Joint beamforming optimization for reconfigurable intelligent surface-enabled MISO-OFDM systems

In this paper, we investigate the reconfigurable intelligent surface (RIS)-enabled multiple-input-single-output orthogonal frequency division multiplexing (MISO-OFDM) system under frequency-selective channels, and propose a low-complexity alternating optimization (AO) based joint beamforming and RIS phase shifts optimization algorithm to maximize the achievable rate. First, with fixed RIS phase shifts, we devise the optimal closed- form transmit beamforming vectors corresponding to different subcarriers. Then, with given active beamforming vectors, near-optimal RIS reflection coefficients can be determined efficiently leveraging fractional programming (FP) combined with manifold optimization (MO) or majorization-minimization (MM) framework. Additionally, we also propose a heuristic RIS phase shifts design approach based on the sum of subcarrier gain maximization (SSGM) criterion requiring lower complexity. Numerical results indicate that the proposed MO/MM algorithm can achieve almost the same rate as the upper bound achieved by the semidefinite relaxation (SDR) algorithm, and the proposed SSGM based scheme is only slightly inferior to the upper bound while has much lower complexity. These results demonstrate the effectiveness of the proposed algorithms.