On the MIMO Channel Capacity for Reconfigurable Intelligent Surface Assisted Symbiotic Radios

In reconfigurable intelligent surface (RIS)-assisted symbiotic radios (SRs), an RIS delivers its information by periodically reflecting the signal from a primary transmitter (PTx) and simultaneously the RIS assists this primary transmission by passive beamforming. In this paper, we are interested in characterizing the multiple-input multiple-output (MIMO) channel capacity for RIS-assisted SR, where both PTx and RIS transmit multiple data streams. To this end, we propose a novel multi-data-stream delivery scheme for the RIS, where the reflecting elements of the RIS are divided into multiple parts and each part is used to transmit one data stream by periodically adjusting its reflection coefficients. With this information delivery scheme, we derive an upper bound on the MIMO channel capacity of an RIS-assisted SR system. To achieve it, we then jointly design the transmit covariance matrix at the PTx and the passive beamforming vector at the RIS by using an alternating optimization algorithm. Considering the high computational complexity of solving the transmit covariance matrix, we further propose two low-complexity algorithms, i.e., upper bound-based algorithm and singular value-based algorithm. Finally, extensive numerical results are presented to show the effectiveness of the proposed algorithms and demonstrate the advantages of the RIS to transmit multiple data streams.