Threshold Analysis of Non-Binary Spatially-Coupled LDPC Codes with Windowed Decoding

In this paper we study the iterative decoding threshold performance of non-binary spatially-coupled low-density parity-check (NB-SC-LDPC) code ensembles for both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (BIAWGNC), with particular emphasis on windowed decoding (WD). We consider both (2,4)-regular and (3,6)-regular NB-SC-LDPC code ensembles constructed using protographs and compute their thresholds using protograph versions of NB density evolution and NB extrinsic information transfer analysis. For these code ensembles, we show that WD of NB-SC-LDPC codes, which provides a significant decrease in latency and complexity compared to decoding across the entire parity-check matrix, results in a negligible decrease in the near-capacity performance for a sufficiently large window size W on both the BEC and the BIAWGNC. Also, we show that NB-SC-LDPC code ensembles exhibit gains in the WD threshold compared to the corresponding block code ensembles decoded across the entire parity-check matrix, and that the gains increase as the finite field size q increases. Moreover, from the viewpoint of decoding complexity, we see that (3,6)-regular NB-SC-LDPC codes are particularly attractive due to the fact that they achieve near-capacity thresholds even for small q and W.Comment: 6 pages, 8 figures; submitted to 2014 IEEE International Symposium on Information Theor

Challenges and Some New Directions in Channel Coding

Three areas of ongoing research in channel coding are surveyed, and recent developments are presented in each area: spatially coupled Low-Density Parity-Check (LDPC) codes, nonbinary LDPC codes, and polar coding.This is the author accepted manuscript. The final version is available from IEEE via http://dx.doi.org/10.1109/JCN.2015.00006

Challenges and some new directions in channel coding

Three areas of ongoing research in channel coding are surveyed, and recent developments are presented in each area: Spatially coupled low-density parity-check (LDPC) codes, nonbinary LDPC codes, and polar coding. © 2015 KICS

How to Achieve the Capacity of Asymmetric Channels

We survey coding techniques that enable reliable transmission at rates that approach the capacity of an arbitrary discrete memoryless channel. In particular, we take the point of view of modern coding theory and discuss how recent advances in coding for symmetric channels help provide more efficient solutions for the asymmetric case. We consider, in more detail, three basic coding paradigms. The first one is Gallager's scheme that consists of concatenating a linear code with a non-linear mapping so that the input distribution can be appropriately shaped. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Furthermore, we derive a scaling law between the gap to capacity, the cardinality of the input and output alphabets, and the required size of the mapper. The second one is an integrated scheme in which the code is used both for source coding, in order to create codewords distributed according to the capacity-achieving input distribution, and for channel coding, in order to provide error protection. Such a technique has been recently introduced by Honda and Yamamoto in the context of polar codes, and we show how to apply it also to the design of sparse graph codes. The third paradigm is based on an idea of B\"ocherer and Mathar, and separates the two tasks of source coding and channel coding by a chaining construction that binds together several codewords. We present conditions for the source code and the channel code, and we describe how to combine any source code with any channel code that fulfill those conditions, in order to provide capacity-achieving schemes for asymmetric channels. In particular, we show that polar codes, spatially coupled codes, and homophonic codes are suitable as basic building blocks of the proposed coding strategy.Comment: 32 pages, 4 figures, presented in part at Allerton'14 and published in IEEE Trans. Inform. Theor

Analysis and Design of Spatially-Coupled Codes with Application to Fiber-Optical Communications

