Eigenvalue Dynamics of a Central Wishart Matrix with Application to MIMO Systems

TL;DR

This paper analyzes the eigenvalue dynamics of central Wishart matrices and applies these findings to understand the behavior of MIMO communication channels under Rayleigh fading, providing exact formulas for eigenvalue distributions and channel SNR variations.

Contribution

It derives exact closed-form expressions for the joint distributions of eigenvalues of Wishart matrices and applies these to characterize MIMO channel SNR dynamics under fading.

Findings

The SNR of the best channel changes slower than the worst channel.

Eigenvalue distributions are expressed in closed-form using special functions.

Channel dynamics depend on the number of antennas, with similar rates when increasing transmit antennas.

Abstract

We investigate the dynamic behavior of the stationary random process defined by a central complex Wishart (CW) matrix ${\bf{W}}(t)$ as it varies along a certain dimension $t$. We characterize the second-order joint cdf of the largest eigenvalue, and the second-order joint cdf of the smallest eigenvalue of this matrix. We show that both cdfs can be expressed in exact closed-form in terms of a finite number of well-known special functions in the context of communication theory. As a direct application, we investigate the dynamic behavior of the parallel channels associated with multiple-input multiple-output (MIMO) systems in the presence of Rayleigh fading. Studying the complex random matrix that defines the MIMO channel, we characterize the second-order joint cdf of the signal-to-noise ratio (SNR) for the best and worst channels. We use these results to study the rate of change of MIMO parallel channels, using different performance metrics. For a given value of the MIMO channel correlation coefficient, we observe how the SNR associated with the best parallel channel changes slower than the SNR of the worst channel. This different dynamic behavior is much more appreciable when the number of transmit ($N_T$) and receive ($N_R$) antennas is similar. However, as $N_T$ is increased while keeping $N_R$ fixed, we see how the best and worst channels tend to have a similar rate of change.