Channel Capacity of RIS-Assisted Symbiotic Radios with Imperfect Knowledge of Channels

In reconfigurable intelligent surface (RIS)-assisted symbiotic radio (SR) systems, an RIS transmits a secondary signal by modulating its information bits over the radio frequency (RF) signals emitted from a primary transmitter (PTx), and simultaneously, the RIS assists the primary transmission by passive beamforming. Considering the inevitable channel estimation errors arising in practice, in this paper, we are interested in quantifying the effects of imperfect channel knowledge on the channel capacity for both primary and secondary transmissions in RIS-assisted SR. For the primary transmission, we first derive upper and lower bounds on the achievable rate considering channel estimation errors. Based on the derived lower bound, we investigate an approximate condition on the number of reflecting elements of an RIS that assures the performance enhancement of the primary transmission as compared to the case without the RIS. For the secondary transmission, exact and asymptotic achievable rates are derived, which are not only related to the channel estimation error of the reflecting link but also are related to that of the direct link. Furthermore, we analyze the effects of system parameters, including the number of pilot symbols and the symbol period ratio between the primary and secondary signals, on the achievable rates for both primary and secondary transmissions. Finally, extensive numerical results are presented to verify the effectiveness of the theoretical analysis and show the impact of the channel estimation errors.