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 rate performance gain over conventional (diagonal) RIS. However, whether BD-RIS aided network is able to deliver more data volume compared to conventional RIS aided network is still questionable, because far more time may be wasted to estimate the massive channel coefficients associated with the off-diagonal entries of the BD-RIS scattering matrix. Somehow counter intuitively, for the first time in the literature, this paper rigorously proves that the channel estimation overhead in fully connected BD-RIS aided network is actually of the same order as that in the conventional RIS aided network, which was characterized in Wang et al. 2020. 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 overhead to perfectly estimate all the channels in the ideal case without noise at the BS, and propose a two-phase estimation framework for the practical case with noise at the BS. The main message of this paper is that we can benefit from the non-diagonal scattering matrix design at a channel estimation cost similar to that in conventional RIS aided network.