A Simple Proof of Maxwell Saturation for Coupled Scalar Recursions

Low-density parity-check (LDPC) convolutional codes (or spatially-coupled codes) were recently shown to approach capacity on the binary erasure channel (BEC) and binary-input memoryless symmetric channels. The mechanism behind this spectacular performance is now called threshold saturation via spatial coupling. This new phenomenon 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 wide class of coupled scalar recursions. Our approach is based on constructing potential functions for both the coupled and uncoupled recursions. Our results actually show that the fixed point of the coupled recursion is essentially determined by the minimum of the uncoupled potential function and we refer to this phenomenon as Maxwell saturation. A variety of examples are considered including the density-evolution equations for: irregular LDPC codes on the BEC, irregular low-density generator matrix codes on the BEC, a class of generalized LDPC codes with BCH component codes, the joint iterative decoding of LDPC codes on intersymbol-interference channels with erasure noise, and the compressed sensing of random vectors with i.i.d. components.Comment: This article is an extended journal version of arXiv:1204.5703 and has now been accepted to the IEEE Transactions on Information Theory. This version adds additional explanation for some details and also corrects a number of small typo

Threshold Saturation for Nonbinary SC-LDPC Codes on the Binary Erasure Channel

We analyze the asymptotic performance of nonbinary spatially-coupled low-density parity-check (SC-LDPC) code ensembles defined over the general linear group on the binary erasure channel. In particular, we prove threshold saturation of belief propagation decoding to the so called potential threshold, using the proof technique based on potential functions introduced by Yedla \textit{et al.}, assuming that the potential function exists. We rewrite the density evolution of nonbinary SC-LDPC codes in an equivalent vector recursion form which is suited for the use of the potential function. We then discuss the existence of the potential function for the general case of vector recursions defined by multivariate polynomials, and give a method to construct it. We define a potential function in a slightly more general form than one by Yedla \textit{et al.}, in order to make the technique based on potential functions applicable to the case of nonbinary LDPC codes. We show that the potential function exists if a solution to a carefully designed system of linear equations exists. Furthermore, we show numerically the existence of a solution to the system of linear equations for a large number of nonbinary LDPC code ensembles, which allows us to define their potential function and thus prove threshold saturation.Comment: To appear in IT Transaction

Analysis of Spatially-Coupled Counter Braids

A counter braid (CB) is a novel counter architecture introduced by Lu et al. in 2007 for per-flow measurements on high-speed links. CBs achieve an asymptotic compression rate (under optimal decoding) that matches the entropy lower bound of the flow size distribution. Spatially-coupled CBs (SC-CBs) have recently been proposed. In this work, we further analyze single-layer CBs and SC-CBs using an equivalent bipartite graph representation of CBs. On this equivalent representation, we show that the potential and area thresholds are equal. We also show that the area under the extended belief propagation (BP) extrinsic information transfer curve (defined for the equivalent graph), computed for the expected residual CB graph when a peeling decoder equivalent to the BP decoder stops, is equal to zero precisely at the area threshold. This, combined with simulations and an asymptotic analysis of the Maxwell decoder, leads to the conjecture that the area threshold is in fact equal to the Maxwell decoding threshold and hence a lower bound on the maximum a posteriori (MAP) decoding threshold. Finally, we present some numerical results and give some insight into the apparent gap of the BP decoding threshold of SC-CBs to the conjectured lower bound on the MAP decoding threshold.Comment: To appear in the IEEE Information Theory Workshop, Jeju Island, Korea, October 201

Proving Threshold Saturation for Nonbinary SC-LDPC Codes on the Binary Erasure Channel

We analyze nonbinary spatially-coupled low-density parity-check (SC-LDPC) codes built on the general linear group for transmission over the binary erasure channel. We prove threshold saturation of the belief propagation decoding to the potential threshold, by generalizing the proof technique based on potential functions recently introduced by Yedla et al.. The existence of the potential function is also discussed for a vector sparse system in the general case, and some existence conditions are developed. We finally give density evolution and simulation results for several nonbinary SC-LDPC code ensembles.Comment: in Proc. 2014 XXXIth URSI General Assembly and Scientific Symposium, URSI GASS, Beijing, China, August 16-23, 2014. Invited pape

The Velocity of the Propagating Wave for General Coupled Scalar Systems

We consider spatially coupled systems governed by a set of scalar density evolution equations. Such equations track the behavior of message-passing algorithms used, for example, in coding, sparse sensing, or constraint-satisfaction problems. Assuming that the "profile" describing the average state of the algorithm exhibits a solitonic wave-like behavior after initial transient iterations, we derive a formula for the propagation velocity of the wave. We illustrate the formula with two applications, namely Generalized LDPC codes and compressive sensing.Comment: 5 pages, 5 figures, submitted to the Information Theory Workshop (ITW) 2016 in Cambridge, U