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

Let $\mathbf{H}=(h_{ij})$ and $\mathbf{G}=(g_{ij})$ be two $m\times n$, $m\leq n$, random matrices, each with i.i.d 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 bi-orthogonal polynomial technique. These PDFs are of interest in multiple-input multiple-output (MIMO) wireless communication channels and systems.