Blending theoretical results with practical applications, this book provides an introduction to random matrix theory and shows how it can be used to tackle a variety of problems in wireless communications. The Stieltjes transform method, free probability theory, combinatoric approaches, deterministic equivalents and spectral analysis methods for statistical inference are all covered from a unique engineering perspective. Detailed mathematical derivations are presented throughout, with thorough explanation of the key results and all fundamental lemmas required for the reader to derive similar calculus on their own. These core theoretical concepts are then applied to a wide range of real-world problems in signal processing and wireless communications, including performance analysis of CDMA, MIMO and multi-cell networks, as well as signal detection and estimation in cognitive radio networks. The rigorous yet intuitive style helps demonstrate to students and researchers alike how to choose the correct approach for obtaining mathematically accurate results.
Author(s): Romain Couillet, Merouane Debbah
Publisher: Cambridge University Press
Year: 2011
Language: English
Pages: 563
Tags: Приборостроение;Теория электросвязи (ТЭС);
Cover......Page 1
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 15
Acknowledgments......Page 17
Acronyms......Page 18
Notation......Page 20
1.1 Motivation......Page 25
1.2 History and book outline......Page 30
Part I Theoretical aspects......Page 39
2.1.1 Definitions and notations......Page 41
2.1.2 Wishart matrices......Page 43
2.2.1 Why go to infinity?......Page 53
2.2.2 Limit spectral distributions......Page 54
3.1 Definitions and overview......Page 59
3.2 The MarĊenko–Pastur law......Page 66
3.2.1 Proof of the MarĊenko–Pastur law......Page 68
3.2.2 Truncation, centralization, and rescaling......Page 78
3.3 Stieltjes transform for advanced models......Page 81
3.4 Tonelli theorem......Page 85
3.5 Central limit theorems......Page 87
4 Free probability theory......Page 95
4.1 Introduction to free probability theory......Page 96
4.2 R- and S-transforms......Page 99
4.3 Free probability and random matrices......Page 101
4.4 Free probability for Gaussian matrices......Page 108
4.5 Free probability for Haar matrices......Page 111
5.1 The method of moments......Page 119
5.2 Free moments and cumulants......Page 122
5.3 Generalization to more structured matrices......Page 129
5.4 Free moments in small dimensional matrices......Page 132
5.5 Rectangular free probability......Page 133
5.6 Methodology......Page 135
6.1 Introduction to deterministic equivalents......Page 137
6.2.1 Bai and Silverstein method......Page 139
6.2.2 Gaussian method......Page 163
6.2.3 Information plus noise models......Page 169
6.2.4 Models involving Haar matrices......Page 177
6.3 A central limit theorem......Page 199
7 Spectrum analysis......Page 203
7.1.1 No eigenvalues outside the support......Page 204
7.1.2 Exact spectrum separation......Page 207
7.1.3 Asymptotic spectrum analysis......Page 210
7.2.1 Exact separation......Page 216
7.2.2 Asymptotic spectrum analysis......Page 219
8.1.1 Girko G-estimators......Page 223
8.1.2 G-estimation of population eigenvalues and eigenvectors......Page 225
8.1.3 Central limit for G-estimators......Page 237
8.2 Moment deconvolution approach......Page 242
9.1 Spiked models......Page 247
9.1.1 Perturbed sample covariance matrix......Page 248
9.1.2 Perturbed random matrices with invariance properties......Page 252
9.2.1 Introduction to the method of orthogonal polynomials......Page 254
9.2.2 Limiting laws of the extreme eigenvalues......Page 257
9.3 Random matrix theory and eigenvectors......Page 261
10 Summary and partial conclusions......Page 267
Part II Applications to wireless communications......Page 273
11.1 Historical account of major results......Page 275
11.1.1 Rate performance of multi-dimensional systems......Page 276
11.1.2 Detection and estimation in large dimensional systems......Page 280
11.1.3 Random matrices and flexible radio......Page 283
12.1 Introduction......Page 287
12.2.1 Random CDMA in uplink frequency flat channels......Page 288
12.2.2 Random CDMA in uplink frequency selective channels......Page 297
12.2.3 Random CDMA in downlink frequency selective channels......Page 305
