SumComp: Coding for Digital Over-the-Air Computation via the Ring of Integers

Communication and computation are traditionally treated as separate entities, allowing for individual optimizations. However, many applications focus on local information’s functionality rather than the information itself. For such cases, harnessing interference for computation in a multiple access channel through digital over-the-air computation can notably increase the computation, as established by the ChannelComp method. However, the coding scheme originally proposed in ChannelComp may suffer from high computational complexity because it is general and is not optimized for specific modulation categories. Therefore, this study considers a specific category of digital modulations for over-the-air computations, quadrature amplitude modulation (QAM) and pulse-amplitude modulation (PAM), for which we introduce a novel coding scheme called SumComp. Furthermore, we derive a mean squared error (MSE) analysis for SumComp coding in the computation of the arithmetic mean function and establish an upper bound on the mean absolute error (MAE) for a set of nomographic functions. Simulation results are presented to affirm the superior performance of SumComp coding compared to traditional analog over-the-air computation and the original coding in ChannelComp approaches in terms of both MSE and MAE over a noisy multiple access channel. Specifically, SumComp coding shows at least 10 dB improvements for computing arithmetic and geometric mean on the normalized MSE for low noise scenarios.