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This paper considers beamforming for a reconfigurable intelligent surface(RIS)-aided multiple input single output (MISO) communication system in thepresence of Rician multipath fading. Our aim is to jointly optimize thetransmit beamformer and RIS phase shift matrix for maximizing the meansignal-to-noise (SNR) of the combined signal received over direct and indirectlinks. While numerical solutions are known for such optimization problems, thisis the first paper to derive closed-form expressions for the optimal beamformerand the phase shifter for a closely related problem. In particular, we maximizea carefully constructed lower bound of the mean SNR, which is more conducive toanalytical treatment. Further, we show that effective channel gain underoptimal beamforming follows Rice distribution. Next, we use these results tocharacterize a closed-form expression for the outage probability under theproposed beamforming scheme, which is subsequently employed to derive ananalytical expression for the ergodic capacity. Finally, we numericallydemonstrate the efficacy of the proposed beamformer solution in comparison withthe existing algorithmically obtained optimal solution for the exact mean SNRmaximization.

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