A Note on Functional Averages over Gaussian Ensembles

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.