  For future Internet of Things (IoT)-based Big Data applications (e.g., smart
cities/transportation), wireless data collection from ubiquitous massive 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
edge fusion centers running computing tasks over large data sets with limited
computation capacity. To tackle these challenges, by exploiting the
superposition property of a multiple-access channel and the functional
decomposition properties, 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 (i.e.,
the fusion center). 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 AirComp systems in terms of the average
computation MSE and the average power consumption under Rayleigh fading
channels with different Tx-Rx policies. For the computation-optimal policy, we
prove that its average computation MSE has a decay rate of $O(1/\sqrt{K})$, and
our numerical results illustrate that the policy also has a vanishing average
power consumption with the increasing $K$, which jointly show the computation
effectiveness and the energy efficiency of the policy with a large number of
sensors.
