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Pre-Decoder Processing Functions for a DMC with Mismatched Decoding

Jonathan Solel, Anelia Somekh-Baruch

TL;DR

This work investigates how a pre-decoding processing function placed before a fixed mismatched decoder can enhance communication over a discrete memoryless channel. It develops symbolwise and vectorwise pre-processing frameworks, deriving single-letter lower bounds based on LM-rate expressions and establishing a separation principle for vectorwise schemes. The paper provides upper bounds on mismatch capacity with pre-processing and proves that a deterministic pre-processing function can maximize the LM rate in the symbolwise case, while identifying optimal-channel constructions for given metrics. Through a suite of examples, it demonstrates cases where pre-processing enables positive rates where none existed and cases where capacity can be fully recovered, highlighting practical benefits for receiver design and latency-conscious implementations.

Abstract

This paper analyzes the effect of adding a pre-decoder processing function to a receiver that contains a fixed mismatched decoder at the output of a discrete memoryless channel. We study properties of the symbolwise pre-processing function and show that it is a simple yet very powerful tool which enables to obtain reliable transmission at a positive rate for almost every metric. We present lower and upper bounds on the capacity of a channel with mismatched decoding and symbolwise(scalar-to-scalar) pre-processing, and show that the optimal pre-processing function for random coding is deterministic. We also characterize achievable error exponents. Finally, we prove that a separation principle holds for vectorwise(vector-to-vector) pre-processing functions and further, that deterministic functions maximize the reliably transmitted rate in this case.

Pre-Decoder Processing Functions for a DMC with Mismatched Decoding

TL;DR

This work investigates how a pre-decoding processing function placed before a fixed mismatched decoder can enhance communication over a discrete memoryless channel. It develops symbolwise and vectorwise pre-processing frameworks, deriving single-letter lower bounds based on LM-rate expressions and establishing a separation principle for vectorwise schemes. The paper provides upper bounds on mismatch capacity with pre-processing and proves that a deterministic pre-processing function can maximize the LM rate in the symbolwise case, while identifying optimal-channel constructions for given metrics. Through a suite of examples, it demonstrates cases where pre-processing enables positive rates where none existed and cases where capacity can be fully recovered, highlighting practical benefits for receiver design and latency-conscious implementations.

Abstract

This paper analyzes the effect of adding a pre-decoder processing function to a receiver that contains a fixed mismatched decoder at the output of a discrete memoryless channel. We study properties of the symbolwise pre-processing function and show that it is a simple yet very powerful tool which enables to obtain reliable transmission at a positive rate for almost every metric. We present lower and upper bounds on the capacity of a channel with mismatched decoding and symbolwise(scalar-to-scalar) pre-processing, and show that the optimal pre-processing function for random coding is deterministic. We also characterize achievable error exponents. Finally, we prove that a separation principle holds for vectorwise(vector-to-vector) pre-processing functions and further, that deterministic functions maximize the reliably transmitted rate in this case.

Paper Structure

This paper contains 21 sections, 8 theorems, 47 equations, 3 figures.

Table of Contents

  1. Introduction
  2. General Problem Formulation
  3. The relationship between the quantities $C_q^{pre}(W)$, $C_q(W)$, and $C(W)$
  4. Examples
  5. A Binary Channel
  6. A DMC with "the worst" metric
  7. Deterministic Quadratic Channel 1
  8. A Deterministic Quadratic Channel 2
  9. A Discrete Input Additive Gaussian Noise Channel
  10. Useless metrics
  11. Achievable Rates and Error Exponents
  12. Main Achievability Theorem
  13. Superposition Coding
  14. An Upper Bound on the Mismatch Capacity with Pre-Processing function
  15. An Optimal Pre-Processing Function which Maximizes the LM Rate
  16. ...and 6 more sections

Key Result

Theorem 1

There exists a channel $W$ from ${\cal X}$ to ${\cal Y}$ s.t $C_q^{pre}(W) >0$ if and only if $\exists x_1,x_2 \in {\cal X}, y_1,y_2 \in {\cal Y}$ s.t.

Figures (3)

  • Figure 1: Channel with a mismatched decoder and pre-processing function
  • Figure 2: A broadcast channel with mismatched decoders $q, \rho$ and a pre-processing function
  • Figure 3: This figure illustrates the construction for lemma 1

Theorems & Definitions (17)

  • Definition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Theorem 1
  • Lemma 1
  • proof
  • Theorem 2
  • Theorem 3
  • ...and 7 more