A Deterministic Equivalent for the Analysis of Correlated MIMO Multiple Access Channels

In this article, novel deterministic equivalents for the Stieltjes transform and the Shannon transform of a class of large dimensional random matrices are provided. These results are used to characterize the ergodic rate region of multiple antenna multiple access channels, when each point-to-point propagation channel is modelled according to the Kronecker model. Specifically, an approximation of all rates achieved within the ergodic rate region is derived and an approximation of the linear precoders that achieve the boundary of the rate region as well as an iterative water-filling algorithm to obtain these precoders are provided. An original feature of this work is that the proposed deterministic equivalents are proved valid even for strong correlation patterns at both communication sides. The above results are validated by Monte Carlo simulations.

[1]  Andrea J. Goldsmith,et al.  Duality, achievable rates, and sum-rate capacity of Gaussian MIMO broadcast channels , 2003, IEEE Trans. Inf. Theory.

[2]  Philippe Loubaton,et al.  MMSE analysis of certain large isometric random precoded systems , 2003, IEEE Trans. Inf. Theory.

[3]  David Tse,et al.  Linear Multiuser Receivers: Effective Interference, Effective Bandwidth and User Capacity , 1999, IEEE Trans. Inf. Theory.

[4]  Jiunn-Tsair Chen,et al.  Asymptotic spectral efficiency of MIMO multiple-access wireless systems exploring only channel spatial correlations , 2005, IEEE Trans. Signal Process..

[5]  R. Clarke.  A statistical theory of mobile-radio reception , 1968 .

[6]  Rodney A. Kennedy,et al.  Antenna saturation effects on dense array MIMO capacity , 2003, 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, 2003. Proceedings. (ICASSP '03)..

[7]  Roland Speicher,et al.  On Slow-Fading MIMO Systems With Nonseparable Correlation , 2008, IEEE Transactions on Information Theory.

[8]  Philippe Loubaton,et al.  On the capacity achieving covariance matrix for frequency selective MIMO channels using the asymptotic approach , 2010, ISIT.

[9]  Rohit U. Nabar,et al.  Introduction to Space-Time Wireless Communications , 2003 .

[10]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[11]  W. Burnside.  Theory of Functions , 1899, Nature.

[12]  Vi︠a︡cheslav Leonidovich Girko,et al.  Theory of random determinants , 1990 .

[13]  J. W. Silverstein,et al.  On the empirical distribution of eigenvalues of large dimensional information-plus-noise-type matrices , 2007 .

[14]  Z. Bai,et al.  Limit of the smallest eigenvalue of a large dimensional sample covariance matrix , 1993 .

[15]  Aris L. Moustakas,et al.  On the Outage Capacity of Correlated Multiple-Path MIMO Channels , 2005, IEEE Transactions on Information Theory.

[16]  M. J. Gans,et al.  On Limits of Wireless Communications in a Fading Environment when Using Multiple Antennas , 1998, Wirel. Pers. Commun..

[17]  Wei Yu,et al.  Iterative water-filling for Gaussian vector multiple-access channels , 2001, IEEE Transactions on Information Theory.

[18]  M. Kreĭn,et al.  The Markov Moment Problem and Extremal Problems , 1977 .

[19]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[20]  V. Marčenko,et al.  DISTRIBUTION OF EIGENVALUES FOR SOME SETS OF RANDOM MATRICES , 1967 .

[21]  Valerie Isham,et al.  Non‐Negative Matrices and Markov Chains , 1983 .

[22]  Sennur Ulukus,et al.  Optimality of beamforming in fading MIMO multiple access channels , 2009, IEEE Transactions on Communications.

[23]  Aris L. Moustakas,et al.  MIMO capacity through correlated channels in the presence of correlated interferers and noise: a (not so) large N analysis , 2003, IEEE Trans. Inf. Theory.

[24]  Antonia Maria Tulino,et al.  Impact of antenna correlation on the capacity of multiantenna channels , 2005, IEEE Transactions on Information Theory.

