Out-of-Air Computation: Enabling Structured Extraction from Wireless Superposition

TL;DR

This paper introduces out-of-air computation (AirCPU), a novel framework that extracts structured computation directly from wireless superposition using nested lattice coding, bypassing traditional pre-embedding methods.

Contribution

It proposes a joint source-channel coding approach with hierarchical lattice architecture for direct wireless computation, expanding reliable operation regimes especially in fading MACs.

Findings

Decoupled resolution reduces distortion impact of noise and finite constellations.

Hierarchical lattice coding enables progressive resolution in wireless computation.

New reliability conditions and optimization methods improve performance in fading channels.

Abstract

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.