# Computing Functions Over-the-Air Using Digital Modulations

## TL;DR

This paper introduces ChannelComp, a novel digital modulation-based method for over-the-air computation that surpasses traditional analog AirComp, enabling more functions with improved accuracy and robustness.

## Contribution

It proposes a new digital modulation technique for over-the-air computation, overcoming the limitations of analog methods and enabling a wider class of functions.

## Key findings

- ChannelComp outperforms AirComp in computation accuracy.
- Around 10 dB improvement for product function computations.
- Digital modulation enables error correction and synchronization benefits.

## Abstract

Over-the-air computation (AirComp) is a known technique in which wireless devices transmit values by analog amplitude modulation so that a function of these values is computed over the communication channel at a common receiver. The physical reason is the superposition properties of the electromagnetic waves, which naturally return sums of analog values. Consequently, the applications of AirComp are almost entirely restricted to analog communication systems. However, the use of digital communications for over-the-air computations would have several benefits, such as error correction, synchronization, acquisition of channel state information, and easier adoption by current digital communication systems. Nevertheless, a common belief is that digital modulations are generally unfeasible for computation tasks because the overlapping of digitally modulated signals returns signals that seem to be meaningless for these tasks. This paper breaks through such a belief and proposes a fundamentally new computing method, named ChannelComp, for performing over-the-air computations by any digital modulation. In particular, we propose digital modulation formats that allow us to compute a wider class of functions than AirComp can compute, and we propose a feasibility optimization problem that ascertains the optimal digital modulation for computing functions over-the-air. The simulation results verify the superior performance of ChannelComp in comparison to AirComp, particularly for the product functions, with around 10 dB improvement of the computation error.