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Achievable Rates and Error Exponents for a Class of Mismatched Compound Channels

Priyanka Patel, Francesc Molina, Albert Guillén i Fàbregas

TL;DR

The paper addresses reliable communication under channel uncertainty by modeling the true channel as lying in a small relative-entropy ball around the decoding metric. It develops worst-case, second-order expansions for achievable rates and error exponents under mismatched decoding for both discrete and continuous alphabets, using dual expressions of GMI/LM and E0, facilitated by Euclidean information theory. Key contributions include closed-form approximations for small mismatch, analysis of symmetric and nearest-neighbor decoders, and extensive treatment of continuous alphabets with Gaussian codebooks and cost constraints. The results quantify a square-root penalty in the ball radius, illuminating the sensitivity to imperfect channel estimation and offering practical guidance for robust decoder design and code construction in uncertain channels.

Abstract

This paper investigates achievable information rates and error exponents of mismatched decoding when the channel belongs to the class of channels that are close to the decoding metric in terms of relative entropy. For both discrete- and continuous-alphabet channels, we derive approximations of the worst-case achievable information rates and error exponents as a function of the radius of a small relative entropy ball centered at the decoding metric, allowing the characterization of the loss incurred due to imperfect channel estimation. We provide a number of examples including symmetric metrics and modulo- additive noise metrics for discrete systems, and nearest neighbor decoding for continuous-alphabet channels, where we derive the approximation when the channel admits arbitrary statistics and when it is assumed noise-additive with unknown finite second-order moment.

Achievable Rates and Error Exponents for a Class of Mismatched Compound Channels

TL;DR

The paper addresses reliable communication under channel uncertainty by modeling the true channel as lying in a small relative-entropy ball around the decoding metric. It develops worst-case, second-order expansions for achievable rates and error exponents under mismatched decoding for both discrete and continuous alphabets, using dual expressions of GMI/LM and E0, facilitated by Euclidean information theory. Key contributions include closed-form approximations for small mismatch, analysis of symmetric and nearest-neighbor decoders, and extensive treatment of continuous alphabets with Gaussian codebooks and cost constraints. The results quantify a square-root penalty in the ball radius, illuminating the sensitivity to imperfect channel estimation and offering practical guidance for robust decoder design and code construction in uncertain channels.

Abstract

This paper investigates achievable information rates and error exponents of mismatched decoding when the channel belongs to the class of channels that are close to the decoding metric in terms of relative entropy. For both discrete- and continuous-alphabet channels, we derive approximations of the worst-case achievable information rates and error exponents as a function of the radius of a small relative entropy ball centered at the decoding metric, allowing the characterization of the loss incurred due to imperfect channel estimation. We provide a number of examples including symmetric metrics and modulo- additive noise metrics for discrete systems, and nearest neighbor decoding for continuous-alphabet channels, where we derive the approximation when the channel admits arbitrary statistics and when it is assumed noise-additive with unknown finite second-order moment.

Paper Structure

This paper contains 38 sections, 5 theorems, 137 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. Problem Setup
  4. Overview of Achievable Rates
  5. Overview of Error Exponents
  6. Contributions and Paper Outline
  7. Small Mismatch: A Class of Compound Channels
  8. Achievable Rates
  9. Example: Symmetric $\widehat{W}$ and Equiprobable $Q_{\!X\!}$
  10. Example: Modulo-additive $\widehat{W}$ and Equiprobable $Q_{\!X\!}$
  11. Example: Ternary-Input Ternary-Output $\widehat{W}$
  12. Error Exponents
  13. Example: Symmetric $\widehat{W}$ and Equiprobable $Q_{\!X\!}$
  14. Example: Modulo-Additive $\widehat{W}$ and Equiprobable $Q_{\!X\!}$
  15. Example: Ternary-Input Ternary-Output $\widehat{W}$
  16. ...and 23 more sections

Key Result

Theorem 1

Consider a class of DMCs $W(y|x)$, a mismatched decoder based on the channel estimate $\widehat{W}(y|x)$ and a fixed input distribution $Q_{\!X\!}(x)$ satisfying eqn:WorstCaseRatesConstraint. Then, for a sufficiently small $r \ge 0$, the worst-case LM and GMI rates can be expressed as where

Figures (4)

  • Figure 1: Information rates (in nats per channel use) computed for $Q_{\!X\!}$ in \ref{['eqn:TernaryInputTernaryOutputQx']} and estimated channel $\widehat{W}$ in \ref{['eqn:TernaryInputTernaryOutputChannel']}.
  • Figure 2: Matched and mismatched Gallager functions computed for $\rho=0.7$, input distribution $Q_{\!X\!}$ in \ref{['eqn:TernaryInputTernaryOutputQx2']} and estimated channel $\widehat{W}$ in \ref{['eqn:TernaryInputTernaryOutputChannelExponents']}.
  • Figure 3: Worst-case GMI and cost-constraint rates versus ball radius $r$ for Gaussian codewords $Q_{\!X\!}(x) = \mathcal{N}(x;1)$ and the nearest neighbor decoder $\widehat{W}(y|x)=\mathcal{N}(y{-}x;1)$. These computations assume that $o(\sqrt{r})=0$. The optimal values from \ref{['eqn:Cont:minIgauss']} are depicted using cross markers.
  • Figure 4: Worst-case channel (in log scale) in the relative entropy ($W^\ast$ in \ref{['eqn:Cont:TrueWCadd']}) and chi-squared ($\tilde{W}^\ast$ in \ref{['eqn:Cont:AppWCadd']}) balls. Computations assume Gaussian codewords $Q_{\!X\!}(x) = \mathcal{N}(x;1)$ and the nearest neighbor decoder $\widehat{W}(y|x)=\mathcal{N}(y{-}x;1)$. The worst-case GMI rates for $W^\ast$ are shown in Fig. \ref{['fig:cont']}.

Theorems & Definitions (12)

  • Definition 1
  • Theorem 1
  • proof
  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Corollary 2.1
  • proof
  • Definition 2
  • ...and 2 more