Spatially-coupled Low-density Parity-check Codes Research Articles

Window code parameters of spatially-coupled LDPC codes

In this paper, we define a window code to be the portion of a Spatially-coupled low-density parity check (SC-LDPC) code seen by a single iteration of a windowed decoder. We consider the design of SC-LDPC codes for windowed decoding via optimization of the window code. In particular, because iterative decoding is optimal on codes with cycle-free graph representations, we ask fundamental questions about the construction and parameters of cycle-free window codes. We show that it is possible to have an SC-LDPC code with cycles and with cycle-free window codes. We consider the relationship between the distance of the window code and the distance of the SC-LDPC code. Further, we show that SC-LDPC codes with MDS window codes exist, and all such codes are asymptotically bad. This work gives insight into the tradeoffs between window code parameters and performance of the SC-LDPC code.

Effective bi-directional overlapped sliding window decoding of SC-LDPC codes

In this paper, a bi-directional sliding window decoder is proposed for spatially coupled low-density parity-check (SC-LDPC) codes, which improves the decoding complexity and performance compared to the conventional sliding window decoding (SWD) by sharing messages at the overlapped part of forward and backward decoding windows. Moreover, by using proper scaling factors that determine the weight of each message at the overlapped part of two sliding windows, good local decoding effects can be efficiently spread out to both ends of SC-LDPC code during decoding process. Such effective message updates of the proposed bi-directional overlapped sliding window decoding (BO-SWD) improve error floor performance compared to the conventional SWD. The validity of BO-SWD is verified by simulation with various SC-LDPC ensembles.

Finite-Length Scaling of SC-LDPC Codes With a Limited Number of Decoding Iterations

We propose four finite-length scaling laws to predict the frame error rate (FER) performance in the waterfall region of spatially-coupled low-density parity-check code ensembles under full belief propagation (BP) decoding with a limit on the number of decoding iterations and a scaling law for sliding window decoding, also with limited iterations. The laws for full BP decoding provide a choice between accuracy and computational complexity; a good balance between them is achieved by the law that models the number of decoded bits after a certain number of BP iterations by a time-integrated Ornstein-Uhlenbeck process. This framework is developed further to model sliding window decoding as a race between the integrated Ornstein-Uhlenbeck process and an absorbing barrier that corresponds to the left boundary of the sliding window. The proposed scaling laws yield accurate FER predictions for the semi-structured code ensembles proposed by Olmos and Urbanke.

Finite-Length Analysis for Spatially Coupled LDPC Codes Based on Base Matrix.

Spatially coupled low density parity check (SC-LDPC) are prominent candidates for future communication standards due to their "threshold saturation" properties. To evaluate the finite-length performance of SC-LDPC codes, a general and efficient finite-length analysis from the perspective of the base matrix is proposed. We analyze the evolution of the residual graphs resulting at each iteration during the decoding process based on the base matrix and then derive the expression for the error probability. To verify the effectiveness of the proposed finite-length analysis, we consider the SC-LDPC code ensembles constructed by parallelly connecting multiple chains (PC-MSC-LDPC). The analysis results show that the predicted error probabilities obtained by using the derived expression for the error probability match the simulated error probabilities. The proposed finite-length analysis provides a useful engineering tool for practical SC-LDPC code design and for analyzing the effects of the code parameters on the performances.

Free-Ride Coding for Constructions of Coupled LDPC Codes

Free-ride coding, as an approach that admits transmission of a few extra bits over a low-density parity-check (LDPC) coded link without bandwidth expansion, is applied in this paper to construct coupled LDPC codes. Firstly, we present a syndrome channel model and derive the lower and upper bounds on its capacity (referred to as accessible capacity), indicating the feasibility of the reliable transmission of extra bits. Secondly, we present the performance evaluation on both the word error rate (WER) and the bit error rate (BER) for the free-ride codes with simple lower and upper bounds. Then we propose three applications of free-ride coding to construct coupled LDPC codes, including implicit globally-coupled LDPC (GC-LDPC) codes, partial product-LDPC codes, and terminated spatially-coupled LDPC (SC-LDPC) codes, all of which have the figure of merits that they share the same code rates with the basic component LDPC codes. Simulation results show that: 1) the proposed GC-LDPC codes can outperform the component LDPC codes, yielding a coding gain of up to 0.8 dB; 2) the proposed product codes with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$(3,6)$ </tex-math></inline-formula> -regular LDPC component codes of length 1024 can lower the WER from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{-2}$ </tex-math></inline-formula> down to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$10^{-6}$ </tex-math></inline-formula> at the SNR around 2 dB; 3) the proposed terminated SC-LDPC codes can perform as well as the conventional terminated SC-LDPC codes but without any rate loss.

Windowed-based synchronization error-correction for spatially coupled codes

This letter proposes a windowed-based error correction scheme for spatially coupled low-density parity-check (SC-LDPC) codes over channels with insertion and deletion errors. We first introduce the windowed detection and decoding scheme and then examine its asymptotic performance with several types of SC-LDPC codes. Subsequently, we demonstrate that a significant performance gain with lower latency and delay can be obtained using finite-length SC-LDPC codes.

Beyond 100 Gbit/s Pipeline Decoders for Spatially Coupled LDPC Codes

Low-density parity-check (LDPC) codes are a well-established class of forward error correction codes that provide excellent error correction performance for large code block sizes. However, for throughputs toward 1 Tbit/s, as expected for B5G systems, state-of-the-art LDPC soft decoders are restricted to short code block sizes of several hundred to thousands of bits due to routing congestion challenges, limiting the overall communications performance of the transmission system. Spatially coupled LDPC (SC-LDPC) codes and respective sliding window decoding methods show the potential to overcome these block size restrictions. However, in contrast to conventional LDPC codes little literature exists on the efficient hardware implementation of respective high-throughput decoders. In this work, we present the first in-depth investigation on the implementation of SC-LDPC decoders for throughputs beyond 100 Gbit/s. For an N=51328, R=0.8 terminated SC-LDPC code with sub-block size c=640 and coupling width m_s=1, we explore various design trade-offs, including row- and column-wise decoding, non-overlapping and overlapping window scheduling, and processor pipelining. To the best of our knowledge, this is the first description of a column-wise SC-LDPC decoding architecture in the literature. We complement the algorithmic investigation with the virtual silicon implementation of all presented decoders in a 22nm FD-SOI technology.

