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

Abstract

Let $\mathbf{H}=(h_{ij})$ and $\mathbf{G}=(g_{ij})$ be two $m\times n$, $m\leq n$, rectangular random matrices, each with independently and identically distributed complex zero-mean unit-variance Gaussian entries, with correlation between any two elements given by $\mathbb{E}[h_{ij}g_{pq}^\star]=\rho\,\delta_{ip} \delta_{jq}$ such that $|\rho|<1$, where ${}^\star$ denotes the complex conjugate and $\delta_{ij}$ is the Kronecker delta. Assume $\{s_k\}_{k=1}^m$ and $\{r_l\}_{l=1}^m$ are unordered singular values of $\mathbf{H}$ and $\mathbf{G}$, respectively, and s and r are randomly selected from $\{s_k\}_{k=1}^m$ and $\{r_l\}_{l=1}^m$, respectively. In this paper, exact analytical closed-form expressions are derived for the joint probability distribution function (PDF) of $\{s_k\}_{k=1}^m$ and $\{r_l\}_{l=1}^m$ using an Itzykson–Zuber-type integral as well as the joint marginal PDF of s and r by a biorthogonal polynomial technique. These PDFs are of interest in multiple-input multiple-output wireless communication channels and systems.