On the Eigenvalue Distribution of Ricean MIMO Channels by Character Expansion of Groups

Joint eigenvalue distribution of the noncentral complex Wishart matrix, i.e. HH* where H is the nonzero-mean complex Gaussian random channel matrix of a multiple-input multiple-output (MIMO) system, is required for the analysis of Ricean MIMO channels from different aspects, including the average of mutual information between the transmitter and the receiver (ergodic capacity), when the channel gains are known to the receiver only. Previous works rely on the available results in mathematics for the joint eigenvalue distribution, obtained by integration over unitary matrices using classic integration methods. In this paper, we present a powerful integration method over unitary matrices which exploits the representation theory and characters of groups. The method was originally proposed for square matrices. We modify the approach from square matrices to rectangular matrices to solve a more general integral over unitary matrices and obtain the joint eigenvalue distribution of the noncentral Wishart matrix. Our result is the generalization of the previous classical integral over unitary matrices so that the result is not restricted to diagonal and/or real matrices, particularly.