Optimal Structure of Receive Beamforming for Over-the-Air Computation

We investigate fast data aggregation via over-the-air computation (AirComp) over wireless networks. In this scenario, an access point (AP) with multiple antennas aims to recover the arithmetic mean of sensory data from multiple wireless devices. To minimize estimation distortion, we formulate a mean-squared-error (MSE) minimization problem that considers joint optimization of transmit scalars at wireless devices, denoising factor, and receive beamforming vector at the AP. We derive closed-form expressions for the transmit scalars and denoising factor, resulting in a non-convex quadratic constrained quadratic programming (QCQP) problem concerning the receive beamforming vector. To tackle the computational complexity of the beamforming design, particularly relevant in massive multiple-input multiple-output (MIMO) AirComp systems, we explore the optimal structure of receive beamforming using successive convex approximation (SCA) and Lagrange duality. By leveraging the proposed optimal beamforming structure, we develop two efficient algorithms based on SCA and semi-definite relaxation (SDR). These algorithms enable fast wireless aggregation with low computational complexity and yield almost identical mean square error (MSE) performance compared to baseline algorithms. Simulation results validate the effectiveness of our proposed methods.