On the Diagonal Distribution of a Complex Wishart Matrix and its Application to the Analysis of MIMO Systems

Abstract—The statistical properties of Wishart matrices have been extensively used to analyze the performance of multiple-input multiple-output (MIMO) systems. In particular, the signal-to-noise ratio (SNR) output statistics of several MIMO systems depends on the diagonal distribution of a complex Wishart matrix. In this paper, we derive the joint density of the diagonal elements of a complex Wishart matrix, which follows a multivariate chi-square distribution. The density expression is in the form of an infinite series representation which converges rapidly and is easy to compute. This expression is used to obtain the distribution of the maximum of the diagonal elements, which allows analyzing the performance of two different MIMO systems under practical conditions. First, our statistical results are applied to the outage probability characterization of MIMO systems with receive antenna selection in spatially correlated Rayleigh fading. Then, the same results are used to analyze the outage probability of transmit beamforming systems under limited-rate feedback. Index Terms—Complex Wishart matrix, multivariate chi-square distribution, receive antenna selection, spatial correlation, transmit beamforming. I