The rapid advancement of artificial intelligence technologies has given rise to diversified intelligent services, which place unprecedented demands on massive connectivity and gigantic data aggregation. However, the scarce radio resources and stringent latency requirement make it challenging to meet these demands. To tackle these challenges, over-the-air computation (AirComp) emerges as a potential technology. Specifically, AirComp seamlessly integrates the communication and computation procedures through the superposition property of multiple-access channels, which yields a revolutionary multiple-access paradigm shift from "compute-after-communicate" to "compute-when-communicate". By this means, AirComp enables spectral-efficient and low-latency wireless data aggregation by allowing multiple devices to occupy the same channel for transmission. In this paper, we aim to present the recent advancement of AirComp in terms of foundations, technologies, and applications. The mathematical form and communication design are introduced as the foundations of AirComp, and the critical issues of AirComp over different network architectures are then discussed along with the review of existing literature. The technologies employed for the analysis and optimization on AirComp are reviewed from the information theory and signal processing perspectives. Moreover, we present the existing studies that tackle the practical implementation issues in AirComp systems, and elaborate the applications of AirComp in Internet of Things and edge intelligent networks. Finally, potential research directions are highlighted to motivate the future development of AirComp.

翻译：人工智能技术的飞速发展催生了多样化的智能服务，这些服务对大规模连接和海量数据聚合提出了前所未有的需求。然而，稀缺的无线资源和严格的时延要求使得满足这些需求面临挑战。为应对这些挑战，空中计算作为一种潜在技术应运而生。具体而言，空中计算通过多址信道的叠加特性，无缝集成了通信与计算过程，从而实现了从"先通信后计算"到"通信即计算"的革命性多址范式转变。通过这种方式，空中计算允许多个设备占用同一信道进行传输，从而实现高谱效、低时延的无线数据聚合。本文旨在从基础、技术和应用三个层面介绍空中计算的最新进展。首先介绍了空中计算的数学形式和通信设计基础，随后结合现有文献综述，讨论了不同网络架构下空中计算的关键问题。从信息论和信号处理的角度回顾了用于空中计算分析与优化的相关技术。此外，我们介绍了现有研究中针对空中计算系统实际实施问题的解决方案，并详细阐述了空中计算在物联网和边缘智能网络中的应用。最后，本文提出了潜在的研究方向，以推动空中计算的未来发展。