Topics in complex random matrices and information theory

Abstract

The eigenvalue distribution of both central and noncentral complex Wishart matrices are investigated with the objective of studying several open problems in information theory and numerical analysis, etc. Specifically, the largest, kth largest, and the smallest eigenvalue distributions of complex Wishart matrices and the condition number distribution of complex Gaussian random matrices are derived. These distributions are represented by complex hypergeometric functions of matrix arguments, which can be expressed in terms of complex zonal polynomials. We derive several results on complex hypergeometric functions and zonal polynomials that are used to evaluate these distributions. We also give a method to compute these complex hypergeometric functions. Then the connection between the complex Wishart matrix theory and information theory is formulated. This facilitates the evaluation of the most important information-theoretic measure, the so-called channel capacity. The capacity of the communication channel expresses the maximum rate at which information can be reliably conveyed by the channel. In particular, the capacities of the multiple input, multiple output Rayleigh and Rician distributed channels are fully investigated. We consider both correlated and uncorrelated channels and derive the corresponding channel capacity formulas. These studies show how the channel correlations degrade the capacity of the communication system.