  Over-the-air computation (AirComp) is a well-known technique by which several
wireless devices transmit by analog amplitude modulation to achieve a sum of
their transmit signals at a common receiver. The underlying physical principle
is the superposition property of the radio waves. Since such superposition is
analog and in amplitude, it is natural that AirComp uses analog amplitude
modulations. Unfortunately, this is impractical because most wireless devices
today use digital modulations. It would be highly desirable to use digital
communications because of their numerous benefits, such as error correction,
synchronization, acquisition of channel state information, and widespread use.
However, when we use digital modulations for AirComp, a general belief is that
the superposition property of the radio waves returns a meaningless overlapping
of the digital signals. In this paper, we break through such beliefs and
propose an entirely new digital channel computing method named ChannelComp,
which can use digital as well as analog modulations. We propose a feasibility
optimization problem that ascertains the optimal modulation for computing
arbitrary functions over-the-air. Additionally, we propose pre-coders to adapt
existing digital modulation schemes for computing the function over the
multiple access channel. The simulation results verify the superior performance
of ChannelComp compared to AirComp, particularly for the product functions,
with more than 10 dB improvement of the computation error.
