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.

翻译：中文摘要：可重构智能表面（RIS）通过调节多个反射单元，能够控制无线网络中的传播环境。传统RIS采用单连接架构，其散射矩阵在数学上表现为对角形式。近年来，超越对角RIS（BD-RIS）被提出作为RIS的一个新分支，其散射矩阵不限于对角形式，这为RIS带来了新的优势与机遇。基于分组连接和全连接的可重构阻抗网络，已实现高效的BD-RIS架构。然而，针对这些架构的全局最优散射矩阵，目前尚缺乏闭式解。本文提出了这样一种闭式解，并证明在任何信道实现下均可精确达到理论性能上界。我们首先考虑在单用户单输入单输出（SISO）系统中，由工作于反射或透射模式的BD-RIS辅助实现的接收信号功率最大化问题。随后，将解推广至单用户多输入多输出（MIMO）及多用户多输入单输出（MISO）系统。结果表明，所提算法的复杂度低于现有文献中采用的迭代优化算法。对于分组连接（全连接）架构，该算法的复杂度随RIS单元数量线性（三次方）增长。