Joint Beamforming Optimization in Multi-Relay Assisted MIMO Over-the-Air Computation for Multi-Modal Sensing Data Aggregation

In this letter, we investigate a multi-relay assisted over-the-air computation network for multi-modal sensing with direct links, where each node is equipped with multiple antennas and all the relays are operated in an amplify-and-forward mode. Specifically, the whole transmission is divided into two phases and all the sensors transmit symbols during both two phases. In particular, we are interested in minimizing the computation distortion measured by the mean-squared error (MSE) via jointly optimizing beamforming matrices at all nodes, subject to individual power constraints at the sensors and relays. The major difficulty lies in the strong coupling of beamforming matrices in the objective function and the non-convex transmit power constraints at the relays. To tackle this problem, a low-complexity locally optimal method based on alternating optimization is proposed, where closed-form expressions are obtained in each iteration. Furthermore, simulation results show that our proposed beamforming design can substantially enhance the computation MSE performance, as compared to other benchmark schemes.