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Achievable Second-Order Asymptotics for MAC and RAC with Additive Non-Gaussian Noise

Yiming Wang, Lin Bai, Zhuangfei Wu, Lin Zhou

TL;DR

The work tackles finite-blocklength performance for a two-user MAC under additive arbitrary noise and for a RAC with unknown activity. By employing spherical codebooks with either joint nearest neighbor (JNN) or successive interference cancellation (SIC) decoding, it derives second-order achievable rate regions, showing identical first-order regions but a clear second-order advantage for JNN in MAC, and a strictly larger first-order region for JNN in RAC. The analysis extends to non-Gaussian noise via moment constraints and recovers known Gaussian results as special cases, while introducing unified second-order forms and new lemmas (g-function bounds) to handle the mismatched multi-user setting. The RAC results leverage a rateless, time-adaptive decoding structure to cope with unknown active-user patterns, highlighting the practical robustness of mismatched coding with spherical codebooks. Overall, the paper advances finite-blocklength understanding for mismatched multi-user channels and sets the stage for future development of ensemble converse results and higher-order characterizations.

Abstract

We first study the two-user additive noise multiple access channel (MAC) where the noise distribution is arbitrary. For such a MAC, we use spherical codebooks and either joint nearest neighbor (JNN) or successive interference cancellation (SIC) decoding. Under both decoding methods, we derive second-order achievable rate regions and compare the finite blocklength performance between JNN and SIC decoding. Our results indicate that although the first-order rate regions of JNN and SIC decoding are identical, JNN decoding has better second-order asymptotic performance. When specialized to the Gaussian noise, we provide an alternative achievability proof to the result by MolavianJazi and Laneman (T-IT, 2015). Furthermore, we generalize our results to the random access channel (RAC) where neither the transmitters nor the receiver knows the user activity pattern. We use spherical-type codebooks and a rateless transmission scheme combining JNN/SIC decoding, and derive second-order achievability bounds. Comparing second-order achievability results of JNN and SIC decoding in a RAC, we show that JNN decoding achieves strictly larger first-order asymptotic rate. When specialized to Gaussian noise, our second-order asymptotic results recover the corresponding results of Yavas, Kostina, and Effros (T-IT, 2021) up to second-order.

Achievable Second-Order Asymptotics for MAC and RAC with Additive Non-Gaussian Noise

TL;DR

The work tackles finite-blocklength performance for a two-user MAC under additive arbitrary noise and for a RAC with unknown activity. By employing spherical codebooks with either joint nearest neighbor (JNN) or successive interference cancellation (SIC) decoding, it derives second-order achievable rate regions, showing identical first-order regions but a clear second-order advantage for JNN in MAC, and a strictly larger first-order region for JNN in RAC. The analysis extends to non-Gaussian noise via moment constraints and recovers known Gaussian results as special cases, while introducing unified second-order forms and new lemmas (g-function bounds) to handle the mismatched multi-user setting. The RAC results leverage a rateless, time-adaptive decoding structure to cope with unknown active-user patterns, highlighting the practical robustness of mismatched coding with spherical codebooks. Overall, the paper advances finite-blocklength understanding for mismatched multi-user channels and sets the stage for future development of ensemble converse results and higher-order characterizations.

Abstract

We first study the two-user additive noise multiple access channel (MAC) where the noise distribution is arbitrary. For such a MAC, we use spherical codebooks and either joint nearest neighbor (JNN) or successive interference cancellation (SIC) decoding. Under both decoding methods, we derive second-order achievable rate regions and compare the finite blocklength performance between JNN and SIC decoding. Our results indicate that although the first-order rate regions of JNN and SIC decoding are identical, JNN decoding has better second-order asymptotic performance. When specialized to the Gaussian noise, we provide an alternative achievability proof to the result by MolavianJazi and Laneman (T-IT, 2015). Furthermore, we generalize our results to the random access channel (RAC) where neither the transmitters nor the receiver knows the user activity pattern. We use spherical-type codebooks and a rateless transmission scheme combining JNN/SIC decoding, and derive second-order achievability bounds. Comparing second-order achievability results of JNN and SIC decoding in a RAC, we show that JNN decoding achieves strictly larger first-order asymptotic rate. When specialized to Gaussian noise, our second-order asymptotic results recover the corresponding results of Yavas, Kostina, and Effros (T-IT, 2021) up to second-order.

Paper Structure

This paper contains 27 sections, 10 theorems, 151 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Main Contributions
  3. Organization for the Rest of the Paper
  4. problem formulation
  5. Two-user MAC
  6. System Model and Code Definition
  7. Performance Metrics
  8. RAC with Unknown User Activity Pattern
  9. Main Results and Discussions
  10. Preliminaries
  11. Two-User MAC
  12. Comparison of Second-order Achievable Rate Regions of JNN and SIC Decoding for MAC
  13. RAC with Unknown User Activity Pattern
  14. Proof of the Achievable Rate Regions for Two-User MAC (Theorems \ref{['theo: mac jnn achie']} and \ref{['theorem:MAC-SIC ach']})
  15. Lemmas
  16. ...and 12 more sections

Key Result

Theorem 1

For any $\varepsilon \in (0,1)$, there exists an $(n,M_1,$$M_2,P_1,P_2)$-JNN code such that for $\mathrm{P}_\mathrm{e}^n\leq \varepsilon$, where

Figures (7)

  • Figure 1: System model of a two-user MAC with arbitrary additive channel noise.
  • Figure 2: Illustration of a spherical codebook in \ref{['eq:sphericaldist']} when $n=3$.
  • Figure 3: Illustration of the first-order rate region for a two-user MAC, where for each $j\in[2]$, $R_j$ and $P_j$ are the first-order rate and transmitter power for user $j$, respectively.
  • Figure 4: Illustration of a RAC applying rateless code with JNN/SIC decoder.
  • Figure 5: Illustration of the spherical-type codebook in \ref{['eq:first n0 codewords distribution']} and \ref{['eq:nj-1 to nj codewords distribution']} when $n_1=2$ and $n_2=3$ (reproduced from yavas2021gaussianmac).
  • ...and 2 more figures

Theorems & Definitions (18)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Theorem 1
  • Corollary 2
  • Theorem 3
  • Theorem 4
  • ...and 8 more