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Towards Optimal Constellation Design for Digital Over-the-Air Computation

Saeed Razavikia, Deniz Gündüz, Carlo Fischione

TL;DR

This work addresses the problem of computing sums over a wireless MAC using digital modulation by designing optimal two-dimensional constellations under a power constraint to minimize the mean-squared error. It develops a rigorous optimization framework for encoder/decoder design, derives nonlinear optimality conditions for ML and MAP decoding, and proves uniqueness of the solutions. In the high-SNR regime, it provides closed-form solutions based on a generalized Lambert function, offering deep insight into the constellation structure and asymptotic behavior, including convergence of ML and MAP. Extensions to heavy-tailed noise, real-domain computation via hybrid modulation, and higher-dimensional grids demonstrate the framework’s broad applicability, and numerical results show substantial performance gains over standard QAM-based schemes. The approach thus enables robust, hardware-friendly digital OAC with theoretically grounded design rules and practical implications for edge computing and distributed inference.

Abstract

Over-the-air computation (OAC) has emerged as a key technique for efficient function computation over multiple-access channels (MACs) by exploiting the waveform superposition property of the wireless domain. While conventional OAC methods rely on analog amplitude modulation, their performance is often limited by noise sensitivity and hardware constraints, motivating the use of digital modulation schemes. This paper proposes a novel digital modulation framework optimized for computation over additive white Gaussian noise (AWGN) channels. The design is formulated as an additive mapping problem to determine the optimal constellation that minimizes the mean-squared error (MSE) under a transmit power constraint. We express the optimal constellation design as a system of nonlinear equations and establish the conditions guaranteeing the uniqueness of its solution. In the high signal-to-noise-ratio (SNR) regime, we derive closed-form expressions for the optimal modulation parameters using the generalized Lambert function, providing analytical insight into the system's behavior. Furthermore, we discuss extensions of the framework to higher-dimensional grids corresponding to multiple channel uses, to non-Gaussian noise models, and to computation over real-valued domains via hybrid digital-analog modulation.

Towards Optimal Constellation Design for Digital Over-the-Air Computation

TL;DR

This work addresses the problem of computing sums over a wireless MAC using digital modulation by designing optimal two-dimensional constellations under a power constraint to minimize the mean-squared error. It develops a rigorous optimization framework for encoder/decoder design, derives nonlinear optimality conditions for ML and MAP decoding, and proves uniqueness of the solutions. In the high-SNR regime, it provides closed-form solutions based on a generalized Lambert function, offering deep insight into the constellation structure and asymptotic behavior, including convergence of ML and MAP. Extensions to heavy-tailed noise, real-domain computation via hybrid modulation, and higher-dimensional grids demonstrate the framework’s broad applicability, and numerical results show substantial performance gains over standard QAM-based schemes. The approach thus enables robust, hardware-friendly digital OAC with theoretically grounded design rules and practical implications for edge computing and distributed inference.

Abstract

Over-the-air computation (OAC) has emerged as a key technique for efficient function computation over multiple-access channels (MACs) by exploiting the waveform superposition property of the wireless domain. While conventional OAC methods rely on analog amplitude modulation, their performance is often limited by noise sensitivity and hardware constraints, motivating the use of digital modulation schemes. This paper proposes a novel digital modulation framework optimized for computation over additive white Gaussian noise (AWGN) channels. The design is formulated as an additive mapping problem to determine the optimal constellation that minimizes the mean-squared error (MSE) under a transmit power constraint. We express the optimal constellation design as a system of nonlinear equations and establish the conditions guaranteeing the uniqueness of its solution. In the high signal-to-noise-ratio (SNR) regime, we derive closed-form expressions for the optimal modulation parameters using the generalized Lambert function, providing analytical insight into the system's behavior. Furthermore, we discuss extensions of the framework to higher-dimensional grids corresponding to multiple channel uses, to non-Gaussian noise models, and to computation over real-valued domains via hybrid digital-analog modulation.

