Bethe Free Energy Approach to LDPC Decoding on Memory Channels

TL;DR

This paper introduces a novel belief propagation framework based on Bethe free energy minimization for joint detection and decoding of LDPC codes on partial-response channels, improving error performance.

Contribution

It derives explicit BP equations for joint detection and decoding in PR channels using a statistical mechanics approach, a first in the field.

Findings

BP equations are explicit and optimal for certain polynomial channels

Proposed algorithm outperforms turbo equalization in simulations

Provides a new inference method for combined detection and decoding

Abstract

We address the problem of the joint sequence detection in partial-response (PR) channels and decoding of low-density parity-check (LDPC) codes. We model the PR channel and the LDPC code as a combined inference problem. We present for the first time the derivation of the belief propagation (BP) equations that allow the simultaneous detection and decoding of a LDPC codeword in a PR channel. To accomplish this we follow an approach from statistical mechanics, in which the Bethe free energy is minimized with respect to the beliefs on the nodes of the PR-LDPC graph. The equations obtained are explicit and are optimal for decoding LDPC codes on PR channels with polynomial $h(D) = 1 - a D^n$ (a real, n positive integer) in the sense that they provide the exact inference of the marginal probabilities on the nodes in a graph free of loops. A simple algorithmic solution to the set of BP equations is proposed and evaluated using numerical simulations, yielding bit-error rate performances that surpass those of turbo equalization.