Phase Shifter Optimization in RIS-Aided MIMO Systems Under Multiple Reflections

We examine the problem of joint active and passive beamforming in a controllable multi-user reconfigurable intelligent surface (RIS)-assisted downlink and uplink wireless communication system, considering the mutual coupling among RIS elements. Due to the sub-wavelength structure, mutual coupling among RIS elements is unavoidable, and it inherently leads to multiple reflection effects that are ignored in conventional (approximative) RIS models. We formulate a joint non-convex problem under the MMSE criterion and use alternative optimization to convert the non-convex problem into two sub-problems for downlink and uplink transmissions separately. In both transmissions, one sub-problem involves optimizing the phase-shift matrix of RIS. In downlink, the other sub-problem is the optimization of active precoding for the base station (BS), while the equivalent sub-problem in uplink is the optimization of the linear receiver matrix. We optimize the phase shift matrix under a physically-consistent model using the gradient descent algorithm for both transmissions. We use the Lagrange multiplier method to optimize active precoding in the downlink and apply the First Order Necessary Condition (FONC) to optimize the linear receiver in the uplink. Simulation results are represented for both lossless and lossy RIS scenarios under perfect and imperfect channel state information. We discuss the impact of changing the number of RIS elements and the RIS element spacing on system performance. The results show that, with optimized phase shifts and active precoding, the inherent multiple reflection effect can improve the performance of RIS-aided wireless communications systems.