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

2025-11-09

AI Summary

This work addresses the optimal constellation design problem for function computation via digital over-the-air computation (OAC) over additive white Gaussian noise (AWGN) channels under a power constraint, with the objective of minimizing mean squared error (MSE). We formulate the optimal constellation as a system of nonlinear equations and, for the first time, derive a closed-form solution at high signal-to-noise ratio (SNR) using the generalized Lambert W function—thereby establishing an analytical relationship among MSE, transmit power, and constellation structure. The framework is further extended to multidimensional lattice constellations, non-Gaussian noise scenarios, and hybrid digital-analog modulation. The proposed approach significantly enhances both computational accuracy and robustness of digital OAC. It provides the first analytically tractable and scalable theoretical framework for signal design in high-SNR digital OAC, along with explicit parameter design guidelines.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Systems and Infrastructure for Web, Mobile and WoT: Energy management for devices in mobile Web and WoT environments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

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.

Problem

• Optimizing digital modulation for wireless function computation
• Designing constellations to minimize mean-squared error in AWGN
• Extending framework to non-Gaussian noise and multi-channel scenarios

Innovation

• Digital modulation framework for over-the-air computation
• Optimal constellation design minimizing mean-squared error
• Closed-form solutions using generalized Lambert function