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On the Information Loss of the Max-Log Approximation in BICM Systems

Mikhail Ivanov, Christian Häger, Fredrik Brännström, Alexandre Graell i Amat, Alex Alvarado, Erik Agrell

TL;DR

This work analyzes the information rate implications of using max-log L-values in BICM with M-PAM constellations. It proves that max-log L-values are information-lossless only for two specific one-dimensional patterns with symmetric constellations, and otherwise incur loss; it also shows that the harmonized GMI is achievable with the standard BICM decoder without explicit L-value correction. The authors examine bit-level and full BICM formulations, deriving inequalities that relate MI and GMI under exact and max-log L-values, and demonstrate that bit-level symmetrization and channel mixing preserve GMI while potentially reducing MI. Numerical results corroborate the theoretical insights, showing negligible practical gaps at moderate-to-high rates, and provide guidance on when max-log L-values are acceptable in practice. The findings offer a rigorous understanding of the trade-offs in using max-log approximations for BICM systems and clarify the role of processing steps in preserving achievable rates.

Abstract

We present a comprehensive study of the information rate loss of the max-log approximation for $M$-ary pulse-amplitude modulation (PAM) in a bit-interleaved coded modulation (BICM) system. It is widely assumed that the calculation of L-values using the max-log approximation leads to an information loss. We prove that this assumption is correct for all $M$-PAM constellations and labelings with the exception of a symmetric 4-PAM constellation labeled with a Gray code. We also show that for max-log L-values, the BICM generalized mutual information (GMI), which is an achievable rate for a standard BICM decoder, is too pessimistic. In particular, it is proved that the so-called "harmonized" GMI, which can be seen as the sum of bit-level GMIs, is achievable without any modifications to the decoder. We then study how bit-level channel symmetrization and mixing affect the mutual information (MI) and the GMI for max-log L-values. Our results show that these operations, which are often used when analyzing BICM systems, preserve the GMI. However, this is not necessarily the case when the MI is considered. Necessary and sufficient conditions under which these operations preserve the MI are provided.

On the Information Loss of the Max-Log Approximation in BICM Systems

TL;DR

This work analyzes the information rate implications of using max-log L-values in BICM with M-PAM constellations. It proves that max-log L-values are information-lossless only for two specific one-dimensional patterns with symmetric constellations, and otherwise incur loss; it also shows that the harmonized GMI is achievable with the standard BICM decoder without explicit L-value correction. The authors examine bit-level and full BICM formulations, deriving inequalities that relate MI and GMI under exact and max-log L-values, and demonstrate that bit-level symmetrization and channel mixing preserve GMI while potentially reducing MI. Numerical results corroborate the theoretical insights, showing negligible practical gaps at moderate-to-high rates, and provide guidance on when max-log L-values are acceptable in practice. The findings offer a rigorous understanding of the trade-offs in using max-log approximations for BICM systems and clarify the role of processing steps in preserving achievable rates.

Abstract

We present a comprehensive study of the information rate loss of the max-log approximation for -ary pulse-amplitude modulation (PAM) in a bit-interleaved coded modulation (BICM) system. It is widely assumed that the calculation of L-values using the max-log approximation leads to an information loss. We prove that this assumption is correct for all -PAM constellations and labelings with the exception of a symmetric 4-PAM constellation labeled with a Gray code. We also show that for max-log L-values, the BICM generalized mutual information (GMI), which is an achievable rate for a standard BICM decoder, is too pessimistic. In particular, it is proved that the so-called "harmonized" GMI, which can be seen as the sum of bit-level GMIs, is achievable without any modifications to the decoder. We then study how bit-level channel symmetrization and mixing affect the mutual information (MI) and the GMI for max-log L-values. Our results show that these operations, which are often used when analyzing BICM systems, preserve the GMI. However, this is not necessarily the case when the MI is considered. Necessary and sufficient conditions under which these operations preserve the MI are provided.

Paper Structure

This paper contains 32 sections, 10 theorems, 78 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Notation
  3. System Model
  4. Modulator
  5. AWGN Channel
  6. Demappers and L-values
  7. Coding Scheme
  8. Bit-Level Analysis
  9. Bit-Level Coding Scheme
  10. Achievable Rates
  11. Generalized Mutual Information
  12. Mutual Information
  13. L-values
  14. Exact L-values
  15. Max-Log L-values
  16. ...and 17 more sections

Key Result

Lemma 1

The exact L-value $l^{\mathsf{ex}}_j(y)$ is symmetric with respect to $y_0 \in \mathbb{R}$, i.e., $l^{\mathsf{ex}}_j(y_0+y) = l^{\mathsf{ex}}_j(y_0-y)$ for $y \in \mathbb{R}$, if and only if the constellation is symmetric around $y_0$ and the pattern corresponding to the $j$th bit position satisfi

Figures (12)

  • Figure 1: Block diagram of the analyzed system.
  • Figure 2: Examples of equally spaced PAM constellations with different binary labelings.
  • Figure 3: Normalized exact (solid) and max-log (dashed) L-values as functions of the observation $y$ assuming an equally spaced 4-PAM constellation labeled with the BRGC and NBC (see Fig. \ref{['fig:4-pam_brgc']}(a)--(b)).
  • Figure 4: Bit-level channel, illustrated for the first bit position.
  • Figure 5: The correction function $g(\cdot)$ (solid) and the exact L-value versus the max-log L-value (dashed) for the three non-equivalent patterns for an equally spaced 4-PAM constellation. The values on the x- and y-axes are normalized by $4\rho/5$.
  • ...and 7 more figures

Theorems & Definitions (24)

  • Lemma 1
  • proof
  • Remark 1
  • Remark 2
  • Lemma 2
  • proof
  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • ...and 14 more