  In this paper, a novel reconfigurable intelligent surface (RIS)-assisted
multiple-input multiple-output (MIMO) symbiotic radio (SR) system is proposed,
in which an RIS, operating as a secondary transmitter (STx), sends messages to
a multi-antenna secondary receiver (SRx) by using cognitive backscattering
communication, and simultaneously, it enhances the primary transmission from a
multi-antenna primary transmitter (PTx) to a multi-antenna primary receiver
(PRx) by intelligently reconfiguring the wireless environment. We are
interested in the joint design of active transmit beamformer at the PTx and
passive reflecting beamformer at the STx to minimize the total transmit power
at the PTx, subject to the signal-to-noise-ratio (SNR) constraint for the
secondary transmission and the rate constraint for the primary transmission.
Due to the non-convexity of the formulated problem, we decouple the original
problem into a series of subproblems using the alternating optimization method
and then iteratively solve them. The convergence performance and computational
complexity of the proposed algorithm are analyzed. Furthermore, we develop a
low-complexity algorithm to design the reflecting beamformer by solving a
backscatter link enhancement problem through the semi-definite relaxation (SDR)
technique. Then, theoretical analysis is performed to reveal the insights of
the proposed system. Finally, simulation results are presented to validate the
effectiveness of the proposed algorithms and the superiority of the proposed
system.
