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：传统上，通信与计算被视为相互独立的实体，因而可分别进行优化。然而，许多应用关注的是局部信息的功能性而非信息本身。针对此类场景，通过数字空中计算在多址信道中利用干扰进行计算，如ChannelComp方法所确立的，可显著提升计算效率。然而，ChannelComp最初提出的编码方案可能因通用性强且未针对特定调制类别优化而导致较高的计算复杂度。因此，本研究考虑一类特定的数字调制方案（QAM与PAM）用于空中计算，并提出了名为SumComp的新型编码方案。进一步地，我们推导了SumComp编码在算术平均函数计算中的均方误差分析，并建立了一组拟图函数的平均绝对误差上界。仿真结果证实，在存在噪声的多址信道中，无论是均方误差还是平均绝对误差，SumComp编码均优于传统模拟空中计算及ChannelComp的原始编码方案。具体而言，在低噪声场景下，SumComp编码在计算算术平均与几何平均时，其归一化均方误差可获得约$10$ dB的性能提升。