Function Computation Over Multiple Access Channels via Hierarchical Constellations

TL;DR

This paper introduces a hierarchical constellation-based coded-modulation framework for reliable function computation over Gaussian MACs, achieving near-optimal rates and robustness through novel shielding techniques.

Contribution

It proposes a new hierarchical constellation design and shielding mechanism for over-the-air computation, improving reliability and asymptotic optimality in multi-transmitter networks.

Findings

Achieves computation with error probability epsilon at rates close to the optimal.

The gap to optimal rate scales as O(log(1/epsilon)/K) and vanishes as network size increases.

Shielding improves reliability, with the gap scaling as O(log log(1/epsilon)).

Abstract

We study function computation over a Gaussian multiple-access channel (MAC), where multiple transmitters aim at computing a function of their values at a common receiver. To this end, we propose a novel coded-modulation framework for over-the-air computation (OAC) based on hierarchical constellation design, which supports reliable computation of multiple function outputs using a single channel use. Moreover, we characterize the achievable computation rate and show that the proposed hierarchical constellations can compute R output functions with decoding error probability epsilon while the gap to the optimal computation rate scales as O(\log_2(1/\epsilon)/K) for independent source symbols, where K denotes the number of transmitters. Consequently, this gap vanishes as the network size grows, and the optimal rate is asymptotically attained. Furthermore, we introduce a shielding mechanism based on variable-length block coding that mitigates noise-induced error propagation across constellation levels while preserving the superposition structure of the MAC. We show that the shielding technique improves reliability, yielding a gap that scales optimally as O(\log_2\ln{(1/\epsilon)}), regardless of the source distribution. Together, these results identify the regimes in which uncoded or lightly coded OAC is information-theoretically optimal, providing a unified framework for low-latency, channel-agnostic function computation.