Over-the-Air Computation Systems: Optimization, Analysis and Scaling Laws

For future Internet-of-Things based Big Data applications, data collection from ubiquitous smart sensors with limited spectrum bandwidth is very challenging. On the other hand, to interpret the meaning behind the collected data, it is also challenging for an edge fusion center running computing tasks over large data sets with a limited computation capacity. To tackle these challenges, by exploiting the superposition property of multiple-access channel and the functional decomposition, the recently proposed technique, over-the-air computation (AirComp), enables an effective joint data collection and computation from concurrent sensor transmissions. In this paper, we focus on a single-antenna AirComp system consisting of K sensors and one receiver. We consider an optimization problem to minimize the computation mean-squared error (MSE) of the K sensors' signals at the receiver by optimizing the transmitting-receiving (Tx-Rx) policy, under the peak power constraint of each sensor. Although the problem is not convex, we derive the computation-optimal policy in closed form. Also, we comprehensively investigate the ergodic performance of the AirComp system, and the scaling laws of the average computation MSE (ACM) and the average power consumption (APC) of different Tx-Rx policies with respect to K. For the computation-optimal policy, we show that the policy has a vanishing ACM and a vanishing APC with the increasing K.