Random matrices, free probability and the replica method

This survey gives an overview of analytic tools to the design, analysis, and modelling of communication systems which can be described by linear vector channels such as where the number of components in each vector is large. Tools from probability theory, operator algebra, and statistical physics are reviewed. Asymptotic eigenvalue distributions of some classes of random matrices are given in terms of densities, moments and/or Stieltjes transforms. Free probability theory which evolved from non-commutative operator algebras is explained from a probabilistic point of view in order to better fit the engineering community. For that purpose freeness is defined without reference to noncommutative algebras. The treatment includes additive and multiplicative free convolution, the R-transform, and the S-transform. The replica method developed in statistical physics for the purpose of analyzing spin glasses is reviewed from the view point of its applications in communications engineering. Correspondences between free energy and mutual information as well as energy functions and detector metrics are established.

[1]  I. Miller.  Probability, Random Variables, and Stochastic Processes , 1966 .

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

[3]  J.E. Mazo,et al.  Digital communications , 1985, Proceedings of the IEEE.

[4]  Y. Yin.  Limiting spectral distribution for a class of random matrices , 1986 .

[5]  P. Hilton,et al.  Catalan Numbers, Their Generalization, and Their Uses , 1991 .

[6]  D. Voiculescu.  Limit laws for Random matrices and free products , 1991 .

[7]  James L. Massey,et al.  Proper complex random processes with applications to information theory , 1993, IEEE Trans. Inf. Theory.

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

[9]  J. W. Silverstein.  Strong convergence of the empirical distribution of eigenvalues of large dimensional random matrices , 1995 .

[10]  Bernard Fino,et al.  Multiuser detection: , 1999, Ann. des Télécommunications.

[11]  David Tse.  Multiuser Receivers, Random Matrices and Free Probability , 1999 .

[12]  S. Thorbjjrnsen.  Mixed Moments of Voiculescu's Gaussian Random Matrices , 1999 .

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

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

[15]  Jamie S. Evans,et al.  Large system performance of linear multiuser receivers in multipath fading channels , 2000, IEEE Trans. Inf. Theory.

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

[17]  David Tse,et al.  Effective interference and effective bandwidth of linear multiuser receivers in asynchronous CDMA systems , 2000, IEEE Trans. Inf. Theory.

[18]  Chen-Nee Chuah,et al.  Capacity scaling in dual-antenna-array wireless systems , 2000, Conference Record of the Thirty-Fourth Asilomar Conference on Signals, Systems and Computers (Cat. No.00CH37154).

[19]  David Tse,et al.  Linear multiuser receivers in random environments , 2000, IEEE Trans. Inf. Theory.

[20]  Edwin K. P. Chong,et al.  Output MAI distributions of linear MMSE multiuser receivers in DS-CDMA systems , 2001, IEEE Trans. Inf. Theory.

[21]  Michael L. Honig,et al.  Performance of reduced-rank linear interference suppression , 2001, IEEE Trans. Inf. Theory.

[22]  Shlomo Shamai,et al.  The impact of frequency-flat fading on the spectral efficiency of CDMA , 2001, IEEE Trans. Inf. Theory.

[23]  David Tse,et al.  Resource pooling and effective bandwidths in CDMA networks with multiuser receivers and spatial diversity , 2001, IEEE Trans. Inf. Theory.

[24]  西森 秀稔.  Statistical physics of spin glasses and information processing : an introduction , 2001 .

[25]  S. Verdú,et al.  Asymptotic eigenvalue moments for linear multiuser detection , 2001, Conference Record of Thirty-Fifth Asilomar Conference on Signals, Systems and Computers (Cat.No.01CH37256).

[26]  Alexander Vardy,et al.  The turbo decoding algorithm and its phase trajectories , 2001, IEEE Trans. Inf. Theory.

[27]  Giuseppe Caire,et al.  Iterative multiuser joint decoding: Unified framework and asymptotic analysis , 2002, IEEE Trans. Inf. Theory.

[28]  Toshiyuki Tanaka,et al.  A statistical-mechanics approach to large-system analysis of CDMA multiuser detectors , 2002, IEEE Trans. Inf. Theory.

[29]  Giuseppe Caire,et al.  Density evolution and power profile optimization for iterative multiuser decoders based on individually optimum multiuser detectors , 2002 .

[30]  Sergio Verdú,et al.  Asymptotic normality of linear multiuser receiver outputs , 2002, IEEE Trans. Inf. Theory.

[31]  Xiaodong Wang,et al.  Large-system performance analysis of blind and group-blind multiuser receivers , 2002, IEEE Trans. Inf. Theory.

[32]  Ralf R. Müller,et al.  On the asymptotic eigenvalue distribution of concatenated vector-valued fading channels , 2002, IEEE Trans. Inf. Theory.

[33]  Wolfgang H. Gerstacker,et al.  On the capacity loss due to separation of detection and decoding in large CDMA systems , 2002, Proceedings of the IEEE Information Theory Workshop.

[34]  G. Caire,et al.  Efficient implementation of iterative multiuser decoding , 2002, Proceedings IEEE International Symposium on Information Theory,.

[35]  Ralf R. Müller,et al.  A random matrix model of communication via antenna arrays , 2002, IEEE Trans. Inf. Theory.

[36]  Edwin K. P. Chong,et al.  Linear MMSE Multiuser receivers: MAI Conditional weak convergence and network capacity , 2002, IEEE Trans. Inf. Theory.

[37]  Michael L. Honig,et al.  Performance of coded DS-CDMA with pilot-assisted channel estimation and linear interference suppression , 2002, IEEE Trans. Commun..

[38]  Antonia Maria Tulino,et al.  Performance of space-time codes for a large number of antennas , 2002, IEEE Trans. Inf. Theory.

[39]  L. Cottatellucci,et al.  Multiuser interference mitigation with multistage detectors: design and analysis for unequal powers , 2002, Conference Record of the Thirty-Sixth Asilomar Conference on Signals, Systems and Computers, 2002..

[40]  Yoshiyuki Kabashima,et al.  A statistical mechanical approach to cdma multiuser detection: propagating beliefs in a dense graph , 2002, IEEE International Symposium on Information Theory, 2003. Proceedings..

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

[42]  S. Verdú,et al.  Multiuser Detection and Statistical Mechanics , 2003 .

[43]  Toshiyuki TANAKA,et al.  A statistical-mechanical analysis of coded CDMA with regular LDPC codes , 2003, IEEE International Symposium on Information Theory, 2003. Proceedings..

[44]  H. Vincent Poor,et al.  Asymptotic spectral efficiency of multicell MIMO systems with frequency-flat fading , 2003, IEEE Trans. Signal Process..

[45]  Dongning Guo,et al.  Decoupling of CDMA multiuser detection via the replica method , 2003 .

[46]  Dongning Guo,et al.  Spectral efficiency of large-system CDMA via statistical physics , 2003 .

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

[48]  Ralf R. Müller,et al.  A systematic approach to multistage detectors in multipath fading channels , 2005, IEEE Transactions on Information Theory.