  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.
