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

TL;DR

This paper analyzes the information loss caused by the max-log approximation in BICM systems with M-PAM, proving it is generally lossy except for specific symmetric 4-PAM cases, and explores how certain operations affect mutual information and GMI.

Contribution

It provides a rigorous proof of when the max-log approximation causes information loss in BICM systems and clarifies the conditions under which GMI remains achievable.

Findings

Max-log approximation causes information loss for all M-PAM except symmetric 4-PAM with Gray coding.

The harmonized GMI is achievable without decoder modifications.

Bit-level symmetrization and mixing preserve GMI but may not preserve MI.

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.