  This paper investigates multiuser multi-input single-output downlink
symbiotic radio communication systems assisted by an intelligent reflecting
surface (IRS). Different from existing methods ideally assuming the secondary
user (SU) can jointly decode information symbols from both the access point
(AP) and the IRS via multiuser detection, we consider a more practical SU that
only non-coherent detection is available. To characterize the non-coherent
decoding performance, a practical upper bound of the average symbol error rate
(SER) is derived. Subsequently, we jointly optimize the beamformer at the AP
and the phase shifts at the IRS to maximize the average sum-rate of the primary
system taking into account the maximum tolerable SER constraint for the SU. To
circumvent the couplings of variables, we exploit the Schur complement that
facilitates the design of a suboptimal beamforming algorithm based on
successive convex approximation. Our simulation results show that compared with
various benchmark algorithms, the proposed scheme significantly improves the
average sum-rate of the primary system, while guaranteeing the decoding
performance of the secondary system.
