Capacity and Character Expansions: Moment-Generating Function and Other Exact Results for MIMO Correlated Channels

Abstract

We apply a promising new method from the field of representations of Lie groups to calculate integrals over unitary groups, which are important for multi-antenna communications. To demonstrate the power and simplicity of this technique, we first re-derive a number of results that have been used recently in the community of wireless information theory, using only a few simple steps. In particular, we derive the joint probability distribution of eigenvalues of the matrix GG † , with G a semicorrelated Gaussian random matrix or a Gaussian random matrix with a non-zero mean. These joint probability distribution functions can then be used to calculate the moment generating function of the mutual information for Gaussian MIMO channels with these probability distribution of their channel matrices G. We then turn to the previously unsolved problem of calculating the moment generating function of the mutual information of MIMO channels, which are correlated at both the receiver and the transmitter. From this moment generating function we obtain the ergodic average of the mutual information and study the outage probability. These methods can be applied to a number of other problems. As a particular example, we examine unitary encoded space-time transmission of MIMO systems and we derive the received signal distribution when the channel matrix is correlated at the transmitter end.

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

“…The authors in [13] try to solve the integral in (10) by using the character expansion of groups. We show that the attempt in [13] is not correct and solve (11) in its general form, which is our main contribution in this paper.…”

“…We emphasize that the second limit in (19) is critical. The authors in [13] does not consider this point and by adding rows and columns to matrices, they practically take both integrations over U(N ) and thus the final result is not correct. Interested readers may examine equation (52) in [13] for a one transmit and two receive antenna system or refer to [19] and [7] for more details.…”

“…Recently, the character expansion method, introduced by Balantekin [12], has been exploited in [13] to calculate the joint pdf of the eigenvalues of a correlated Wishart matrix for capacity analysis of full correlated MIMO channels. In [12], the character expansions are used for integration over unitary matrices where the coefficient matrices are nonzerodeterminant square matrices.…”

“…However, when the channel matrix is not a square matrix, the integrations are over unitary matrices with rectangular coefficient matrices. To handle the integrations with non-square coefficient matrices, a framework is proposed in [13]. We believe that the integration steps presented in [13] are not accurate, which result in incorrect joint eigenvalue distributions.…”

“…To handle the integrations with non-square coefficient matrices, a framework is proposed in [13]. We believe that the integration steps presented in [13] are not accurate, which result in incorrect joint eigenvalue distributions. In this paper, after a brief introduction to the character expansion of groups, we present a modified framework to use the character expansions for integrations over unitary matrices.…”