A Simple Proof of Threshold Saturation for Coupled Scalar Recursions

TL;DR

This paper provides a straightforward proof of the threshold saturation phenomenon in coupled scalar recursions, explaining how spatially-coupled codes achieve near-capacity performance on various channels.

Contribution

It introduces a simple, broad-application proof of threshold saturation using potential functions, applicable to multiple LDPC code scenarios.

Findings

Proof applies to irregular LDPC codes on BEC

Validates for generalized LDPC codes and intersymbol-interference channels

Simplifies understanding of threshold saturation phenomenon

Abstract

Low-density parity-check (LDPC) convolutional codes (or spatially-coupled codes) have been shown to approach capacity on the binary erasure channel (BEC) and binary-input memoryless symmetric channels. The mechanism behind this spectacular performance is the threshold saturation phenomenon, which is characterized by the belief-propagation threshold of the spatially-coupled ensemble increasing to an intrinsic noise threshold defined by the uncoupled system. In this paper, we present a simple proof of threshold saturation that applies to a broad class of coupled scalar recursions. The conditions of the theorem are verified for the density-evolution (DE) equations of irregular LDPC codes on the BEC, a class of generalized LDPC codes, and the joint iterative decoding of LDPC codes on intersymbol-interference channels with erasure noise. Our approach is based on potential functions and was motivated mainly by the ideas of Takeuchi et al. The resulting proof is surprisingly simple when compared to previous methods.