  Symbiotic radio (SR) backscatter systems are possible techniques for the
future low-power wireless communications for Internet of Things devices. In
this paper, we propose a multiple-input-multiple-output (MIMO) SR backscatter
system, where the secondary multi-antenna transmission from the backscatter
device (BD) to the receiver is riding on the primary multi-antenna transmission
from the transmitter to the receiver. We investigate the beamforming design
optimization problem which maximizes the achievable rate of secondary
transmission under the achievable rate constraint of primary transmission. In
the MIMO SR backscatter system, each antenna of the SR BD reflects its received
ambient radio frequency signals from all the transmitting antennas of the
transmitter, which causes the globally optimal solution is difficult to obtain.
In this paper, we propose a method to obtain the achievable rate upper bound.
Furthermore, considering both primary and secondary transmissions, we propose
an exact penalty method based locally optimal solution. Simulation results
illustrate that our proposed exact penalty method based locally optimal
solution performs close to the upper bound.
