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Empirical Coordination over Markov Channel with Independent Source

Mengyuan Zhao, Maël Le Treust, Tobias J. Oechtering

TL;DR

This work addresses empirical coordination for joint source–channel coding over Markov channels with strictly causal encoders and noncausal decoders. It derives single-letter inner and outer bounds on the achievable joint distribution by introducing an auxiliary variable $W$ and an information constraint $I(X;Y|Y') - I(U;W|X) \ge 0$, capturing how the encoder conveys compressed source information to the decoder under Markov memory. The analysis introduces input-driven Markov typicality, along with its AEP and packing properties, to exploit channel structure beyond independence-based arguments. The results generalize coordination coding beyond DMCs, and special cases recover Shannon-style source–channel separation when variables are independent, highlighting the framework's practical relevance for JSCC with memory.

Abstract

We study joint source-channel coding over Markov channels through the empirical coordination framework. More specifically, we aim at determining the empirical distributions of source and channel symbols that can be induced by a coding scheme. We consider strictly causal encoders that generate channel inputs, without access to the past channel states, henceforth driving the current Markov state evolution. Our main result is the single-letter inner and outer bounds of the set of achievable joint distributions, coordinating all the symbols in the network. To establish the inner bound, we introduce a new notion of typicality, the input-driven Markov typicality, and develop its fundamental properties. Contrary to the classical block-Markov coding schemes that rely on blockwise independence for discrete memoryless channels, our analysis directly exploits the Markov channel structure and improves beyond the independence-based arguments.

Empirical Coordination over Markov Channel with Independent Source

TL;DR

This work addresses empirical coordination for joint source–channel coding over Markov channels with strictly causal encoders and noncausal decoders. It derives single-letter inner and outer bounds on the achievable joint distribution by introducing an auxiliary variable and an information constraint , capturing how the encoder conveys compressed source information to the decoder under Markov memory. The analysis introduces input-driven Markov typicality, along with its AEP and packing properties, to exploit channel structure beyond independence-based arguments. The results generalize coordination coding beyond DMCs, and special cases recover Shannon-style source–channel separation when variables are independent, highlighting the framework's practical relevance for JSCC with memory.

Abstract

We study joint source-channel coding over Markov channels through the empirical coordination framework. More specifically, we aim at determining the empirical distributions of source and channel symbols that can be induced by a coding scheme. We consider strictly causal encoders that generate channel inputs, without access to the past channel states, henceforth driving the current Markov state evolution. Our main result is the single-letter inner and outer bounds of the set of achievable joint distributions, coordinating all the symbols in the network. To establish the inner bound, we introduce a new notion of typicality, the input-driven Markov typicality, and develop its fundamental properties. Contrary to the classical block-Markov coding schemes that rely on blockwise independence for discrete memoryless channels, our analysis directly exploits the Markov channel structure and improves beyond the independence-based arguments.
Paper Structure (10 sections, 6 theorems, 49 equations, 1 figure)

This paper contains 10 sections, 6 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. System Model
  3. Main Results
  4. Input-driven Markov Typicality
  5. Achievability Proof of Theorem \ref{['thm: main']}
  6. Joint Packing Lemma
  7. Converse Proof of Theorem \ref{['thm: main']}
  8. Property of the Input-Driven Markov Typicality
  9. Proof for Theorem \ref{['thm: conditional controlled Markov typicality lemma']}
  10. Proof of Theorem \ref{['lemma: AEP and cardinality bound']}

Key Result

Lemma 1

The Markov chain $Y^n$ admits a unique equilibrium distribution $\pi_Y \in \Delta(\mathcal{R}_Y)$, which is uniquely determined by the stationary input distribution $\mathcal{P}_X$ and the channel transition kernel $\mathcal{W}_{Y|X,Y'}$ by

Figures (1)

  • Figure 1: Joint source-channel coding with strictly causal encoder and noncausal decoder over Markov channel

Theorems & Definitions (11)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Theorem 1: Main Result
  • Remark 1: Source-Channel Separation
  • Definition 3: Input-driven Markov Typicality
  • Theorem 2: Ergodicity
  • Theorem 3: AEP and Cardinality Bound
  • Lemma 2: Joint Packing Lemma
  • Proposition 1
  • ...and 1 more