  Beyond diagonal reconfigurable intelligent surface (BD-RIS) is a new advance
and generalization of the RIS technique. BD-RIS breaks through the isolation
between RIS elements by creatively introducing inter-element connections,
thereby enabling smarter wave manipulation and enlarging coverage. However,
exploring proper channel estimation schemes suitable for BD-RIS aided
communication systems still remains an open problem. In this paper, we study
channel estimation and beamforming design for BD-RIS aided multi-antenna
systems. We first describe the channel estimation strategy based on the least
square (LS) method, derive the mean square error (MSE) of the LS estimation,
and formulate the joint pilot sequence and BD-RIS design problem with unique
constraints induced by BD-RIS architectures. Specifically, we propose an
efficient pilot sequence and BD-RIS design which theoretically guarantees to
achieve the minimum MSE. With the estimated channel, we then consider two
BD-RIS scenarios and propose beamforming design algorithms. Finally, we provide
simulation results to verify the effectiveness of the proposed channel
estimation scheme and beamforming design algorithms. We also show that more
interelement connections in BD-RIS improves the performance while increasing
the training overhead for channel estimation.
