On the Separability of Beamforming for Reconfigurable Intelligent Surfaces Under Statistical CSI

Reconfigurable Intelligent Surfaces (RIS) have channel models with many parameters, motivating the study and use of statistical channel state information (CSI) in this context. This paper investigates the question of separability of RIS/transmitter beamforming under spatially correlated Rician fading with statistical channel state information. Subject to any spatial correlation, as long as the transmitter-RIS channel is Rayleigh, we show that the beamforming optimization decomposes into two independent optimizations involving the RIS correlation matrix, and the transmitter array correlation matrix. The tools and techniques of this paper also lead to novel and useful optimizations for beamforming optimization when the transmitter-RIS channel is Rician with non-zero mean, even though in this case separability is not established. Numerical results support our findings and provide insights into the proposed algorithm.