Min-Max Fairness-Based Beamforming Design for Coupled Phase Shift STAR-RIS-Assisted MIMO System

Simultaneously transmitting and reflecting reconfigurable intelligent surface (STAR-RIS) represents a novel technology that extends communication coverage to the entire space, compared to the conventional reconfigurable intelligent surface (RIS) covering only half space. However, previous studies mostly focused solely on best-effort services, potentially resulting in lower communication quality for users with poor channel conditions or even communication failure. Consequently, the fairness among users is investigated in the STAR-RIS-assisted multi-input-multi-output system to improve the system performance. To minimize the maximum average mean-square error of each user, a fair beamforming design is utilized to jointly optimize the transmit precoding matrices, reflecting and transmitting coefficients (RTCs), as well as receive combining matrices. To tackle this problem, we first decompose it into three tractable subproblems and then utilize an alternating-optimization algorithm to optimize each one alternately. For the transmit precoding and receive combining matrices, each matrices are obtained by transforming the nonconvex subproblem into a second-order cone programming problem. For the RTCs, considering a low-cost coupled phase shift STAR-RIS model, an element-wise two-stage search algorithm is proposed to optimize the phase shift and amplitude for both energy splitting and mode switching STAR-RIS protocols. Also, the proposed algorithms are extended to the robust beamforming design under imperfect channel state information. Simulation results demonstrate that the proposed algorithm outperforms the other benchmark schemes in the STAR-RIS-assisted MIMO system. Moreover, its performance is closest to the lower bound under various differences in channel conditions among users.