  In reconfigurable intelligent surface (RIS)-assisted symbiotic radio (SR), an
RIS is exploited to assist the primary system and to simultaneously operate as
a secondary transmitter by modulating its own information over the incident
primary signal from the air. Such an operation is called over-the-air
modulation. The existing modulation schemes such as on-off keying and binary
phase-shift keying suffer from two problems for joint detection of the primary
and secondary signals in RIS-assisted SR, i.e., one is the detection ambiguity
problem when the direct link is blocked, and the other is the bit error rate
(BER) error-floor problem when the direct link is weak. To address the two
problems, we propose a novel modulation scheme by dividing the phase-shift
matrix into two parts: one is the assistance beamforming matrix for assisting
the primary system and the other is the transmission beamforming matrix for
delivering the secondary signal. To optimize the assistance and transmission
beamforming matrices, we first introduce an assistance factor that describes
the performance requirement of the primary system and then formulate a problem
to minimize the BER of the secondary system, while guaranteeing the BER
requirement of the primary system controlled by the assistance factor. To solve
this non-convex problem, we resort to the successive convex approximation
technique to obtain a suboptimal solution. Furthermore, to draw more insights,
we propose a low-complexity assistance-transmission beamforming structure by
borrowing the idea from the classical maximum ratio transmission and zero
forcing techniques. Finally, simulation results reveal an interesting tradeoff
between the BER performance of the primary and secondary systems by adjusting
the assistance factor.
