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In this paper, a novel beamforming technique is proposed as the over-the-air computation (AirComp) framework in a multiple-input multiple-output (MIMO) Internet-of-things (IoT) network consisting of multiple IoT sensors (STAs) and a single access point (AP). We assume that each IoT device has the channel state information (CSI) from itself to the AP and the AP has the global CSI of all IoT devices. We consider the mean squared error (MSE), which represents the reliability of function computation, as a performance metric. In short, each IoT device exploits maximum-ratio transmission (MRT) as a transmit beamforming technique to improve MSE performance by taking full advantage of multiple transmit antennae. Moreover, for the receive beamforming, we first consider a receive antenna selection (RAS) technique as the simplest beamforming method at the AP. Then, a semi-definite relaxation (SDR) method and a successive convex approximation (SCA) algorithm are considered and compared with each other in terms of MSE. Finally, we propose a novel two-step beamforming algorithm to further improve the MSE performance of the aforementioned techniques. We have numerically verified through computer simulations that the proposed framework has an improved MSE performance of about 6dB compared to the conventional single-input multiple-output (SIMO) AirComp, even with only two transmit antennae, and the modified MRT outperforms the other transmit beamforming techniques. Furthermore, the proposed receive beamforming technique, a two-step algorithm, shows the best performance in terms of MSE compared to prior studies.

Beamforming Techniques for Over-the-Air Computation in MIMO IoT Networks