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，能够显著提升计算效率，ChannelComp 方法已证实了这一点。然而，ChannelComp 最初提出的编码方案可能面临较高的计算复杂度，因为其具有通用性，未针对特定调制类别进行优化。因此，本研究考虑了一类特定的数字调制方案——QAM 与 PAM——用于空中计算，并为此提出了一种称为 SumComp 的新型编码方案。此外，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，并建立了一组可表示函数在平均绝对误差上的上界。仿真结果证实，在存在噪声的多址信道中，SumComp 编码在均方误差和平均绝对误差方面均优于传统的模拟空中计算及 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，其归一化均方误差约有 $10$ dB 的性能提升。