Distribution and characteristic functions for correlated complex Wishart matrices

Abstract

Let A ( t ) be a complex Wishart process defined in terms of the M × N complex Gaussian matrix X ( t ) by A ( t ) = X ( t ) X ( t ) H . The covariance matrix of the columns of X ( t ) is Σ . If X ( t ) , the underlying Gaussian process, is a correlated process over time, then we have dependence between samples of the Wishart process. In this paper, we study the joint statistics of the Wishart process at two points in time, t 1 , t 2 , where t 1 < t 2 . In particular, we derive the following results: the joint density of the elements of A ( t 1 ) , A ( t 2 ) , the joint density of the eigenvalues of Σ - 1 A ( t 1 ) , Σ - 1 A ( t 2 ) , the characteristic function of the elements of A ( t 1 ) , A ( t 2 ) , the characteristic function of the eigenvalues of Σ - 1 A ( t 1 ) , Σ - 1 A ( t 2 ) . In addition, we give the characteristic functions of the eigenvalues of a central and non-central complex Wishart, and some applications of the results in statistics, engineering and information theory are outlined.