Reconfigurable intelligent surfaces (RISs) can improve the capacity of wireless communication links by passively beamforming the impinging signals in desired directions. This feature has been demonstrated both analytically and experimentally for conventional RISs, consisting of independently reflecting elements. To further enhance reconfigurability, a new architecture called beyond-diagonal RIS (BD-RIS) has been proposed. It allows for controllable signal flows between RIS elements, resulting in a non-diagonal reflection matrix, unlike the conventional RIS architecture. Previous studies on BD-RIS-assisted communications have predominantly considered single-antenna transmitters/receivers. One recent work provides an iterative capacity-improving algorithm for multiple-input multiple-output (MIMO) setups but without providing geometrical insights. In this paper, we derive the first closed-form capacity-maximizing BD-RIS reflection matrix for a MIMO channel. We describe how this solution pairs together propagation paths, how it behaves when the signal-to-noise ratio is high, and what capacity is achievable with ideal semi-unitary channel matrices. The analytical results are corroborated numerically.

翻译：可重构智能表面（RIS）能够通过将入射信号被动波束成形至期望方向，从而提升无线通信链路的容量。这一特性已在传统RIS（由独立反射单元构成）中得到了理论与实验的验证。为进一步增强可重构性，一种称为非对角可重构智能表面（BD-RIS）的新型架构被提出。该架构允许RIS单元之间存在可控的信号流，从而产生非对角反射矩阵，这与传统RIS架构不同。先前关于BD-RIS辅助通信的研究主要考虑单天线发射机/接收机。近期一项工作为多输入多输出（MIMO）系统提供了一种迭代式容量提升算法，但未给出几何层面的深入分析。本文中，我们首次推导了MIMO信道下容量最大化的BD-RIS反射矩阵的闭式解。我们阐述了该解如何配对传播路径，分析了其在高信噪比下的行为，并给出了在理想半酉信道矩阵下可达到的容量。分析结果通过数值仿真得到了验证。