A Simple Proof of Threshold Saturation for Coupled Vector Recursions

Arvind Yedla; Yung-Yih Jian; Phong S. Nguyen; Henry D. Pfister

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arXiv · 1208.4080

A Simple Proof of Threshold Saturation for Coupled Vector Recursions

Arvind Yedla·Yung-Yih Jian·Phong S. Nguyen·Henry D. Pfister

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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.

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Arvind Yedla, Yung-Yih Jian, Phong S. Nguyen, Henry D. Pfister. 2012-08-20. A Simple Proof of Threshold Saturation for Coupled Vector Recursions. https://arxiv.org/abs/1208.4080

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