The theme of this thesis is the analysis and design of error-correcting codes that are suitable for high-speed fiber-optical communication systems. In particular, we consider two code classes. The codes in the first class are protograph-based low-density parity-check (LDPC) codes which are decoded using iterative soft-decision decoding. The codes in the second class are generalized LDPC codes with degree-2 variable nodes—henceforth referred to as generalized product codes (GPCs)—which are decoded using iterative bounded-distance decoding (BDD). Within each class, our focus is primarily on spatially-coupled codes. Spatially-coupled codes possess a convolutional structure and are characterized by a wave-like decoding behavior caused by a termination boundary effect. The contributions of this thesis can then be categorized into two topics, as outlined below.First, we consider the design of systems operating at high spectral efficiency. In particular, we study the optimization of the mapping of the coded bits to the modulation bits for a polarization-multiplexed system that is based on the bit-interleaved coded modulation paradigm. As an example, for the (protograph-based) AR4JA code family, the transmission reach can be extended by roughly up to 8% by using an optimized bit mapper, without significantly increasing the system complexity. For terminated spatially-coupled codes with long spatial length, the bit mapper optimization only results in marginal performance improvements, suggesting that a sequential allocation is close to optimal. On the other hand, an optimized allocation can significantly improve the performance of tail-biting spatially-coupled codes which do not possess an inherent termination boundary. In this case, the unequal error protection offered by the modulation bits of a nonbinary signal constellation can be exploited to create an artificial termination boundary that induces a wave-like decoding for tail-biting spatially-coupled codes.As a second topic, we study deterministically constructed GPCs. GPCs are particularly suited for high-speed applications such as optical communications due to the significantly reduced decoding complexity of iterative BDD compared to iterative soft-decision decoding of LDPC codes. We propose a code construction for GPCs which is sufficiently general to recover several well-known classes of GPCs as special cases, e.g., irregular product codes (PCs), block-wise braided codes, and staircase codes. Assuming transmission over the binary erasure channel, it is shown that the asymptotic performance of the resulting codes can be analyzed by means of a recursive density evolution (DE) equation. The DE analysis is then applied to study three different classes of GPCs: spatially-coupled PCs, symmetric GPCs, and GPCs based on component code mixtures

Analysis and Design of Spatially-Coupled Codes with Application to Fiber-Optical Communications

The theme of this thesis is the analysis and design of error-correcting codes that are suitable for high-speed fiber-optical communication systems. In particular, we consider two code classes. The codes in the first class are protograph-based low-density parity-check (LDPC) codes which are decoded using iterative soft-decision decoding. The codes in the second class are generalized LDPC codes with degree-2 variable nodes—henceforth referred to as generalized product codes (GPCs)—which are decoded using iterative bounded-distance decoding (BDD). Within each class, our focus is primarily on spatially-coupled codes. Spatially-coupled codes possess a convolutional structure and are characterized by a wave-like decoding behavior caused by a termination boundary effect. The contributions of this thesis can then be categorized into two topics, as outlined below.First, we consider the design of systems operating at high spectral efficiency. In particular, we study the optimization of the mapping of the coded bits to the modulation bits for a polarization-multiplexed system that is based on the bit-interleaved coded modulation paradigm. As an example, for the (protograph-based) AR4JA code family, the transmission reach can be extended by roughly up to 8% by using an optimized bit mapper, without significantly increasing the system complexity. For terminated spatially-coupled codes with long spatial length, the bit mapper optimization only results in marginal performance improvements, suggesting that a sequential allocation is close to optimal. On the other hand, an optimized allocation can significantly improve the performance of tail-biting spatially-coupled codes which do not possess an inherent termination boundary. In this case, the unequal error protection offered by the modulation bits of a nonbinary signal constellation can be exploited to create an artificial termination boundary that induces a wave-like decoding for tail-biting spatially-coupled codes.As a second topic, we study deterministically constructed GPCs. GPCs are particularly suited for high-speed applications such as optical communications due to the significantly reduced decoding complexity of iterative BDD compared to iterative soft-decision decoding of LDPC codes. We propose a code construction for GPCs which is sufficiently general to recover several well-known classes of GPCs as special cases, e.g., irregular product codes (PCs), block-wise braided codes, and staircase codes. Assuming transmission over the binary erasure channel, it is shown that the asymptotic performance of the resulting codes can be analyzed by means of a recursive density evolution (DE) equation. The DE analysis is then applied to study three different classes of GPCs: spatially-coupled PCs, symmetric GPCs, and GPCs based on component code mixtures

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

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