Optimal Phase Shift Solution and Diversity Analysis for Discrete RIS-assisted Systems under Rank Deficient Channels

This paper investigates the design of finite discrete control for each reflecting element of a reconfigurable intelligent surface (RIS) that assists a point-to-point multiple-input single-output communication. Under the assumption that the propagation channel is rank deficient, we propose an optimal algorithm that retrieves the same solution as the exhaustive search method, yet with a polynomial complexity with respect to the number of reflecting elements. Interestingly, we establish that under some specified conditions, the proposed design achieves a signal-to-noise ratio that increases quadratically with the number of reflecting elements, just as the continuously controllable phase shift control. Moreover, we remark that the performance gap between the discrete RIS and the continuous one becomes negligible when the phase shifts of the cascaded channel paths become close to each other. Simulation results confirm the superiority of the proposed design, verify the derived conclusion, and provide us with useful implementation insights.