12.3 Performance of orthogonal CDMA technologies......Page 308
12.3.2.1 Matched-flter......Page 309
12.3.3 Orthogonal CDMA in downlink frequency selective channels......Page 310
12.3.3.1 Matched-flter......Page 311
12.3.3.2 MMSE decoder......Page 312
13.1 Quasi-static MIMO fading channels......Page 317
13.2 Time-varying Rayleigh channels......Page 319
13.2.1 Small dimensional analysis......Page 320
13.2.2 Large dimensional analysis......Page 321
13.2.3 Outage capacity......Page 322
13.3 Correlated frequency flat fading channels......Page 324
13.3.1 Communication in strongly correlated channels......Page 329
13.3.2 Ergodic capacity in strongly correlated channels......Page 333
13.3.3 Ergodic capacity in weakly correlated channels......Page 335
13.3.4 Capacity maximizing precoder......Page 336
13.4.1 Quasi-static mutual information and ergodic capacity......Page 340
13.4.2 Capacity maximizing power allocation......Page 342
13.4.3 Outage mutual information......Page 344
13.5 Frequency selective channels......Page 346
13.5.1 Ergodic capacity......Page 348
13.5.2 Capacity maximizing power allocation......Page 349
13.6 Transceiver design......Page 352
13.6.1 Channel matrix model with i.i.d. entries......Page 355
13.6.2 Channel matrix model with generalized variance profile......Page 356
14 Rate performance in multiple access and broadcast channels......Page 359
14.1 Broadcast channels with linear precoders......Page 360
14.1.1 System model......Page 363
14.1.2 Deterministic equivalent of the SINR......Page 365
14.1.3 Optimal regularized zero-forcing precoding......Page 372
14.1.4 Zero-forcing precoding......Page 373
14.1.5 Applications......Page 377
14.2 Rate region of MIMO multiple access channels......Page 379
14.2.1 MAC rate region in quasi-static channels......Page 381
14.2.2 Ergodic MAC rate region......Page 384
14.2.3 Multi-user uplink sum rate capacity......Page 388
15.1 Performance of multi-cell networks......Page 393
15.1.1 Two-cell network......Page 397
15.1.2 Wyner model......Page 400
15.2 Multi-hop communications......Page 402
15.2.1 Multi-hop model......Page 403
15.2.3 Large dimensional analysis......Page 406
15.2.4 Optimal transmission strategy......Page 412
16.1 Cognitive radios and sensor networks......Page 417
16.2 System model......Page 420
16.3 Neyman–Pearson criterion......Page 423
16.3.1.1 Derivation of PY│Hi in the SIMO case......Page 424
16.3.1.2 Multi-source case......Page 429
16.3.2 Unknown signal and noise variances......Page 430
16.3.3 Unknown number of sources......Page 431
16.4 Alternative signal sensing approaches......Page 436
16.4.1 Condition number method......Page 437
16.4.2 Generalized likelihood ratio test......Page 438
16.4.3 Test power and error exponents......Page 440
17 Estimation......Page 445
17.1.1 System model......Page 446
17.1.2 The MUSIC approach......Page 447
17.1.3 Large dimensional eigen-inference......Page 449
17.1.4 The correlated signal case......Page 453
17.2 Blind multi-source localization......Page 456
17.2.1 System model......Page 458
17.2.2 Small dimensional inference......Page 460
17.2.3 Conventional large dimensional approach......Page 462
17.2.4 Free deconvolution approach......Page 464
17.2.5 Analytic method......Page 471
17.2.6 Joint estimation of number of users, antennas and powers......Page 493
17.2.7.1 Method comparison......Page 495
17.2.7.2 Joint estimation of K, nk, Pk......Page 496
18 System modeling......Page 501
18.1 Introduction to Bayesian channel modeling......Page 502
18.2 Channel modeling under environmental uncertainty......Page 504
18.2.1.1 Average channel energy constraint......Page 505
18.2.1.2 Probabilistic average channel energy constraint......Page 506
18.2.1.3 Application to the single antenna channel......Page 507
18.2.2.1 Deterministic knowledge of the correlation matrix......Page 508
18.2.2.2 Knowledge of the existence of a correlation matrix......Page 510
18.2.2.3 Limited-rank covariance matrix......Page 518
18.2.2.4 Discussion......Page 521
19.1 From asymptotic results to finite dimensional studies......Page 525
19.2 The replica method......Page 529
19.3 Towards time-varying random matrices......Page 530
20 Conclusion......Page 535
References......Page 539
Index......Page 561
