In symbiotic radio (SR), also called cognitive backscatter communications, a secondary transmitter (STx) transmits messages by modulating its information over the RF signals from a primary transmitter (PTx), and in return, the secondary transmission provides multipath gain instead of interference to the primary transmission when the spreading factor <tex>$K$</tex> of SR is large enough. In this paper, we are interested in the fundamental mutualistic mechanism between the primary and secondary transmissions in SR, which describes the condition through which the two systems can benefit each other. We first derive the closed-form expressions for the bit error rates (BERs) for both primary and secondary transmissions for general <tex>$K$</tex>, then obtain the condition on <tex>$K$</tex> to enable mutualistic symbiosis in SR. It is observed that the critical point of <tex>$K$</tex> is related to the average strengths of the direct and backscatter links when the number of receiving antennas is large. Extensive simulation and numerical results are provided to verify the accuracy of theoretical analysis and demonstrate the interrelationship between primary and secondary transmissions.
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