A Note on Functional Averages over Gaussian Ensembles

We find a new formula for matrix averages over the Gaussian ensemble. Let H be an n×n Gaussian random matrix with complex, independent, and identically distributed entries of zero mean and unit variance. Given an n×n positive definite matrix A and a continuous function f:ℝ+→ℝ such that ∫0∞e-αt|f(t)|2dt<∞ for every α>0, we find a new formula for the expectation[Tr(f(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