  This paper studies the reconfigurable intelligent surface (RIS)-assisted
symbiotic radio (SR) system, where an RIS acts as a secondary transmitter to
transmit its information by leveraging the primary signal as its RF carrier and
simultaneously assists the primary transmission. Conventionally, all reflecting
elements of the RIS are used to transmit the secondary signal, which, however,
would limit its capability for assisting the primary transmission. To address
this issue, we propose a novel RIS partitioning scheme, where the RIS is
partitioned into two sub-surfaces, one to assist the primary transmission and
the other to transmit the secondary signal. Naturally, there exists a
fundamental tradeoff between the assistance and transmission capabilities of
RIS regarding the surface partitioning strategy. Considering the coupling
effect between the primary and secondary transmissions, we focus on the
detection of the composite signal formed by the primary and secondary ones,
based on which we propose a novel two-step detector. Then, we formulate the
assistance-transmission tradeoff problem to minimize the bit error rate (BER)
of the composite signal by jointly optimizing the surface partitioning strategy
and the phase shifts of the two sub-surfaces, such that the overall BER of
RIS-assisted SR is minimized. By solving this problem, we show that the
optimized surface partitioning strategy depends on the channel strength ratio
of the direct link to the reflected link. Finally, extensive simulations show
that our proposed RIS partitioning scheme outperforms the conventional schemes
which use all reflecting elements for either assistance or transmission.
