A Note on Functional Averages over Gaussian Ensembles. (11th July 2013) - Tucci, Gabriel H.; Vega, Maria V.
Abstract : 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 - α t | f ( t ) | 2 d t < ∞ for every α > 0, we find a new formula for the expectation 𝔼 [ Tr ( f ( H A H * ) ) ] . Taking f ( x ) = l o g ( 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.