Ergodic Capacity and Beamforming Optimality for Multi-Antenna Relaying with Statistical CSI

We investigate the capacity and beamforming optimality of multi-antenna relaying systems, given access to statistical channel information at the relay and source. Multi-antenna relay configurations are considered, for which the source as well as either the relay or destination have multiple antennas, and the relay operates with amplify-and-forward. We first compute the optimal transmission strategies at both the source and relay, and derive necessary and sufficient conditions for which beamforming achieves capacity. To gain more insight, we then employ tools from stochastic majorization theory to characterize the impact of the destination correlation and the relay gain on the capacity and beamforming optimality range. These results demonstrate some intriguing behavior, which arises due to the joint interplay between the transmit power, relay gain, and level of correlation at the destination. It is shown, among other things, that for certain transmit powers and relay gains, the beamforming optimality region becomes independent of the destination correlation. By relaxing the beamforming optimality condition, we also derive a simple explicit upper bound which gives further insights into the joint effect of the SNR and the transmit correlation on the beamforming optimality range.