Multi-RIS-Assisted mmWave MIMO Systems Exploiting Statistical CSI: Ergodic Capacity Analysis and Joint Beamforming Design

With a large number of tunable reflecting elements, reconfigurable intelligent surface (RIS) can shape a favorable electromagnetic wave propagation environment, significantly benefiting wireless communication. This paper focuses on a multi-RIS-aided millimeter wave (mmWave) multiple-input multiple-output (MIMO) system, which involves passive reflection beamforming at the RIS and active hybrid beamforming at the transmitter and receiver. The goal is to optimally design the RIS reflection matrix and the transmitter/reciever hybrid beamforming matrices, when only the statistical channel state information (CSI) is available. To achieve this, we first derive the closed-form ergodic capacity by the replica method originated in statistical physics. Then, we propose an alternating optimization (AO) algorithm to maximize the ergodic capacity, thereby fulfilling the design of reflection and beamforming matrices. To obtain the AO algorithm, we leverage several frameworks such as weighted minimum mean square error (WMMSE), majorization-minimization (MM) and manifold gradient. Simulation results corroborate that: (i) The derived analytical ergodic capacity can match well with the Monte Carlo averaging results in various scenarios. (ii) The proposed scheme remarkably outperforms the counterparts which are designed using instantaneous CSI.