Statistically Optimal Beamforming and Ergodic Capacity for RIS-Aided MISO Systems

This paper focuses on optimal beamforming to maximize the mean signal-to-noise ratio (SNR) for a passive reconfigurable intelligent surface (RIS)-aided multiple-input single-output (MISO) downlink system. We consider a realistic setting where both the direct and indirect (through RIS) links to the user equipment (UE) experience correlated Rician fading. Such a general fading model is particularly important to capture the impact of line-of-sight (LoS) and correlated multipath fading that may occur due to the compact placement of a large number of RIS elements. The assumption of passive RIS imposes the unit modulus constraint, which makes the beamforming problem non-convex. To tackle this issue, we apply semidefinite relaxation (SDR) to obtain the optimal phase-shift matrix and propose an iterative algorithm to obtain a statistically optimal solution for the transmit beamforming vector and RIS-phase shift matrix. Further, to measure the performance of the proposed beamforming scheme, we analyze key system performance metrics such as outage probability (OP) and ergodic capacity (EC). Similar to the existing works, the OP and EC evaluations rely on the numerical computation of the proposed iterative algorithm. However, this is not conducive to revealing the functional dependence of system performance on key parameters such as line-of-sight (LoS) components, correlated fading, number of reflecting elements, number of antennas at the base station (BS), and fading factor. To overcome this major limitation, we derive closed-form expressions for the optimal beamforming vector and phase shift matrix for various special cases of the above general fading model. These fixed-point solutions aid in deriving a closed-form solution for OP that further provides a direct evaluation of EC. These mathematical expressions are then used to gain useful insights into the system’s performance. We analytically establish the fact that correlated fading is more beneficial than the independent and identically distributed (i.i.d.) case when the LoS components are blocked. Further, we also analytically demonstrate that the maximum mean SNR improves linearly/quadratically with the number of RIS elements in the absence/presence of LoS component under i.i.d. fading.