Paper Structure

This paper contains 32 sections, 14 theorems, 140 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Paper Organization
  3. Notation
  4. Problem Formulation
  5. Channel Model
  6. Problem Statement
  7. Encoding
  8. Decoding Using Maximum Likelihood (ML) Estimator
  9. Decoding Using Maximum a Posteriori (MAP) Estimator
  10. Main Results
  11. Optimal Modulation Diagram for ML Decoding
  12. Optimal Modulation Diagram for MAP Decoding
  13. Closed-form Solution for High SNR Region
  14. Computation Performance
  15. ML vs. MAP Decoding
  16. ...and 17 more sections

Key Result

Theorem 1

For a network with $K$ transmitters using an encoding function $\mathscr{E}_{q}$ defined in eq:energy_conste, with modulation order $Q=q\times n$, let $N_{1,K}:= K(q-1)+1, N_{2,K}:= K(n-1)+1$ and $\bar{N}_{1,K}=\lfloor 2N_{1,K}/3\rfloor, \bar{N}_{2,K}=\lfloor 2N_{2,K}/3\rfloor$. Then, under an ML de For any SNR $\xi$, when $3\leq N_{1,K}, N_{2,K} \leq8$, the optimal parameters ${d}_1^*$ and ${d}_2

Figures (14)

  • Figure 1: Illustration of the proposed encoding structure for digital OAC with $K=2$ nodes. The two grids on the left represent the individual encoder mappings $\mathscr{E}_3(s_1)$ and $\mathscr{E}_3(s_2)$, with modulation levels $(q_1,n_1) = (3,3)$ and $(q_2,n_1) = (3,2)$. Each node maps its input symbol $s_k \in \{0, \dots, qn_k - 1\}$ onto a structured two-dimensional grid in the complex plane. The rightmost grid depicts the superimposed constellation diagram resulting from the addition $\mathscr{E}_3(s_1) + \mathscr{E}_3(s_2)$ at the CP.
  • Figure 2: Examples of encoded constellation diagrams $\mathscr{E}_q(s)$ for various combinations of modulation order $Q$ and in-phase and quadrature levels $q$ and $n$, respectively. Each diagram represents the mapping of the input symbol $s \in \{0,1,\ldots,q\times n-1\}$ to a multidimensional coordinate space, illustrating how constellation points are arranged. Subfigure \ref{['fig:GrayCode(a)']} shows $q=3, n=2$; \ref{['fig:GrayCode(b)']} depicts $q=2, n=4$; and \ref{['fig:GrayCode(c)']} illustrates $q=4, n=4$ using a 16-QAM constellation with labeled spacing parameters $d_1$ and $d_2$.
  • Figure 3: Plot of $\mathcal{G}_{Q}^{N}(t)$ in \ref{['eq:MathcalG']} with parameters $K=6$, $q=4$, and $n=16$, yielding $N_{1,K}=19$ and $N_{2, K}=91$. The interval $t\in[-0.5,0.5]$ is highlighted.
  • Figure 4: Comparison of the actual approximation error and the final tail‐based bound by Lemma \ref{['lem:apprximation_fg']} as functions of $\xi$ for $t=0.1$ and $t=0.3$. Solid curves denote the computed error; dashed curves denote the analytical bound. The magnitude is plotted on a logarithmic scale. Parameters: $K=2$, and $q=n=4$.
  • Figure 5: Ratio of the optimal scales $d_1^*/ d_2^*$ versus the parameter $\xi$ for two array sizes. (a) $K=2$, showing curves for $(q,n)=(6,4),(4,4),(4,6)$ that asymptotically converge to $1$. (b) $K=10$, showing equal‐dimension cases $(q,n)=(4,4),(6,6),(8,8)$ with the unity asymptote.
  • ...and 9 more figures

Theorems & Definitions (32)

  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1
  • proof
  • Corollary 1
  • Remark 4
  • Theorem 2
  • proof
  • Remark 5
  • ...and 22 more