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

Saeed Razavikia, Deniz Gündüz, Carlo Fischione

TL;DR

This work studies digital OAC with QAM-based signaling under heavy-tailed interference modeled by a Cauchy distribution, and seeks QAM-like constellations that minimize the mean-squared error of sum aggregation subject to an average-power constraint.

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.

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

TL;DR

This work studies digital OAC with QAM-based signaling under heavy-tailed interference modeled by a Cauchy distribution, and seeks QAM-like constellations that minimize the mean-squared error of sum aggregation subject to an average-power constraint.

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.
Paper Structure (11 sections, 2 theorems, 17 equations, 2 figures)

This paper contains 11 sections, 2 theorems, 17 equations, 2 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Multiple Access Channel (MAC)
  4. Decoding Procedure
  5. Encoding Procedure
  6. Problem Statement
  7. Optimal Constellation Design
  8. Numerical Results
  9. Conclusions
  10. Proof of Lemma \ref{['LEM_PROOF']}
  11. Proof Sketch of Theorem \ref{['TH:MAIN']}

Key Result

Lemma 1

For a $K$-user MAC with encoder $\mathscr{E}_q(\cdot)$ in eq:encoding_qam, ML decoder $\mathscr{D}$ in eq:ml_decoder, and Cauchy noise $z \sim \mathcal{C}(0,\gamma)$, assume the induced constellation points of $\sum_k s_k$ are uniformly distributed over $\mathcal{Y}$. Then, the MSE is where $\mu(x) = \frac{2}{\pi} \sum\nolimits_{m=1}^{N-1} \alpha_m \arctan(\frac{\gamma}{(2m-1)x})$, with $\alpha_m

Figures (2)

  • Figure 1: Gray code vs SumComp code for QAM $Q=16$ modulation. The right constellation diagram uses spacing parameters $(d_1^{*},d_2^{*})$ are determined by Theorem \ref{['TH:MAIN']}.
  • Figure 2: Monte Carlo evaluation of the MSE for the sum function with \ref{['fig:SUMQAM']}$K=10$ and $K=100$ transmitter over $5\times10^4$ independent trials: Solid curves denote the optimized distance parameters $({d}_1^*,{d}_2^*)$ obtained by, whereas dashed curves correspond to equal‐distance ${d}_1={d}_2=\sqrt{6/(Q-1)}$.

Theorems & Definitions (7)

  • Remark 1
  • Remark 2
  • Remark 3
  • Lemma 1
  • proof
  • Theorem 1
  • proof