Capacity of multivariate channels with multiplicative noise: I.Random matrix techniques and large-N expansions for full transfer matrices

TL;DR

This paper derives exact formulas for the capacity of large multivariate Gaussian channels with random full transfer matrices, using random matrix theory and large-N expansions, relevant for multi-antenna wireless systems.

Contribution

It provides novel exact expressions for the distribution and moments of the channel capacity in large random matrix channels with correlated entries, including partially known channel matrices.

Findings

Exact expectation and variance of the capacity in large matrix limit.

Full moment generating function for identity correlation matrices in high SNR.

Capacity formulas for channels with partially known transfer matrices.

Abstract

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.