Function Computation Over Multiple Access Channels via Hierarchical Constellations

TL;DR

The paper tackles computing functions of distributed data over a Gaussian MAC by introducing a hierarchical constellation framework for over-the-air computation. It develops a shift-map based encoding and a digit-extraction decoder, enabling reliable computation of multiple function outputs from a single channel use, and couples this with a shielding mechanism via variable-length block coding to curb error propagation across constellation levels. The authors characterize the achievable computation rate, showing that for independent source symbols the gap to the optimum scales as $\mathcal{O}(\log_2(1/\epsilon)/K)$ and vanishes as the network grows, while shielding with guards yields a tighter $\mathcal{O}(\log_2\ln(1/\epsilon))$ gap; variable-length coding further achieves near-optimal $\epsilon$-scaling with a rate of $\approx \frac{1}{2}\log_2(\mathrm{SNR})/\log_2 B$. Collectively, these results provide a channel-agnostic, low-latency framework for function computation in large-scale networks and illuminate regimes where uncoded or lightly coded OAC is information-theoretically optimal. Future work includes extending to fading channels and exploring polynomial signaling approaches to enhance robustness.

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/ε)/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/ε)}), 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.

Figures (2)

• Figure 1: System model for analog function computation over a noisy MAC. Transmitter $k$ encodes its source block $\mathbf{s}_k$ using the shift-map encoder $\mathscr{E}_q(\cdot)$ to generate a channel input $x_k$. The transmitted signals are superimposed over the channel and corrupted by additive Gaussian noise. The receiver applies decoder $\mathscr{D}_{q, K}(\cdot)$ to recover the desired function outputs $\hat{u}[m]$.
• Figure 2: Schematic illustration of the proposed shielding mechanism. The superposition of source symbols is mapped by the encoder $\mathscr{E}(\cdot)$ into a single digit $t_k[m]$ using an enlarged base $B+\alpha_m$, where $\beta_m$ introduces guard intervals and the effective information-bearing region has size $B$. This structure prevents carry-over and limits noise propagation across significance levels in the superimposed constellation.

Theorems & Definitions (12)

• Definition 1: Decoding Error Probability
• Definition 2: Computation Rate
• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Theorem 1
• proof
• Remark 6