  We study memoryless, discrete time, matrix channels with additive white
Gaussian noise and input power constraints of the form $Y_i = \sum_j H_{ij} X_j
+ Z_i$, where $Y_i$ ,$X_j$ and $Z_i$ are complex, $i=1..m$, $j=1..n$, and $H$
is a complex $m\times n$ matrix with some degree of randomness in its entries.
The additive Gaussian noise vector is assumed to have uncorrelated entries. Let
$H$ be a full matrix (non-sparse) with pairwise correlations between matrix
entries of the form $ E[H_{ik} H^*_{jl}] = {1\over n}C_{ij} D_{kl} $, where
$C$,$D$ are positive definite Hermitian matrices. Simplicities arise in the
limit of large matrix sizes (the so called large-N limit) which allow us to
obtain several exact expressions relating to the channel capacity. We study the
probability distribution of the quantity $ f(H) = \log \det (1+P H^{\dagger}S
H) $. $S$ is non-negative definite and hermitian, with $Tr S=n$. Note that the
expectation $E[f(H)]$, maximised over $S$, gives the capacity of the above
channel with an input power constraint in the case $H$ is known at the receiver
but not at the transmitter. For arbitrary $C$,$D$ exact expressions are
obtained for the expectation and variance of $f(H)$ in the large matrix size
limit. For $C=D=I$, where $I$ is the identity matrix, expressions are in
addition obtained for the full moment generating function for arbitrary
(finite) matrix size in the large signal to noise limit. Finally, we obtain the
channel capacity where the channel matrix is partly known and partly unknown
and of the form $\alpha I+ \beta H$, $\alpha,\beta$ being known constants and
entries of $H$ i.i.d. Gaussian with variance $1/n$. Channels of the form
described above are of interest for wireless transmission with multiple
antennae and receivers.
