Closed-Form Global Optimization of Beyond Diagonal Reconfigurable Intelligent Surfaces

TL;DR

This paper derives a closed-form global optimization solution for beyond diagonal reconfigurable intelligent surfaces (BD-RISs), enabling optimal performance in various wireless system configurations with reduced computational complexity.

Contribution

It provides the first closed-form solution for the optimal scattering matrix of BD-RIS architectures, achieving performance bounds efficiently.

Findings

Closed-form solution exactly achieves performance upper bounds.

Algorithm complexity is lower than previous iterative methods.

Complexity scales linearly or cubically with the number of RIS elements.

Abstract

Reconfigurable intelligent surfaces (RISs) allow controlling the propagation environment in wireless networks by tuning multiple reflecting elements. RISs have been traditionally realized through single connected architectures, mathematically characterized by a diagonal scattering matrix. Recently, beyond diagonal RISs (BD-RISs) have been proposed as a novel branch of RISs whose scattering matrix is not limited to be diagonal, which creates new benefits and opportunities for RISs. Efficient BD-RIS architectures have been realized based on group and fully connected reconfigurable impedance networks. However, a closed-form solution for the global optimal scattering matrix of these architectures is not yet available. In this paper, we provide such a closed-form solution proving that the theoretical performance upper bounds can be exactly achieved for any channel realization. We first consider the received signal power maximization in single-user single-input single-output (SISO) systems aided by a BD-RIS working in reflective or transmissive mode. Then, we extend our solution to single-user multiple-input multiple-output (MIMO) and multi-user multiple-input single-output (MISO) systems. We show that our algorithm is less complex than the iterative optimization algorithms employed in the previous literature. The complexity of our algorithm grows linearly (resp. cubically) with the number of RIS elements in the case of group (resp. fully) connected architectures.