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. Specifically, we begin by exploring the ideal RIS model and propose a joint optimization of the transceiver design and RIS configuration to minimize the mean squared error (MSE) between the target and estimated function values. To manage the resultant tri-convex optimization problem, we employ the alternating optimization (AO) technique to decompose it into three convex subproblems, each solvable optimally. Subsequently, we 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 design to account for the self-interference of the active RIS. To handle the resultant highly non-convex problem, we further devise a two-layer AO framework. Simulation results demonstrate the superiority of the active RIS in enhancing AirComp performance compared to its passive counterpart.

翻译：空中计算（AirComp）作为无线数据聚合的一项有前景技术正在兴起。然而，其性能受到信道条件较差的用户的限制。为缓解这一性能瓶颈，本文将有源可重构智能表面（RIS）引入AirComp系统。具体而言，我们首先探索理想RIS模型，并提出收发器设计与RIS配置的联合优化，以最小化目标函数值与估计函数值之间的均方误差（MSE）。为处理由此产生的三凸优化问题，我们采用交替优化（AO）技术将其分解为三个可最优求解的凸子问题。随后，我们研究两种特例并分析其各自渐近性能，以揭示有源RIS在降低MSE方面相对于无源RIS的优越性。最后，我们调整收发器与RIS配置设计方案以考虑有源RIS的自干扰。为处理由此产生的高度非凸问题，我们进一步设计了一个双层AO框架。仿真结果表明，与无源RIS相比，有源RIS在增强AirComp性能方面具有显著优势。