[25]  J. W. Silverstein,et al.  ON THE SIGNAL-TO-INTERFERENCE RATIO OF CDMA SYSTEMS IN WIRELESS COMMUNICATIONS , 2007, math/0702888.

[26]  Moustakas,et al.  Communication through a diffusive medium: coherence and capacity , 2000, Science.

[27]  E. Seneta.  Non-negative Matrices and Markov Chains , 2008 .

[28]  S. Verdu,et al.  Multiple-access channels with memory with and without frame synchronism , 1989, IEEE Trans. Inf. Theory.

[29]  Shlomo Shamai,et al.  Information-theoretic considerations for symmetric, cellular, multiple-access fading channels - Part II , 1997, IEEE Trans. Inf. Theory.

[30]  M. Mézard,et al.  Spin Glass Theory and Beyond , 1987 .

[31]  J. W. Silverstein,et al.  On the empirical distribution of eigenvalues of a class of large dimensional random matrices , 1995 .

[32]  Ralf R. Müller,et al.  MIMO channel modeling and the principle of maximum entropy , 2005, IEEE Transactions on Information Theory.

[33]  A. Paulraj,et al.  Capacity optimization for Rician correlated MIMO wireless channels , 2005, Conference Record of the Thirty-Ninth Asilomar Conference onSignals, Systems and Computers, 2005..

[34]  Shlomo Shamai,et al.  The Capacity Region of the Gaussian Multiple-Input Multiple-Output Broadcast Channel , 2006, IEEE Transactions on Information Theory.

[35]  Philippe Loubaton,et al.  A CLT FOR INFORMATION-THEORETIC STATISTICS OF GRAM RANDOM MATRICES WITH A GIVEN VARIANCE PROFILE , 2007, 0706.0166.

[36]  K. Fan,et al.  Maximum Properties and Inequalities for the Eigenvalues of Completely Continuous Operators. , 1951, Proceedings of the National Academy of Sciences of the United States of America.

[37]  Toshiyuki TANAKA,et al.  Generic Multiuser Detection and Statistical Physics , 2009 .

[38]  Chen-Nee Chuah,et al.  Capacity scaling in MIMO Wireless systems under correlated fading , 2002, IEEE Trans. Inf. Theory.

[39]  Stefania Sesia,et al.  LTE - The UMTS Long Term Evolution, Second Edition , 2011 .

[40]  R. Bass,et al.  Review: P. Billingsley, Convergence of probability measures , 1971 .

[41]  W. Hachem,et al.  Deterministic equivalents for certain functionals of large random matrices , 2005, math/0507172.

[42]  Andrea J. Goldsmith,et al.  Capacity limits of MIMO channels , 2003, IEEE J. Sel. Areas Commun..

[43]  Shlomo Shamai,et al.  Multicell uplink spectral efficiency of coded DS-CDMA with random signatures , 2001, IEEE J. Sel. Areas Commun..

[44]  Emre Telatar,et al.  Capacity of Multi-antenna Gaussian Channels , 1999, Eur. Trans. Telecommun..

[45]  J. W. Silverstein,et al.  Spectral Analysis of Large Dimensional Random Matrices , 2009 .

[46]  Stefan Rolewicz,et al.  On a problem of moments , 1968 .

[47]  Iain B. Collings,et al.  Eigenvalue Distributions of Sums and Products of Large Random Matrices Via Incremental Matrix Expansions , 2005, IEEE Transactions on Information Theory.

[48]  J. Norris.  Appendix: probability and measure , 1997 .

[49]  Philippe Loubaton,et al.  On the Capacity Achieving Covariance Matrix for Frequency Selective MIMO Channels Using the Asymptotic Approach , 2010, IEEE Transactions on Information Theory.

[50]  Philippe Loubaton,et al.  On the Capacity Achieving Covariance Matrix for Rician MIMO Channels: An Asymptotic Approach , 2007, IEEE Transactions on Information Theory.