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Worst-Case Per-User Error Bound for Asynchronous Unsourced Multiple Access

Jyun-Sian Wu, Pin-Hsun Lin, Marcel A. Mross, Eduard A. Jorswieck

TL;DR

This paper tackles worst-case asynchronicity in unsourced multiple access by introducing a bounded-delay model $\alpha n$ and deriving a uniform PUPE bound that holds across all delay patterns. Using saddlepoint approximations and a maximal information density decoder, it provides a finite-blocklength analysis (Theorem 1) for a given delay realization and a uniform bound (Theorem 2) across delays, avoiding enumeration of all erroneous subsets. The results show that asynchronous operation can reduce interference but requires more energy per bit to meet the PUPE constraint, with energy-per-bit rising as the effective decoding window shortens ($\alpha$ increases). The findings quantify the energy-delay tradeoffs and position the synchronous case ($\alpha=0$) as a special instance within the broader AUMAC framework, offering practical insight for scalable IoT-type networks.

Abstract

This work considers an asynchronous $\textsf{K}_\text{a}$-active-user unsourced multiple access channel (AUMAC) with the worst-case asynchronicity. The transmitted messages must be decoded within $n$ channel uses, while some codewords are not completely received due to asynchronicities. We consider a constraint of the largest allowed delay of the transmission. The AUMAC lacks the permutation-invariant property of the synchronous UMAC since different permutations of the same codewords with a fixed asynchronicity are distinguishable. Hence, the analyses require calculating all $2^{\textsf{K}_\text{a}}-1$ combinations of erroneously decoded messages. Moreover, transmitters cannot adapt the corresponding codebooks according to asynchronicity due to a lack of information on asynchronicities. To overcome this challenge, a uniform bound of the per-user probability of error (PUPE) is derived by investigating the worst-case of the asynchronous patterns with the delay constraint. Numerical results show the trade-off between the energy-per-bit and the number of active users for different delay constraints. In addition, although the asynchronous transmission reduces interference, the required energy-per-bit increases as the receiver decodes with incompletely received codewords, compared to the synchronous case.

Worst-Case Per-User Error Bound for Asynchronous Unsourced Multiple Access

TL;DR

This paper tackles worst-case asynchronicity in unsourced multiple access by introducing a bounded-delay model and deriving a uniform PUPE bound that holds across all delay patterns. Using saddlepoint approximations and a maximal information density decoder, it provides a finite-blocklength analysis (Theorem 1) for a given delay realization and a uniform bound (Theorem 2) across delays, avoiding enumeration of all erroneous subsets. The results show that asynchronous operation can reduce interference but requires more energy per bit to meet the PUPE constraint, with energy-per-bit rising as the effective decoding window shortens ( increases). The findings quantify the energy-delay tradeoffs and position the synchronous case () as a special instance within the broader AUMAC framework, offering practical insight for scalable IoT-type networks.

Abstract

This work considers an asynchronous -active-user unsourced multiple access channel (AUMAC) with the worst-case asynchronicity. The transmitted messages must be decoded within channel uses, while some codewords are not completely received due to asynchronicities. We consider a constraint of the largest allowed delay of the transmission. The AUMAC lacks the permutation-invariant property of the synchronous UMAC since different permutations of the same codewords with a fixed asynchronicity are distinguishable. Hence, the analyses require calculating all combinations of erroneously decoded messages. Moreover, transmitters cannot adapt the corresponding codebooks according to asynchronicity due to a lack of information on asynchronicities. To overcome this challenge, a uniform bound of the per-user probability of error (PUPE) is derived by investigating the worst-case of the asynchronous patterns with the delay constraint. Numerical results show the trade-off between the energy-per-bit and the number of active users for different delay constraints. In addition, although the asynchronous transmission reduces interference, the required energy-per-bit increases as the receiver decodes with incompletely received codewords, compared to the synchronous case.
Paper Structure (8 sections, 3 theorems, 53 equations, 2 figures)

This paper contains 8 sections, 3 theorems, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. System model and preliminaries
  3. Main Results
  4. Numerical Results
  5. Conclusions
  6. Proof of Theorem 1
  7. Proof of Theorem 2
  8. Proof of Lemma \ref{['lemma: the product of 2 functions are non-decreasing']}

Key Result

Theorem 1

Fix $0<\mathsf{P}<\mathsf{P}'$. There exists an $(n,{\normalfont \textsf{M}},\epsilon,\normalfont \textsf{K}_{\text{a}},\alpha,D^{\normalfont \textsf{K}_{\text{a}}}\!)-$code for an AUMAC such that the PUPE can be upper bounded by the following: if there exists a $t_0(a^n)\in(0,1)$ such that $E^{(1)}_t(a^n, t_0(a^n))=|\mathcal{S}|{\normalfont \log\textsf{M}}$, where and $p_0:=\frac{\normalfont \t

Figures (2)

  • Figure 1: A $\normalfont \textsf{K}_{\text{a}}$-active-user AUMAC with $D^{\normalfont \textsf{K}_{\text{a}}}\!=[0,1,3,5,...,5]$.
  • Figure 2: $\frac{\text{E}_{\text{b}}}{\text{N}_{\text{0}}}$ of AUMAC compared to synchronous UMAC for different numbers of active users.

Theorems & Definitions (8)

  • Definition 1
  • Remark 1
  • Definition 2
  • Theorem 1
  • Definition 3
  • Theorem 2
  • Remark 2
  • Lemma 1