Reciprocal Beyond-Diagonal Reconfigurable Intelligent Surface (BD-RIS): Scattering Matrix Design via Manifold Optimization

Abstract

Beyond-diagonal reconfigurable intelligent surfaces (BD-RISs) are emerging as a transformative technology in wireless communications, enabling enhanced performance and quality of service (QoS) of wireless systems in harsh urban environments due to their relatively low cost and advanced signal processing capabilities. Generally, BD-RIS systems are employed to improve robustness, increase achievable rates, and enhance energy efficiency of wireless systems in both direct and indirect ways. The direct way is to produce a favorable propagation environment via the design of optimized scattering matrices, while the indirect way is to reap additional improvements via the design of multiple-input multiple-output (MIMO) beamformers that further exploit the latter "engineered" medium. In this article, the problem of sum-rate maximization via BD-RIS is examined, with a focus on feasibility, namely low-complexity physical implementation, by enforcing reciprocity in the BD-RIS design in a manner that adheres to the geometry of the manifold of symmetric matrices. To that end, the sum-rate objective is transformed into a quadratic function via fractional programming (FP), augmented via the also quadratic reciprocity constraint in the form of a regularization term, while the unitary constraint is dealt with via a manifold optimization framework. Simulation results demonstrate the effectiveness of the proposed method in outperforming current state-of-the-art (SotA) approaches in terms of sum-rate maximization.