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A Simple Proof of Threshold Saturation for Coupled Scalar Recursions
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In this paper, a simple proof of threshold saturation that applies to a broad class of coupled scalar recursions is presented, which is based on potential functions and was motivated mainly by the ideas of Takeuchi et al.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.read more
Citations
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Spatially Coupled Ensembles Universally Achieve Capacity Under Belief Propagation
TL;DR: The key technical result is a proof that, under belief-propagation decoding, spatially coupled ensembles achieve essentially the area threshold of the underlying uncoupled ensemble.
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Spatially coupled ensembles universally achieve capacity under belief propagation
TL;DR: The key technical result is a proof that, under belief-propagation decoding, spatially coupled ensembles achieve essentially the area threshold of the underlying uncoupled ensemble.
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Status and Recent Advances on Forward Error Correction Technologies for Lightwave Systems
Andreas Leven,Laurent Schmalen +1 more
TL;DR: This paper gives a tutorial-style introduction of one class of commonly used codes, namely low-density parity-check (LDPC) codes, and discusses new developments such as convolutional LDPC codes and show how they can be employed as potential candidates for future optical communication systems.
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Threshold Saturation for Spatially-Coupled LDPC and LDGM Codes on BMS Channels
TL;DR: In this article, the authors used potential functions to prove threshold saturation for irregular LDPC and low-density generator-matrix (LDGM) codes on binary memoryless symmetric (BMS) channels, extending the simplified proof technique to BMS channels.
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Mutual Information and Optimality of Approximate Message-Passing in Random Linear Estimation
TL;DR: In this article, the authors consider the estimation of a signal from the knowledge of its noisy linear random Gaussian projections and show that approximate message-passing always reaches the minimal-mean-square error.
References
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Matrix Differential Calculus with Applications in Statistics and Econometrics
Jan R. Magnus,Heinz Neudecker +1 more
TL;DR: In this article, the authors discuss the properties of Vectors and Matrices, the Vec-Operator, the Moore-Penrose Inverse Miscellaneous Matrix Results, and the Linear Regression Model.
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TL;DR: This summary of the state-of-the-art in iterative coding makes this decision more straightforward, with emphasis on the underlying theory, techniques to analyse and design practical iterative codes systems.
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Time-varying periodic convolutional codes with low-density parity-check matrix
TL;DR: A class of convolutional codes defined by a low-density parity-check matrix and an iterative algorithm for decoding these codes is presented, showing that for the rate R=1/2 binary codes, the performance is substantially better than for ordinary convolutionian codes with the same decoding complexity per information bit.
Low-Density Parity-Check (LDPC) Codes Constructed from Protographs
TL;DR: This work introduces a new class of low-density parity-check codes constructed from a template called a protograph, which serves as a blueprint for constructing LDPC codes of arbitrary size whose performance can be predicted by analyzing the protograph.
Journal ArticleDOI
Threshold Saturation via Spatial Coupling: Why Convolutional LDPC Ensembles Perform So Well over the BEC
TL;DR: The fundamental mechanism that explains why “convolutional-like” or “spatially coupled” codes perform so well is described, and it is conjecture that for a large range of graphical systems a similar saturation of the “dynamical” threshold occurs once individual components are coupled sufficiently strongly.