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An Achievability Bound for Variable-Length Stop-Feedback Coding over the Gaussian Channel

Ioannis Papoutsidakis, Robert J. Piechocki, Angela Doufexi

TL;DR

A general achievability bound is derived, that employs a random coding ensemble in combination with minimum distance decoding for variable-length coding with stop-feedback and confirms the significant value of feedback compared to transmission with fixed blocklength coding and without feedback.

Abstract

Feedback holds a pivotal role in practical communication schemes, even though it does not enhance channel capacity. Its main attribute includes adaptability in transmission that allows for a higher rate of convergence of the error probability to zero with respect to blocklength. Motivated by this fact, we present a non-asymptotic achievability bound for variable-length coding with stop-feedback. Specifically, a general achievability bound is derived, that employs a random coding ensemble in combination with minimum distance decoding. The general bound is particularized for the Gaussian channel. Numerical evaluation of the bound confirms the significant value of feedback compared to transmission with fixed blocklength coding and without feedback.

An Achievability Bound for Variable-Length Stop-Feedback Coding over the Gaussian Channel

TL;DR

A general achievability bound is derived, that employs a random coding ensemble in combination with minimum distance decoding for variable-length coding with stop-feedback and confirms the significant value of feedback compared to transmission with fixed blocklength coding and without feedback.

Abstract

Feedback holds a pivotal role in practical communication schemes, even though it does not enhance channel capacity. Its main attribute includes adaptability in transmission that allows for a higher rate of convergence of the error probability to zero with respect to blocklength. Motivated by this fact, we present a non-asymptotic achievability bound for variable-length coding with stop-feedback. Specifically, a general achievability bound is derived, that employs a random coding ensemble in combination with minimum distance decoding. The general bound is particularized for the Gaussian channel. Numerical evaluation of the bound confirms the significant value of feedback compared to transmission with fixed blocklength coding and without feedback.
Paper Structure (8 sections, 2 theorems, 49 equations, 1 figure)

This paper contains 8 sections, 2 theorems, 49 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and Definitions
  3. General Achievability Bound
  4. The Gaussian Channel
  5. Numerical Analysis
  6. Conclusions
  7. Complement to the proof of (\ref{['makrinar']})
  8. Sampling of $V_{2,n}$

Key Result

Theorem 1

Fix a threshold $0 \leq \epsilon \leq 1$ on the probability of error and a memoryless channel $\{P_{Y_i|X_i}\}_{i=1}^\infty$. Let the arbitrary mutually independent and identical processes $\textbf{X}_i= (X_{i1},X_{i2},...,X_{in},...)$ for $i=1,...,M$ where $X_{in} \in\mathcal{A}$. Let the marginal and a stopping time where Then for any $M$ there exists an $(l,M,\epsilon)$ VLSF code with

Figures (1)

  • Figure 1: Numerical evaluation of Theorem \ref{['theor2']} for signal-to-noise ratio $\gamma=1$ (0 dB) and average probability of error $\epsilon=10^{-3}$. The capacity of the channel and the normal approximation for fixed-length coding without feedback are also presented.

Theorems & Definitions (5)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2
  • proof