Multi-Functional RIS for Distributed Over-the-Air Computation in Base Station Free Environments

Distributed over-the-air computation (AirComp) is a promising technology for fast data aggregation in wireless networks by leveraging multiple access channel to achieve communication and computation simultaneously. However, device-to-device (D2D) links applied are vulnerable to obstacles, and the performance of distributed AirComp is restricted by the device with the worst channel condition. To tackle these issues, we introduce a multi-functional reconfigurable intelligent surface (MF-RIS) to reconstruct the wireless propagation environment, where the MF-RIS can achieve signal reflection, refraction, and amplification simultaneously. Specifically, we propose an MF-RIS-aided distributed AirComp framework, where MF-RIS receives the aggregated data from all devices and then transmits it to each device for post-processing. We formulate a mean-squared error (MSE) minimization problem by jointly optimizing transmit scalar, receive scalar, and MF-RIS coefficients. To address this non-convex problem, we employ an alternating optimization (AO) technique to decompose it into three subproblems, where semi-closed form or closed form solutions are obtained. Then, we extend the single-input single-output (SISO) system into multiple-input multiple-output (MIMO) one. Next, we derive the asymptotic MSE performance for SISO and MIMO cases when the number of RIS elements and that of transmit/receive antennas are very large. Numerical results demonstrate the superiority of MF-RIS in improving MSE performance compared to the baseline without RIS. Additionally, the MF-RIS outperforms its passive counterparts, which reveals the advantages of deploying MF-RIS in distributed AirComp systems to reduce data aggregation error.