Statistical CSI-Enabled Optimization for Beyond Diagonal SIM

Stacked intelligent metasurfaces (SIM) represent an innovative design comprising multiple metasurface layers, enabling complex signal processing tasks directly in the wave domain. This paradigm not only enhances signal processing efficiency but also reduces hardware costs. However, the traditional single-connected reconfigurable intelligent surface (RIS) architecture limits the design flexibility of SIM. In this letter, we propose a beyond diagonal SIM (BD-SIM) framework, constructed by stacking beyond diagonal RIS (BD-RIS) units, and obtain a large-system approximation of the achievable rate that enables optimal design based on statistical rather than instantaneous channel state information (CSI), thereby reducing signaling overhead. Then, we optimize the phase-shifting matrices for both fully-connected and group-connected BD-SIM by employing a minorize-majorization (MM) algorithm. Results show that BD-SIM, by integrating the strengths of SIM and BD-RIS, further enhances wave-domain signal processing capabilities.