  Over-the-air computation (AirComp) has been recognized as a promising
technique in Internet-of-Things (IoT) networks for fast data aggregation from a
large number of wireless devices. However, as the number of devices becomes
large, the computational accuracy of AirComp would seriously degrade due to the
vanishing signal-to-noise ratio (SNR). To address this issue, we exploit the
massive multiple-input multiple-output (MIMO) with hybrid beamforming, in order
to enhance the computational accuracy of AirComp in a cost-effective manner. In
particular, we consider the scenario with a large number of multi-antenna
devices simultaneously sending data to an access point (AP) equipped with
massive antennas for functional computation over the air. Under this setup, we
jointly optimize the transmit digital beamforming at the wireless devices and
the receive hybrid beamforming at the AP, with the objective of minimizing the
computational mean-squared error (MSE) subject to the individual transmit power
constraints at the wireless devices. To solve the non-convex hybrid beamforming
design optimization problem, we propose an alternating-optimization-based
approach. In particular, we propose two computationally efficient algorithms to
handle the challenging receive analog beamforming problem, by exploiting the
techniques of successive convex approximation (SCA) and block coordinate
descent (BCD), respectively. It is shown that for the special case with a
fully-digital receiver at the AP, the achieved MSE of the massive MIMO AirComp
system is inversely proportional to the number of receive antennas.
Furthermore, numerical results show that the proposed hybrid beamforming design
substantially enhances the computation MSE performance as compared to other
benchmark schemes, while the SCA-based algorithm performs closely to the
performance upper bound achieved by the fully-digital beamforming.
