On Designing Modulation for Over-the-Air Computation -- Part I: Noise-Aware Design

Over-the-air computation (OAC) leverages the physical superposition property of wireless multiple access channels (MACs) to compute functions while communication occurs, enabling scalable and low-latency processing in distributed networks. While analog OAC methods suffer from noise sensitivity and hardware constraints, existing digital approaches are often limited in design complexity, which may hinder scalability and fail to exploit spectral efficiency fully. This two-part paper revisits and extends the ChannelComp framework, a general methodology for computing arbitrary finite-valued functions using digital modulation. In Part I, we develop a novel constellation design approach that is aware of the noise distribution and formulates the encoder design as a max-min optimization problem using noise-tailored distance metrics. Our design supports noise models, including Gaussian, Laplace, and heavy-tailed distributions. We further demonstrate that, for heavy-tailed noise, the optimal ChannelComp setup coincides with the solution to the corresponding max-min criterion for the channel noise with heavy-tailed distributions. Numerical experiments confirm that our noise-aware design achieves a substantially lower mean-square error than leading digital OAC methods over noisy MACs. In Part II, we consider a constellation design with a quantization-based sampling scheme to enhance modulation scalability and computational accuracy for large-scale digital OAC.