Recently, the ChannelComp framework has proposed digital over-the-air computation by designing digital modulations that enable the computation of arbitrary functions. Unlike traditional analog over-the-air computation, which is restricted to nomographic functions, ChannelComp enables a broader range of computational tasks while maintaining compatibility with digital communication systems. This framework is intended for applications that favor local information processing over the mere acquisition of data. However, ChannelComp is currently designed for scalar function computation, while numerous data-centric applications necessitate vector-based computations, and it is susceptible to channel fading. In this work, we introduce a generalization of the ChannelComp framework, called VecComp, by integrating ChannelComp with multiple-antenna technology. This generalization not only enables vector function computation but also ensures scalability in the computational complexity, which increases only linearly with the vector dimension. As such, VecComp remains computationally efficient and robust against channel impairments, making it suitable for high-dimensional, data-centric applications. We establish a non-asymptotic upper bound on the mean squared error of VecComp, affirming its computation efficiency under fading channel conditions. Numerical experiments show the effectiveness of VecComp in improving the computation of vector functions and fading compensation over noisy and fading multiple-access channels.

翻译：近期，ChannelComp框架通过设计能够计算任意函数的数字调制方案，提出了数字空中计算方法。与仅适用于标称函数的传统模拟空中计算不同，ChannelComp在保持与数字通信系统兼容的同时，实现了更广泛的计算任务。该框架适用于偏好本地信息处理而非单纯数据采集的应用场景。然而，ChannelComp当前仅针对标量函数计算设计，而众多以数据为中心的应用需要基于向量的计算，且该框架易受信道衰落影响。本研究通过将ChannelComp与多天线技术相结合，提出了一种称为VecComp的ChannelComp泛化框架。该泛化不仅实现了向量函数计算，还确保了计算复杂度的可扩展性——其复杂度仅随向量维度线性增长。因此，VecComp在保持计算高效性的同时，具备对信道损伤的鲁棒性，适用于高维数据为中心的应用场景。我们建立了VecComp均方误差的非渐近上界，证实了其在衰落信道条件下的计算效率。数值实验表明，VecComp在提升向量函数计算性能及衰落补偿方面，在噪声和衰落多址信道中表现出显著有效性。