A Simple Proof of Threshold Saturation for Coupled Vector Recursions

TL;DR

This paper provides a simple proof demonstrating that threshold saturation occurs in a broad class of coupled vector recursions, confirming capacity-achieving performance for various LDPC code systems.

Contribution

It extends the proof of threshold saturation from scalar to vector recursions, applicable to multiple coding scenarios including LDPC and protograph codes.

Findings

Threshold saturation is proven for coupled vector recursions.

The proof applies to joint decoding in Slepian-Wolf and multiple-access channels.

Results confirm capacity achievement for several LDPC code configurations.

Abstract

Convolutional low-density parity-check (LDPC) codes (or spatially-coupled codes) have now been shown to achieve capacity on binary-input memoryless symmetric channels. The principle behind this surprising result is the threshold-saturation phenomenon, which is defined by the belief-propagation threshold of the spatially-coupled ensemble saturating to a fundamental threshold defined by the uncoupled system. Previously, the authors demonstrated that potential functions can be used to provide a simple proof of threshold saturation for coupled scalar recursions. In this paper, we present a simple proof of threshold saturation that applies to a wide class of coupled vector recursions. The conditions of the theorem are verified for the density-evolution equations of: (i) joint decoding of irregular LDPC codes for a Slepian-Wolf problem with erasures, (ii) joint decoding of irregular LDPC codes on an erasure multiple-access channel, and (iii) general protograph codes on the BEC. This proves threshold saturation for these systems.