
8.2.3 Probability of error 227
8.2.4 Hardening effect 229
8.3 Differential time-space modulation (DTSM) and an effective solution for the non-coherent MIMO channel 233
8.3.1 Introduction to DTSM 233
8.3.2 Performance of autocorrelation receiver of DSTM in generalized Gaussian channels 234
8.3.3 Comments on MIMO channel model 235
8.3.4 Differential space-time modulation 235
8.3.5 Performance of DTSM 237
8.3.6 Numerical results and discussions 243
8.3.7 Some comments 243
Bibliography 245
Index 257
