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Second Order Analysis for Joint Source-Channel Coding with Markovian Source

Ryo Yaguchi, Masahito Hayashi

TL;DR

The second order rates of joint source-channel coding, whose source obeys an irreducible and ergodic Markov process, are derived by introducing new distribution family, switched Gaussian convolution distribution, when the channel is a discrete memoryless.

Abstract

We derive the second order rates of joint source-channel coding, whose source obeys an irreducible and ergodic Markov process when the channel is a discrete memoryless, while a previous study solved it only in a special case. We also compare the joint source-channel scheme with the separation scheme in the second order regime while a previous study made a notable comparison only with numerical calculation. To make these two notable progress, we introduce two kinds of new distribution families, switched Gaussian convolution distribution and *-product distribution, which are defined by modifying the Gaussian distribution.

Second Order Analysis for Joint Source-Channel Coding with Markovian Source

TL;DR

The second order rates of joint source-channel coding, whose source obeys an irreducible and ergodic Markov process, are derived by introducing new distribution family, switched Gaussian convolution distribution, when the channel is a discrete memoryless.

Abstract

We derive the second order rates of joint source-channel coding, whose source obeys an irreducible and ergodic Markov process when the channel is a discrete memoryless, while a previous study solved it only in a special case. We also compare the joint source-channel scheme with the separation scheme in the second order regime while a previous study made a notable comparison only with numerical calculation. To make these two notable progress, we introduce two kinds of new distribution families, switched Gaussian convolution distribution and *-product distribution, which are defined by modifying the Gaussian distribution.

Paper Structure

This paper contains 35 sections, 18 theorems, 173 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notations and Information quantities
  3. Single shot
  4. Markovian process
  5. New Probability Distribution Families
  6. Switched Gaussian convolution distribution
  7. $*$-product distribution
  8. Single Shot Setting
  9. Problem formulation
  10. Direct part
  11. General case
  12. Conditional additive case
  13. Converse part
  14. General case
  15. Conditional additive case
  16. ...and 20 more sections

Key Result

Proposition 2

When $X^n=(X_1, \ldots, X_n)$ and $Z^n=(Z_1, \ldots, Z_n)$ are subject to the Markovian process generated by a non-hidden transition matrix $W$, the random variable $\frac{1}{\sqrt{n}} (- \log P_{X^n|Z^n}(X^n|Z^n)-n H^W(X|Z))$ asymptotically obeys the Gaussian distribution with variance $V^W(X|Z)$T

Figures (3)

  • Figure 1: Graphs of ${\Psi}[1,1,v_3](x)$. Black line: $v_3=1$, Red dashed line: $v_3=1/9$, Red normal line: $v_3 \to 0$, Blue dashed line: $v_3=4$, Blue normal line: $v_3=25$, Blue dashed thick line: $v_3=25^2$, Blue thick line: $v_3 \to \infty$.
  • Figure 2: Graphs of functions in \ref{['sep,so,th1']}. Red line: $2 \Phi_{4}(x)-\Phi_{4}(x)^2$. Blue dotted line: $\tilde{\Phi}[1.5,0.5](x)$. Blue normal line: $\tilde{\Phi}[1.9,0.1](x)$. Blue dashed line: $\tilde{\Phi}[1.99,0.01](x)$. Black line: $\Phi_{2}(x)$.
  • Figure 3: Graphs of the ratio $\frac{\varepsilon_{KV}(R) }{\varepsilon(R)}$ with $\frac{ C }{ H^{W_s}(M) } V^{ W_s } (M) =1$. The origin is $(0,1)$. Blue line expresses the upper bound given in \ref{['UPP7']}. Red line expresses the case with $V^*_{-} (W_{Y|X}) =0.1$ and $V^*_{+} (W_{Y|X}) =10$. Black line expresses the case with $V^*_{-} (W_{Y|X}) =0.5$ and $V^*_{+} (W_{Y|X}) =1.5$.

Theorems & Definitions (33)

  • Definition 1: non-hidden
  • Proposition 2: Central limit theorem for Markovian Process (Aetc.)
  • Lemma 1
  • Remark 3
  • Proposition 4
  • Corollary 1
  • Lemma 2
  • proof
  • Lemma 3
  • Remark 5
  • ...and 23 more