On the Application of Character Expansions for MIMO Capacity Analysis

To evaluate the unitary integrals, such as the well-known Harish–Chandra–Itzykson–Zuber integral, character expansions were developed by Balantekin, where the matrix integrand is a group member; i.e., a square matrix with a nonzero determinant. Recently, this method has been exploited to derive the joint eigenvalue distributions of the Wishart matrices; i.e.,
${\bf H}{\bf H}^{\ast}$
where
${\bf H}$
is the complex Gaussian random channel matrix of a multiple-input multiple-output (MIMO) system. The joint eigenvalue distributions are used to calculate the moment generating function of the mutual information (ergodic capacity) of a MIMO channel. In this paper, we show that the previous integration framework presented in the literature is not correct, and results in incorrect joint eigenvalue distributions for the Ricean and full-correlated Rayleigh MIMO channels. We develop a new framework to apply the character expansions for integrations over the unitary group, involving general rectangular complex matrices in the integrand. We derive the correct distribution functions and use them to obtain the capacity of the Ricean and correlated Rayleigh MIMO systems in a unified and straightforward approach. The integration technique proposed in this paper is general enough to be used for other unitary integrals in engineering, mathematics, and physics.