Correlated MIMO Rayleigh fading channels: capacity and optimal signaling

The capacity of the MIMO channel is investigated under the assumption that the elements of the channel matrix are zero mean proper complex Gaussian random variables with a general correlation structure. It is assumed that the receiver knows the channel perfectly but that the transmitter knows only the channel statistics. The analysis is carried out using an equivalent virtual representation of the channel that is obtained via a spatial discrete Fourier transform. The non-vanishing virtual channel coefficients are approximately independent and their variances correspond to samples of the underlying two-dimensional spatial scattering function that reflects the channel gain in different receive and transmit angular directions. It is shown that, in the virtual domain, the capacity achieving input vector is a zero mean proper complex Gaussian vector, with a diagonal covariance matrix Λ°, whose components can be computed numerically. Furthermore, in the asymptotic regime of low signal-to-noise ratio (SNR), it is shown that only one element of Λ° is non-zero, i.e., that beamforming along one virtual transmit angle is asymptotically optimum. In the more general cast of arbitrary SNR, a necessary and sufficient condition for the optimality of beamforming is also derived.