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, QAM and PAM, for which we introduce a novel coding scheme called SumComp. Furthermore, we derive an MSE analysis for SumComp coding in the computation of the arithmetic mean function and establish an upper bound on the MAE for a set of nomographic functions. Simulation results affirm the superior performance of SumComp coding compared to traditional analog over-the-air computation and the original coding in ChannelComp approaches regarding both MSE and MAE over a noisy multiple access channel. Specifically, SumComp coding shows approximately $10$ dB improvements for computing arithmetic and geometric mean on the normalized MSE for low noise scenarios.

翻译：通信与计算传统上被视为独立实体，可实现各自的优化。然而，许多应用关注的是本地信息的功能而非信息本身。在此类场景中，通过数字空中计算利用多址信道中的干扰进行计算，可显著提升计算效率——这已通过ChannelComp方法得到验证。但ChannelComp最初提出的编码方案因具有通用性且未针对特定调制类型优化，存在计算复杂度高的问题。因此，本研究聚焦于空中计算中一类特定数字调制方式（QAM与PAM），提出了一种名为SumComp的新型编码方案。此外，我们针对算术均值函数的计算推导了SumComp编码的MSE分析，并为一系列列线图函数建立了MAE的上界。仿真结果证实，相较于传统模拟空中计算及ChannelComp原始编码方法，SumComp编码在含噪多址信道的MSE与MAE性能上均表现更优。具体而言，在低噪声场景下，SumComp编码计算算术均值与几何均值时，归一化MSE可实现约10 dB的性能提升。