A Note on Functional Averages over Gaussian Ensembles

TL;DR

This paper derives a new formula for matrix averages over Gaussian ensembles, enabling easier computation of expectations related to MIMO channel capacity and MMSE performance.

Contribution

It introduces a novel formula for expectations of functions of Gaussian matrix products, applicable to capacity and MMSE calculations in communication systems.

Findings

New formula for matrix averages over Gaussian ensembles

Application to MIMO channel capacity calculation

Application to MMSE performance evaluation

Abstract

In this work we find a new formula for matrix averages over the Gaussian ensemble. Let ${\bf H}$ be an $n\times n$ Gaussian random matrix with complex, independent, and identically distributed entries of zero mean and unit variance. Given an $n\times n$ positive definite matrix ${\bf A}$, and a continuous function $f:\R^{+}\to\R$ such that $\int_{0}^{\infty}{e^{-\alpha t}|f(t)|^2\,dt}<\infty$ for every $\alpha>0$, we find a new formula for the expectation $\E[\mathrm{Tr}(f({\bf HAH^{*}}))]$. Taking $f(x)=\log(1+x)$ gives another formula for the capacity of the MIMO communication channel, and taking $f(x)=(1+x)^{-1}$ gives the MMSE achieved by a linear receiver.