Optimality of Beamforming for MIMO Multiple Access Channels Via Virtual Representation

In this correspondence, we consider the optimality of beamforming for achieving the ergodic capacity of multiple-input multiple-output (MIMO) multiple access channel (MAC) via virtual representation (VR) model. We assume that the receiver knows the channel state information (CSI) perfectly but that the transmitter knows only partial CSI, i.e., the channel statistics. For the single-user case, we prove that the capacity-achieving beamforming angle (c.b.a.) is unique, and there exists a signal-to-noise ratio (SNR) threshold below which beamforming is optimal and above which beamforming is strictly suboptimal. For the multi-user case, we show that the c.b.a is not unique and we obtain explicit conditions that determine the beamforming angles for a special class of correlated MAC-VR models. Under mild conditions, we show that a large class of power allocation schemes can achieve the sum-capacity within a constant as the number of users in the system becomes large. The beamforming scheme, in particular, is shown to be asymptotically capacity-achieving only for certain MAC-VR models.