Bethe Free Energy Approach to LDPC Decoding on Memory Channels

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.