Joint Beamforming Aided Over-the-Air Computation Systems Relying on Both BS-Side and User-Side Reconfigurable Intelligent Surfaces

Over-the-air computation (AirComp) has received substantial attention, given its ability to aggregate massive amounts of data from distributed wireless devices (WDs). However, the computation accuracy at the fusion center (FC) may be severely affected by receiving data corrupted by the poor channel conditions. To mitigate this issue, we consider the employment of reconfigurable intelligent surfaces (RISs) in the AirComp system considered for improving the quality of received data, and hence improve the computation accuracy. However, most previous contributions on RIS-assisted AirComp systems only employ a single RIS in the resultant single-RIS-assisted (SRIS-assisted) AirComp systems. We develop this concept further for mitigating the deleterious channel effects by conceiving a double-RIS-assisted (DRIS-assisted) AirComp system, where one of the RISs is located near the WDs and the other in the vicinity of the FC. We theoretically prove that the DRIS-assisted AirComp system outperforms its SRIS-assisted counterpart in terms of the resultant computation mean-squared-error (MSE). Furthermore, we propose a pair of algorithms for jointly optimizing the transmit power at the WDs, the receive beamforming vector at the FC, and the passive beamforming matrices at the RISs for minimizing the computational MSE. Specifically, the transmit power is updated by exploiting the Lagrange duality method, while the receive beamforming vector is optimized by utilizing the first-order optimality condition. Furthermore, a pair of techniques are developed for optimizing the passive beamforming matrices at the RISs based on semidefinite relaxation (SDR) and penalty-duality-decomposition (PDD), respectively. Both the complexity and the convergence of the proposed algorithms are analyzed. Finally, simulation results are provided for quantifying the overall performance of the resultant DRIS-assisted AirComp system.