A Note on Functional Averages over Gaussian Ensembles

In this work we find a new formula for matrix averages over the Gaussian ensemble. Let $\mathbf{H}$ 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 $\mathbf{A}$, and a continuous function $f:{\mathbb{R}}^{+}\to \mathbb{R}$ such that $\int_{0}^{\infty}{e^{-\alpha t}|f(t)|^2,dt}<\infty$forevery$\alpha>0$, we find a new formula for the expectation $\text{E}[\mathrm{T}\mathrm{r}(f(\mathbf{H}\mathbf{A}{\mathbf{H}}^{\ast }))]$. 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.