Random Matrix Methods for Wireless Communications

"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"-- Provided by publisher 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 Machine generated contents note: 1. Introduction; Part I. Theoretical Aspects: 2. Random matrices; 3. The Stieltjes transform method; 4. Free probability theory; 5. Combinatoric approaches; 6. Deterministic equivalents; 7. Spectrum analysis; 8. Eigen-inference; 9. Extreme eigenvalues; 10. Summary and partial conclusions; Part II. Applications to Wireless Communications: 11. Introduction to applications in telecommunications; 12. System performance of CDMA technologies; 13. Performance of multiple antenna systems; 14. Rate performance in multiple access and broadcast channels; 15. Performance of multi-cellular and relay networks; 16. Detection; 17. Estimation; 18. System modeling; 19. Perspectives; 20. Conclusion. 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 real-world problems in wireless communications. Intuitive yet rigorous, it demonstrates how to choose the correct approach for obtaining mathematically accurate results.