Optimal Beamforming and Outage Analysis for Max Mean SNR under RIS-aided Communication

This paper investigates beamforming for a reconfigurable intelligent surface (RIS)-aided multiple input single output (MISO) communication system in the presence of Rician multipath fading. Our objective is to jointly optimize the transmit beamformer and RIS phase shift matrix that maximizes the mean signal-to-noise ratio (SNR) of the signal received over direct and indirect links. While numerical approaches have been known to address such optimization problems, our work distinguishes itself by providing closed-form expressions for the optimal transmit beamformer and RIS phase shifter. In particular, we maximize a carefully constructed lower bound on the mean SNR, which is more conducive to analytical treatment. Additionally, we establish that the effective channel envelope, under optimal beamforming, follows the Rice distribution. We leverage this result to derive a closed-form expression for outage probability within the proposed beamforming framework, which is subsequently employed to derive an analytical expression for ergodic capacity. Finally, we numerically validate the effectiveness of our proposed beamforming closed-form solution by demonstrating that it performs very close to the algorithmically obtained optimal solution that maximizes the exact mean SNR.