A Note on Functional Averages over Gaussian Ensembles

We find a new formula for matrix averages over the Gaussian ensemble. Let be an Gaussian random matrix with complex, independent, and identically distributed entries of zero mean and unit variance. Given an positive definite matrix and a continuous function such that for every , we find a new formula for the expectation . Taking gives another formula for the capacity of the MIMO communication channel, and taking gives the MMSE achieved by a linear receiver. 1. Introduction Random matrix theory was introduced to the theoretical physics community by Wigner in his work on nuclear physics in the 1950s [1, 2]. Since that time, the subject is an important and active research area in mathematics, and it finds applications in fields as diverse as the Riemann conjecture, physics, chaotic systems, multivariate statistics, wireless communications, signal processing, compressed sensing, and information theory. In the last decades, a considerable amount of work has emerged in the communications and information theory on the fundamental limits of communication channels that make use of results in random matrix theory [3–5]. For this reason, computing averages over certain matrix ensembles becomes extremely important in many situations. To be more specific, consider the well-known case of the single user MIMO channel with multiple transmitting and receiving antennas. Denoting the number of transmitting antennas by and the number of receiving antennas by , the channel model is where is the transmitted vector, is the received vector, is a complex matrix, and is the zero mean complex Gaussian vector with independent, equal variance entries. We assume that , where denotes the complex conjugate transpose. It is reasonable to put a power constraint where is the total transmitted power. The signal to noise ratio, denoted by snr, is defined as the quotient of the signal power and the noise power and in this case is equal to . Recall that if is an Hermitian matrix, then there exists unitary and such that . Given a continuous function , we define as Naturally, the simplest example is the one where has independent and identically distributed (i.i.d.) Gaussian entries, which constitutes the canonical model for the single user narrow band MIMO channel. It is known that the capacity of this channel is achieved when is a complex Gaussian zero mean and covariance snr vector (see e.g., [4, 6]). For the fast fading channel, assuming the statistical channel state information at the transmitter, the ergodic capacity is given by where in the last equality we use the fact that . We refer