Large $n$ Analysis of Amplify-and-Forward MIMO Relay Channels With Correlated Rayleigh Fading

In this correspondence, the cumulants of the mutual information of the flat Rayleigh fading two-hop amplify-and-forward multiple-input multiple-output (MIMO) relay channel under independent and identically distributed (i.i.d.) Gaussian input vectors are derived in the large array limit. The analysis is based on the
replica

trick
and covers both spatially independent and Kronecker correlated fading. Beamforming at all terminals is restricted to weight matrices that are independent of the channel realization and constant over time. Expressions for mean and variance of the mutual information are obtained. Their parameters are determined by a nonlinear equation system. All higher cumulants are shown to vanish as the number of antennas per terminal,
n
, grows to infinity. In conclusion, the distribution of the mutual information
I
becomes Gaussian in the large
n
limit. In this asymptotic regime, it is completely characterized by the expressions obtained for mean and variance of
I
, which are in
Theta(n
) and
O
(1), respectively. Comparisons with simulation results show that the asymptotic results serve as excellent approximations for systems with only few antennas at each terminal.