Over-the-air computation (AirComp) has traditionally been built on the principle of pre-embedding computation into transmitted waveforms or on exploiting massive antenna arrays, often requiring the wireless multiple-access channel (MAC) to operate under conditions that approximate an ideal computational medium. This paper introduces a new computation framework, termed out-of-air computation (AirCPU), which establishes a joint source-channel coding foundation in which computation is not embedded before transmission but is instead extracted from the wireless superposition by exploiting structured coding. AirCPU operates directly on continuous-valued device data, avoiding the need for a separate source quantization stage, and employs a multi-layer nested lattice architecture that enables progressive resolution by decomposing each input into hierarchically scaled components, all transmitted over a common bounded digital constellation under a fixed power constraint. We formalize the notion of decoupled resolution, showing that in operating regimes where the decoding error probability is sufficiently small, the impact of channel noise and finite constellation constraints on distortion becomes negligible, and the resulting computation error is primarily determined by the target resolution set by the finest lattice. For fading MACs, we further introduce collective and successive computation mechanisms, in addition to the proposed direct computation, which exploit multiple decoded integer-coefficient functions and side-information functions as structural representations of the wireless superposition to significantly expand the reliable operating regime; in this context, we formulate and characterize the underlying reliability conditions and integer optimization problems, and develop a structured low-complexity two-group approximation to address them.

翻译：空中计算（AirComp）传统上建立在计算预嵌入传输波形或利用大规模天线阵列的准则之上，这通常要求无线多址接入信道（MAC）在接近理想计算媒介的条件下运行。本文提出一种新的计算框架——空中计算（AirCPU），它建立了联合信源信道编码基础，其中计算并非在传输前嵌入，而是通过利用结构化编码从无线叠加中提取。AirCPU直接对连续值设备数据进行操作，无需独立的信源量化阶段，并采用多层嵌套格架构，通过将每个输入分解为分层缩放分量来实现渐进分辨率，所有分量均在固定功率约束下通过通用有界数字星座传输。我们形式化了去耦分辨率的概念，证明在解码错误概率足够小的运行区间内，信道噪声和有限星座约束对失真的影响可忽略，由此产生的计算误差主要取决于由最细粒度格设定的目标分辨率。针对衰落MAC，除提出的直接计算外，我们进一步引入集体计算和连续计算机制，利用多个解码的整数系数函数和边信息函数作为无线叠加的结构化表示，显著扩展了可靠运行区间；在此背景下，我们构建并刻画了底层可靠性条件与整数优化问题，并开发了一种结构化的低复杂度双组近似方法来解决这些问题。