Second-order statistics of large isometric matrices and applications to MMSE SIR

In this paper we introduce a diagrammatic method to calculate asymptotic statistics of functions of large random isometric matrices. We have applied this method to calculate the mean and variance of the MMSE SIR for downlink synchronous CDMA systems. We compare our results to numerical simulations using three types of randomly generated isometric matrices, namely complex unitary Haar matrices, real orthogonal Haar matrices and orthogonal matrices generated from random sub- spaces of the N-dimensional real Hadamard matrix. While the first two types of matrices have good agreement with our analytic results, the Hadamard generated matrices give a consistently higher variance when the channel matrix is assumed to have a Toeplitz form. We argue that this discrepancy is due to the structure of eigenvectors of the Hadamard matrix.