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 independent identically distributed components.

Recently, it has been observed that terminated low-density-parity-check (LDPC) convolutional codes (or spatially-coupled codes) appear to approach the capacity universally across the class of binary memoryless channels. This is facilitated by the “threshold saturation” effect whereby the belief-propagation (BP) threshold of the spatially-coupled ensemble is boosted to the maximum a-posteriori (MAP) threshold of the underlying constituent ensemble. In this paper, we consider spatially-coupled codes over intersymbol-interference (ISI) channels under joint iterative decoding where we empirically show that threshold saturation also occurs. This can be observed by first identifying the GEXIT curve that naturally obeys the general area theorem. From this curve, the corresponding MAP and the BP threshold estimates are then numerically obtained. Given the fact that regular LDPC codes can achieve the symmetric information rate (SIR) under MAP decoding, we conjecture that spatially-coupled codes with joint iterative decoding can universally approach the SIR of ISI channels.

A fundamental connection between the belief propagation (BP) and maximum a posteriori (MAP) decoding thresholds was derived by Méasson, Montanari, and Urbanke using the area theorem for extrinsic information transfer (EXIT) curves. This connection allows the MAP threshold, for the binary erasure channel, to be evaluated efficiently via an upper bound that can be shown to be tight in some cases. In this paper, a similar analysis is used to extend these results to several multiuser systems, namely a noisy Slepian-Wolf problem and a multiple-access channel with erasures. The simplicity of these channel models allows for rigorous analysis and enables the derivation of upper bounds on the MAP thresholds using EXIT area theorems. In some cases, one can also show these bounds are tight. One interesting application is that the MAP thresholds can be compared with the BP thresholds of spatially-coupled codes to verify threshold saturation for the corresponding systems.

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) admissible protograph codes on the BEC. This proves threshold saturation for these systems.

One popular approach to soft-decision decoding of Reed-Solomon (RS) codes is based on using multiple trials of a simple RS decoding algorithm in combination with erasing or flipping a set of symbols or bits in each trial. This paper presents a framework based on rate-distortion (RD) theory to analyze these multiple-decoding algorithms. By defining an appropriate distortion measure between an error pattern and an erasure pattern, the successful decoding condition, for a single errors-and-erasures decoding trial, becomes equivalent to distortion being less than a fixed threshold. Finding the best set of erasure patterns also turns into a covering problem that can be solved asymptotically by RD theory. Thus, the proposed approach can be used to understand the asymptotic performance-versus-complexity tradeoff of multiple errors-and-erasures decoding of RS codes. This initial result is also extended a few directions. The rate-distortion exponent (RDE) is computed to give more precise results for moderate blocklengths. Multiple trials of algebraic soft-decision (ASD) decoding are analyzed using this framework. Analytical and numerical computations of the RD and RDE functions are also presented. Finally, simulation results show that sets of erasure patterns designed using the proposed methods outperform other algorithms with the same number of decoding trials.

Algorithms based on multiple decoding attempts of Reed-Solomon (RS) codes have recently attracted new attention. Choosing decoding candidates based on rate-distortion theory, as proposed previously by the authors, currently provides the best performance-versus-complexity trade-off. In this paper, an analysis based on the rate-distortion exponent is used to directly minimize the exponential decay rate of the error probability. This enables rigorous bounds on the error probability for finite-length RS codes and leads to modest performance gains. As a byproduct, a numerical method is derived that computes the rate-distortion exponent for independent non-identical sources. Analytical results are given for errors/erasures decoding.