Diagonal distribution of a complex non-central Wishart matrix: A new trivariate non-central chi-squared density

Abstract

a b s t r a c tThis paper derives the joint density of a particular trivariate non-central χ 2 distribution corresponding to the diagonal elements of a 3 × 3 complex non-central Wishart matrix.This distribution is important for a number of practical statistical signal processing applications, including synthetic aperture radar, extra-solar planet detection, and multiantenna wireless communications. The density expression is in the form of an infinite series representation which converges rapidly and is easy to compute. The joint density of the diagonal elements of a 2 × 2 complex non-central Wishart matrix is also derived by simple reduction of the trivariate result.

Spectral density of the noncentral correlated Wishart ensembles

“…It is easy to see that results (35) and (44) are immediate from the result (55), for the choices L = 1 N and ζ = 0, and L = 1 N and ξ = 1 N , respectively in (55).…”

“…For instance, consider the ξ jk = δ jk . For this choice the Pastur equation (35) yields the resolvent…”

“…Predating the Gaussian ensembles of RMT [1,31], such non-central matrices were introduced in mathematical statistics to better the so-called null hypothesis supplied by CWE [32]. However, the non-central Wishart ensembles (nc-WE) have been revisited recently in the context of signal processing [33] and in mathematical statistics [34][35][36]. In physics, however, nc-CWEs have been used recently in the context of density matrices [37] remarking that the zero-mean condition is a priory not valid for the density matrices.…”

Non-Central Multivariate Chi-Square and Gamma Distributions

“…a bivariate integral is obtained for the cdf of Jensen's bivariate chi-square distribution (see also [5] and [17]). A survey on formulas for the central 2 ( , ) p R  -distribution and some similar distributions is found in the appendix of [3]. [7] or [9] (and also [11]…”

“…( 1) p  -variate integral representation in section 5 is suitable for feasible computations for 3 p  and might be preferred for a numerical evaluation at least for cases with 2 1 k a  or for the here not included case with one vanishing correlation.…”