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Two-Class Joint Source-Channel Coding: Expurgated Exponents with i.i.d. Distributions

Seyed AmirPouya Moeini, Albert Guillén i Fàbregas

TL;DR

This work analyzes expurgated exponents for joint source-channel coding with i.i.d. coding, showing that a two-class partitioning of source sequences with source-type dependent codeword distributions can achieve an exponent not smaller than the optimal single-class coding exponent. The key technique extends CKM-style expurgation to partitioned ensembles, yielding a bound that reduces to the single-class case under appropriate choices and demonstrating a fundamental one-sided advantage of partitioning. Numerically, partitioning can offer gains in some regimes, though not universally, and the results clarify when partitioning helps (e.g., non-binary input alphabets) versus when it does not (binary-input channels). Overall, the paper provides a rigorous bridge between partitioned and non-partitioned expurgation in JSCC and confirms the existence of partition schemes that match or improve the best known exponents for i.i.d. coding.

Abstract

This paper studies expurgated exponents for joint source-channel coding of discrete memoryless sources and channels under i.i.d. random coding. We show that a two-class partitioning of source sequences, where the codeword distribution depends on the source type, achieves an exponent at least as high as that of optimal single-class coding, in which the codeword distribution is independent of the source message.

Two-Class Joint Source-Channel Coding: Expurgated Exponents with i.i.d. Distributions

TL;DR

This work analyzes expurgated exponents for joint source-channel coding with i.i.d. coding, showing that a two-class partitioning of source sequences with source-type dependent codeword distributions can achieve an exponent not smaller than the optimal single-class coding exponent. The key technique extends CKM-style expurgation to partitioned ensembles, yielding a bound that reduces to the single-class case under appropriate choices and demonstrating a fundamental one-sided advantage of partitioning. Numerically, partitioning can offer gains in some regimes, though not universally, and the results clarify when partitioning helps (e.g., non-binary input alphabets) versus when it does not (binary-input channels). Overall, the paper provides a rigorous bridge between partitioned and non-partitioned expurgation in JSCC and confirms the existence of partition schemes that match or improve the best known exponents for i.i.d. coding.

Abstract

This paper studies expurgated exponents for joint source-channel coding of discrete memoryless sources and channels under i.i.d. random coding. We show that a two-class partitioning of source sequences, where the codeword distribution depends on the source type, achieves an exponent at least as high as that of optimal single-class coding, in which the codeword distribution is independent of the source message.
Paper Structure (10 sections, 6 theorems, 38 equations, 3 figures)

This paper contains 10 sections, 6 theorems, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Achievable Expurgated Error Exponents
  4. Comparison with Single-Class Coding
  5. Numerical Examples
  6. Proofs
  7. Proof of Lemma \ref{['lem-main']}
  8. Proof of Theorem \ref{['thm-main-1']}
  9. Proof of Theorem \ref{['thm-single-ach']}
  10. Proof of Theorem \ref{['thm-main']}

Key Result

Lemma 1

Let $\{\mathcal{A}_1, \ldots, \mathcal{A}_m\}$ be a given partition of the source sequences, with each class assigned an i.i.d. codeword distribution from $\mathcal{Q}_m=\{Q_1,\ldots,Q_m\}$. Then, there exists a sequence of codebooks such that, for all choices of $\{\rho_c\}_{c=1}^{m}$ with $\rho_c where and with $d_B(x,\bar{x}) \triangleq -\log \sum_y \sqrt{W(y|x)W(y|\bar{x})}$ denoting the Bh

Figures (3)

  • Figure 1: Example where partitioning strictly improves the exponent, obtained with $Q_1 = (0.4,0.4,0.2)$, and $Q_2 = (0.5,0.5,0)$, $t = 0.75$ and $P_V(0) = 0.025$.
  • Figure 2: Example where single-class coding yields a higher exponent, with the same setting as Fig. \ref{['fig-part']} except $t = 0.5$ and $P_V(0) = 0.02$.
  • Figure 3: The two exponents coincide under optimal distributions, with the same setting as Fig. \ref{['fig-part']} except $Q_1 = (0.4489,0.4489,0.1021)$ and $Q_2 = (0.5,0.5,0)$.

Theorems & Definitions (6)

  • Lemma 1
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Lemma 2
  • Lemma 3