Random Beamforming Over Quasi-Static and Fading Channels: A Deterministic Equivalent Approach

In this work, we study the performance of random isometric precoders over quasi-static and correlated fading channels. We derive deterministic approximations of the mutual information and the signal-to-interference-plus-noise ratio (SINR) at the output of the minimum-mean-square-error (MMSE) receiver and provide simple provably converging fixed-point algorithms for their computation. Although these approximations are only proven exact in the asymptotic regime with infinitely many antennas at the transmitters and receivers, simulations suggest that they closely match the performance of small-dimensional systems. We exemplarily apply our results to the performance analysis of multi-cellular communication systems, multiple-input multiple-output multiple-access channels (MIMO-MAC), and MIMO interference channels. The mathematical analysis is based on the Stieltjes transform method. This enables the derivation of deterministic equivalents of functionals of large-dimensional random matrices. In contrast to previous works, our analysis does not rely on arguments from free probability theory which enables the consideration of random matrix models for which asymptotic freeness does not hold. Thus, the results of this work are also a novel contribution to the field of random matrix theory and applicable to a wide spectrum of practical systems.

[1]  F. Hiai,et al.  The semicircle law, free random variables, and entropy , 2006 .

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

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

[4]  L. Pastur,et al.  Eigenvalue Distribution of Large Random Matrices , 2011 .

[5]  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.

[6]  Iain B. Collings,et al.  Asymptotic analysis of LMMSE multiuser receivers for multi-signature multicarrier CDMA in Rayleigh fading , 2004, IEEE Transactions on Communications.

[7]  Preben E. Mogensen,et al.  A stochastic model of the temporal and azimuthal dispersion seen at the base station in outdoor propagation environments , 2000, IEEE Trans. Veh. Technol..

[8]  Mérouane Debbah,et al.  A Deterministic Equivalent for the Analysis of Correlated MIMO Multiple Access Channels , 2009, IEEE Transactions on Information Theory.

[9]  Walid Hachem,et al.  Simple polynomial detectors for CDMA downlink transmissions on frequency-selective channels , 2004, IEEE Transactions on Information Theory.

[10]  Jeffrey G. Andrews,et al.  Performance of Orthogonal Beamforming for SDMA With Limited Feedback , 2006, IEEE Transactions on Vehicular Technology.

[11]  O. Johnson.  Free Random Variables , 2004 .

[12]  R. Couillet,et al.  Large System Analysis of Linear Precoding in MISO Broadcast Channels with Limited Feedback , 2009 .

[13]  Sebastian Wagner,et al.  MU-MIMO transmission and reception techniques for the next generation of cellular wireless standards (LTE-A) , 2011 .

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

[15]  Jianzhong Zhang,et al.  MIMO Technologies in 3GPP LTE and LTE-Advanced , 2009, EURASIP J. Wirel. Commun. Netw..

[16]  Sergio Verdu,et al.  Multiuser Detection , 1998 .

[17]  R. Heath,et al.  Limited feedback unitary precoding for spatial multiplexing systems , 2005, IEEE Transactions on Information Theory.

[18]  Philippe Loubaton,et al.  On the asymptotic analysis of mutual information of MIMO Rician correlated channels , 2006 .

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

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

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

[22]  Mérouane Debbah,et al.  Large System Analysis of Linear Precoding in Correlated MISO Broadcast Channels Under Limited Feedback , 2009, IEEE Transactions on Information Theory.

[23]  Thomas M. Cover,et al.  Elements of information theory (2. ed.) , 2006 .

[24]  Rick S. Blum.  MIMO capacity with interference , 2003, IEEE J. Sel. Areas Commun..

[25]  Dan Voiculescu,et al.  Free Noncommutative Random Variables, Random Matrices and the II1 Factors of Free Groups , 1991 .

[26]  Mérouane Debbah,et al.  Asymptotic moments for interference mitigation in correlated fading channels , 2011, 2011 IEEE International Symposium on Information Theory Proceedings.

[27]  W. Rudin.  Real and complex analysis, 3rd ed. , 1987 .

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

[29]  R. Couillet,et al.  Uplink capacity of self-organizing clustered orthogonal CDMA networks in flat fading channels , 2009, 2009 IEEE Information Theory Workshop.

[30]  R. Cooke.  Real and Complex Analysis , 2011 .

[31]  Shlomo Shamai,et al.  Spectral Efficiency of CDMA with Random Spreading , 1999, IEEE Trans. Inf. Theory.

[32]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

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

[34]  R. Couillet,et al.  Random Matrix Methods for Wireless Communications: Estimation , 2011 .

[35]  Angel E. Lozano,et al.  Effect of antenna separation on the capacity of BLAST in correlated channels , 2000, IEEE Communications Letters.

[36]  Rick S. Blum,et al.  On the capacity of cellular systems with MIMO , 2002, IEEE Communications Letters.

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

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

[39]  Carlos Vargas,et al.  Free Deterministic Equivalents, Rectangular Random Matrix Models, and Operator-Valued Free Probability Theory , 2011, ArXiv.

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

[41]  J. W. Silverstein,et al.  No eigenvalues outside the support of the limiting spectral distribution of large-dimensional sample covariance matrices , 1998 .

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

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

[44]  Roy D. Yates,et al.  A Framework for Uplink Power Control in Cellular Radio Systems , 1995, IEEE J. Sel. Areas Commun..

[45]  Iain B. Collings,et al.  Asymptotic spectral efficiency of multiuser multisignature CDMA in frequency-selective channels , 2006, IEEE Transactions on Information Theory.

[46]  Cedric E. Ginestet.  Spectral Analysis of Large Dimensional Random Matrices, 2nd edn , 2012 .

[47]  Thomas M. Cover,et al.  Elements of Information Theory , 2005 .

[48]  David Tse,et al.  Opportunistic beamforming using dumb antennas , 2002, IEEE Trans. Inf. Theory.