Beamforming Design for Active RIS-Aided Over-the-Air Computation

Over-the-air computation (AirComp) is emerging as a promising technology for wireless data aggregation. However, its performance is hampered by users with poor channel conditions. To mitigate such a performance bottleneck, this paper introduces an active reconfigurable intelligence surface (RIS) into the AirComp system. We begin by exploring the ideal active RIS model and propose a joint optimization of the transceiver and RIS configuration to minimize the mean squared error (MSE) between the target and estimated function values. To manage the resulting tri-convex optimization problem, we employ the alternating optimization (AO) framework to decompose it into three convex subproblems, each of which can be solved optimally. We then investigate two specific cases and analyze their respective asymptotic performance to reveal the superiority of the active RIS in mitigating the MSE relative to its passive counterpart. Lastly, we adapt our transceiver and RIS configuration optimization approach to account for the self-interference of the active RIS. To handle the resulting highly non-convex problem, we further develop a two-layer AO framework. Simulation results confirm the superiority of the active RIS in enhancing AirComp performance compared to its passive counterpart.