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，可各自进行优化。然而，许多应用关注的是本地信息的功能性而非信息本身。对于此类场景，通过数字空中计算在多址信道中利用干扰进行计算，可显著提升计算效率，这一点已由ChannelComp方法证实。但ChannelComp中原始提出的编码方案因具有通用性且未针对特定调制类别优化，可能存在高计算复杂度的问题。因此，本研究针对空中计算中一类特定的数字调制方式——QAM和PAM，提出了一种名为SumComp的新型编码方案。进一步地，我们推导了算术均值函数计算中SumComp编码的均方误差分析，并为一组列线图函数的平均绝对误差建立了上界。仿真结果证实，在有噪多址信道中，SumComp编码在均方误差和平均绝对误差两方面均优于传统的模拟空中计算方法及ChannelComp中的原始编码方案。具体而言，在低噪声场景下，SumComp编码在计算算术均值与几何均值时，归一化均方误差可提升约10 dB。