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An Achievability Bound for Type-Based Unsourced Multiple Access

Deekshith Pathayappilly Krishnan, Kaan Okumus, Khac-Hoang Ngo, Giuseppe Durisi

TL;DR

This work addresses the problem of estimating the empirical message type in a type-based unsourced GMAC, introducing a numerically computable random-coding achievability bound that bounds the expected total-variation distance between the transmitted type $\mathbf{t}$ and its estimate $\hat{\mathbf{T}}$. The bound is derived via a random-coding construction on a Gaussian MAC using Chernoff bounds and Gallager’s $\rho$-trick, carefully accounting for multiplicity estimation errors with a nuanced error-event partition. Numerical results reveal how the minimum energy per bit, $E_b/N_0$, depends on the distance between the actual type and a uniform-type baseline, showing a nonmonotone behavior with a peak at intermediate multiplicities; a practical CCS-AMP scheme operates within about 3 dB of the bound for the tested parameters. The findings offer design insights for energy-efficient type-based communication in distributed sensing and aggregation tasks and provide a framework for evaluating type-estimation strategies in unsourced access systems.

Abstract

We derive an achievability bound to quantify the performance of a type-based unsourced multiple access system -- an information-theoretic model for grant-free multiple access with correlated messages. The bound extends available achievability results for the per-user error probability in the unsourced multiple access framework, where, different from our setup, message collisions are treated as errors. Specifically, we provide an upper bound on the total variation distance between the type (i.e., the empirical probability mass function) of the transmitted messages and its estimate over a Gaussian multiple access channel. Through numerical simulations, we illustrate that our bound can be used to determine the message type that is less efficient to transmit, because more difficult to detect. We finally show that a practical scheme for type estimation, based on coded compressed sensing with approximate message passing, operates approximately 3 dB away from the bound, for the parameters considered in the paper.

An Achievability Bound for Type-Based Unsourced Multiple Access

TL;DR

This work addresses the problem of estimating the empirical message type in a type-based unsourced GMAC, introducing a numerically computable random-coding achievability bound that bounds the expected total-variation distance between the transmitted type and its estimate . The bound is derived via a random-coding construction on a Gaussian MAC using Chernoff bounds and Gallager’s -trick, carefully accounting for multiplicity estimation errors with a nuanced error-event partition. Numerical results reveal how the minimum energy per bit, , depends on the distance between the actual type and a uniform-type baseline, showing a nonmonotone behavior with a peak at intermediate multiplicities; a practical CCS-AMP scheme operates within about 3 dB of the bound for the tested parameters. The findings offer design insights for energy-efficient type-based communication in distributed sensing and aggregation tasks and provide a framework for evaluating type-estimation strategies in unsourced access systems.

Abstract

We derive an achievability bound to quantify the performance of a type-based unsourced multiple access system -- an information-theoretic model for grant-free multiple access with correlated messages. The bound extends available achievability results for the per-user error probability in the unsourced multiple access framework, where, different from our setup, message collisions are treated as errors. Specifically, we provide an upper bound on the total variation distance between the type (i.e., the empirical probability mass function) of the transmitted messages and its estimate over a Gaussian multiple access channel. Through numerical simulations, we illustrate that our bound can be used to determine the message type that is less efficient to transmit, because more difficult to detect. We finally show that a practical scheme for type estimation, based on coded compressed sensing with approximate message passing, operates approximately 3 dB away from the bound, for the parameters considered in the paper.
Paper Structure (12 sections, 5 theorems, 1 equation, 1 figure)

This paper contains 12 sections, 5 theorems, 1 equation, 1 figure.

Table of Contents

  1. Introduction
  2. System Model
  3. A Random-Coding Bound for TUMA
  4. Codebook Generation
  5. Decoder
  6. Error Analysis
  7. Numerical Results
  8. Conclusion
  9. Proof of Lemma \ref{['Lem:Delta']}
  10. Proof of Lemma \ref{['Lem:Opt_lambda']}
  11. Proof of Lemma \ref{['Lem:Exp_f_c']}
  12. Proof of Lemma \ref{['Lem:Pi_Cardinality']}

Key Result

Theorem 1

For a with ${\mathrm{K}\xspace_{\rm a}}$ active users and ${\mathrm{M}\xspace_{\rm a}}$ active messages, and a fixed message multiplicity vector $\bind\xspace\xspace$ with $|\mathtt{Supp}\mathopen{}\left(\bind\xspace\xspace\right)|={\mathrm{M}\xspace_{\rm a}}$ and $\lVert\bind\xspace\xspace\rVert_1= Here, with $\bm{C}_1\sim \mathcal{N}\xspace\mathopen{}\left(\mathbf{0},\mathrm{P}\xspace'\pmb{\mat

Figures (1)

  • Figure 1: The required $E_{\rm b}/N_0$ (dB) for message types with $({\mathrm{M}\xspace_{\rm a}},{\mathrm{K}\xspace_{\rm a}})$ in ${\{ (100,100), (80,92), (60, 98), (50,102), (30,102), (10,100)\}}$.

Theorems & Definitions (11)

  • Definition 1: TUMA Code
  • Theorem 1: Random-Coding Bound
  • proof : Proof of Theorem \ref{['Th:Main']}
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 1 more