Low-Overhead Channel Estimation Framework for Beyond Diagonal Reconfigurable Intelligent Surface Assisted Multi-User MIMO Communication

Beyond diagonal reconfigurable intelligent surface (BD-RIS) refers to a family of RIS architectures characterized by scattering matrices not limited to being diagonal and enables higher wave manipulation flexibility and large performance gains over conventional (diagonal) RIS. To achieve those promising gains, accurate channel state information (CSI) needs to be acquired in BD-RIS assisted communication systems. However, the number of coefficients in the cascaded channels to be estimated in BD-RIS assisted systems is significantly larger than that in conventional RIS assisted systems, because the channels associated with the off-diagonal elements of the scattering matrix have to be estimated as well. Surprisingly, for the first time in the literature, this paper rigorously shows that the uplink channel estimation overhead in BD-RIS assisted systems is actually of the same order as that in the conventional RIS assisted systems. This amazing result stems from a key observation: for each user antenna, its cascaded channel matrix associated with one reference BD-RIS element is a scaled version of that associated with any other BD-RIS element due to the common RIS-base station (BS) channel. In other words, the number of independent unknown variables is far less than it would seem at first glance. Building upon this property, this paper manages to characterize the minimum overhead to perfectly estimate all the channels in the ideal case without noise at the BS, and propose a twophase estimation framework for the practical case with noise at the BS. Numerical results demonstrate outstanding channel estimation overhead reduction over existing schemes in BD-RIS assisted systems.