  Over-the-air computation (AirComp) has been recognized as a low-latency
solution for wireless sensor data fusion, where multiple sensors send their
measurement signals to a receiver simultaneously for computation. Most existing
work only considered performing AirComp over a single frequency channel.
However, for a sensor network with a massive number of nodes, a single
frequency channel may not be sufficient to accommodate the large number of
sensors, and the AirComp performance will be very limited. So it is highly
desirable to have more frequency channels for large-scale AirComp systems to
benefit from multi-channel diversity. In this letter, we propose an
$M$-frequency AirComp system, where each sensor selects a subset of the $M$
frequencies and broadcasts its signal over these channels under a certain power
constraint. We derive the optimal sensors' transmission and receiver's signal
processing methods separately, and develop an algorithm for joint design to
achieve the best AirComp performance. Numerical results show that increasing
one frequency channel can improve the AirComp performance by threefold compared
to the single-frequency case.
