  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 (and G* its hermitian conjugate) . These
joint probability distribution functions can then be used to calculate the
moment generating function of the mutual information for Gaussian channels with
multiple antennas on both ends with this 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
(multiple input-multiple output) 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.
