Channel capacity and beamforming for multiple transmit and receive antennas with covariance feedback

We consider the capacity of a narrowband point to point communication system employing multiple-element antenna arrays at both the transmitter and the receiver with covariance feedback. Under covariance feedback the receiver is assumed to have perfect channel state information (CSI) while at the transmitter the channel matrix is modeled as consisting of zero mean complex jointly Gaussian random variables with known covariances. Specifically we assume a channel matrix with i.i.d. rows and correlated columns, a common model for downlink transmission. We determine the optimal transmit precoding strategy to maximize the Shannon capacity of such a system. We also derive closed form necessary and sufficient conditions on the spatial covariance for when the maximum capacity is achieved by beamforming. The conditions for optimality of beamforming agree with the notion of water-filling over multiple degrees of freedom