Joint Singular Value Distribution of Two Correlated Rectangular Gaussian Matrices and Its Application

Let H = (hij) and G = (gij) be two m × n, m ≤ n, random matrices, each with i.i.d complex zero-mean unit-variance Gaussian entries, with correlation between any two elements given by E[hijg ⋆ pq ] = ρ δipδjq such that |ρ| < 1, where ⋆ denotes the complex conjugate and δij is the Kronecker delta. Assume {sk} m k=1 and {rl} m l=1 are unordered singular values of H and G, respectively, and s and r are randomly selected from {sk} m k=1 and {rl} m l=1 , respectively. In this paper, exact analytical closed-form expressions are derived for the joint probability distribution function (PDF) of {sk} m k=1 and {rl} m l=1 using an Itzykson-Zuber-type integral, as well as the joint marginal PDF of s and r, by a bi-orthogonal polynomial technique. These PDFs are of interest in multiple-input multiple-output (MIMO) wireless communication channels and systems.