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

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dc.creatorWang, Shuangquan
dc.creatorAbdi, Ali
dc.date2006-03-07
dc.date.accessioned2026-07-07T07:06:37Z
dc.date.available2026-07-07T07:06:37Z
dc.descriptionLet $\mathbf{H}=(h_{ij})$ 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 $\mathbb{E}[h_{ij}g_{pq}^\star]=ρδ_{ip}δ_{jq}$ such that $|ρ|<1$, where ${}^\star$ denotes the complex conjugate and $δ_{ij}$ 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.
dc.description10 pages, 1 figure, submitted to SIAM J. Matrix Anal. Appl
dc.identifierhttps://arxiv.org/abs/math/0603170
dc.identifierhttp://arxiv.org/abs/math/0603170
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/110096
dc.subjectProbability
dc.subject15A52; 15A18; 62E15; 33C45
dc.titleJoint Singular Value Distribution of Two Correlated Rectangular Gaussian Matrices and Its Application
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