Journal of Probability and Statistics (Jan 2013)

A Note on Functional Averages over Gaussian Ensembles

  • Gabriel H. Tucci,
  • Maria V. Vega

DOI
https://doi.org/10.1155/2013/941058
Journal volume & issue
Vol. 2013

Abstract

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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)|2dt0, we find a new formula for the expectation 𝔼[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.