Optimality of beamforming in MIMO multi-access channels via virtual representation

In this paper, we consider the optimality of the beamforming scheme for both the multiple-input multiple-output (MIMO) point-to-point channel and the MIMO multiple access channel (MAC), where all communication terminals are assumed to be equipped with multiple antennas. For both channels, the channel matrices have correlated elements and are modelled by virtual representation. For the point-to-point channel, i.e., the single user case, we show that the optimal beamforming angle is unique and is independent of the signal-to-noise ratio (SNR). We further show that there exists a certain SNR threshold below which beamforming is optimal and above which beamforming is strictly suboptimal. For the MIMO MAC, we show that to achieve sum capacity, the inputs from different users are independent and their covariance matrices are diagonal. We also derive a necessary and sufficient condition for the optimal input distribution to achieve the sum capacity. Based on these results, we investigate the conditions under which beamforming achieves the sum capacity. We show that the optimal beamforming angles are not unique, and are dependent on both the value of SNR and beamforming angles of other users. We further provide explicit conditions to determine the optimal beamforming angles for a special class of correlated MIMO MACs.