Optimal QAM Constellation for Over-the-Air Computation in the Presence of Heavy-Tailed Channel Noise

TL;DR

This paper investigates the design of optimal QAM constellations for over-the-air computation in wireless channels with heavy-tailed, impulsive noise modeled by a Cauchy distribution, aiming to minimize aggregation error.

Contribution

It formulates a constrained optimization for QAM constellation design under heavy-tailed noise, deriving unique optimality conditions and demonstrating effectiveness through numerical results.

Findings

Optimal QAM constellations reduce MSE in heavy-tailed noise environments.

The framework extends to nomographic functions and other noise models.

Numerical results confirm the proposed design's superiority.

Abstract

Over-the-air computation (OAC) enables low-latency aggregation over multiple-access channels (MACs) by exploiting the superposition property of the wireless medium to compute functions efficiently in distributed networks. A critical but often overlooked challenge is that electromagnetic interference in practical radio channels frequently exhibits heavy-tailed behavior, causing strong impulsive noise that severely degrades computation performance. This work studies digital OAC with QAM-based signaling under heavy-tailed interference modeled by a Cauchy distribution (lacking a finite second moment). We seek QAM-like constellations that minimize the mean-squared error (MSE) of sum aggregation subject to an average-power constraint. The problem is formulated as a constrained optimization, whose solution yields unique optimality conditions. Numerical results confirm the effectiveness of the proposed design. Notably, the framework extends naturally to nomographic functions, broader constellation families, and alternative noise models.