Advanced Digital Signal Processing and Variable-Rate Coding for Unrepeatered Optical Transmission

In this paper, we evaluate the selective combination of algorithms for fiber nonlinearity compensation, transceiver impairments mitigation and variable-rate coding in unrepeatered optical transmission. A post-emphasis filter and a two-stage 4×4 multiple-input and multiple-output equalizer compensate the impact of bandwidth limitations and in-phase and quadrature skew. Nonlinear compensation is accomplished by a fast-converging adaptive digital back-propagation algorithm and maximum likelihood sequence estimation. Forward error correction is provided by a variable-rate spatially-coupled low-density parity-check code. The performance and complexity of the proposed digital subsystems are experimentally evaluated by the unrepeatered wavelength division multiplexing transmission of 17×200-Gb/s (DP-16QAM 32-GBd) channels over 350-km of large effective area and low-loss single-mode fibers. The experimental results for different combinations of algorithms elucidate the trade-off of complexity and performance in unrepeatered optical transmission.

Optimization of SC-LDPC Codes for Window Decoding With Target Window Sizes

In this paper, we propose an optimization method for protograph-based spatially coupled low-density parity-check (SC-LDPC) codes under window decoding (WD). Previous works on constructing SC-LDPC codes for WD typically focused on optimizing asymptotic performance metrics such as the WD threshold. However, in this paper, it is observed that the WD threshold is not an appropriate metric to sufficiently explain the finite-length behavior of SC-LDPC codes under WD. Thus, we propose a new performance metric, called the window mean parameter, based on a scaling analysis to capture the WD performance more accurately and formulate a code optimization algorithm that optimizes the proposed performance metric. Since the proposed metric depends on the window size, the optimization algorithm can provide a code family of SC-LDPC codes optimized for various target window sizes. Simulation results confirm that the improvement in the proposed metric leads to a finite-length performance improvement, resulting in one to two orders of the frame error rate gain over the conventional SC-LDPC codes for a wide range of window sizes. Furthermore, we investigate structural characteristics of the proposed codes to provide a supplementary explanation for the performance improvement, which also promotes a better understanding of SC-LDPC codes for WD.

Concatenated Spatially Coupled LDPC Codes With Sliding Window Decoding for Joint Source-Channel Coding

In this paper, a method for joint source-channel coding (JSCC) based on concatenated spatially coupled low-density parity-check (SC-LDPC) codes is investigated. A construction consisting of two SC-LDPC codes is proposed: one for source coding and the other for channel coding, with a joint belief propagation-based decoder. Also, a novel windowed decoding (WD) scheme is presented with significantly reduced latency and complexity requirements. The asymptotic behavior for various graph node degrees is analyzed using a protograph-based Extrinsic Information Transfer (EXIT) chart analysis for both LDPC block codes with block decoding and for SC-LDPC codes with the WD scheme, showing robust performance for concatenated SC-LDPC codes. Simulation results show a notable performance improvement compared to existing state-of-the-art JSCC schemes based on LDPC codes with comparable latency and complexity constraints.

On the Global and Local Decoding of q-Ary SC-LDPC Codes With Moderate Coupling Length

This letter discusses a special type of q-ary spatially coupled low-density parity-check (SC-LDPC) codes with the potential for local decoding (LD). We demonstrate that q-ary SC-LDPCL (suffix “L” stands for locality) code ensembles have better decoding thresholds than the corresponding binary ensembles. We also develop local window decoding (LWD) to improve local decoding performance. The q-ary codes are constructed from low-order finite fields that guarantees relatively low complexity. The provided examples show that q-ary SC-LDPCL codes outperform the corresponding binary codes in both global and local decoding.

On Doped SC-LDPC Codes for Streaming

In streaming applications, doping improves the performance of spatially-coupled low-density parity-check (SC-LDPC) codes by creating reduced-degree check nodes in the coupled chain. We formulate a scaling law to predict the bit and block error rate of periodically-doped semi-infinite SC-LDPC code ensembles streamed over the binary erasure channel under sliding window decoding for a given finite component block length. The scaling law assumes that with some probability doping is equivalent to full termination and triggers two decoding waves; otherwise, decoding performs as if the coupled chain had not been doped at all. We approximate that probability and use the derived scaling laws to predict the error rates of SC-LDPC code ensembles in the presence of doping. The proposed scaling law provides accurate error rate predictions. We further use it to show that in streaming applications periodic doping can yield higher rates than periodic full termination for the same error-correcting performance.

Girth Analysis and Design of Periodically Time-Varying SC-LDPC Codes

Time-varying spatially coupled low-density parity-check (SC-LDPC) codes with very large period are characterized by significantly better error rate performance and girth properties than their time-invariant counterparts, but the number of parameters they require to be described is usually very large and unpractical. Time-invariant SC-LDPC codes, which can be seen as periodically time-varying codes with unitary period, are represented through a small number of parameters and designed exploiting few degrees of freedom, but their error rate performance and girth properties are sub-optimal. In this paper, we show that the limits of time-invariant SC-LDPC codes can be overcome by transforming them into time-varying SC-LDPC codes with very small period. In particular, we show that periodically time-varying SC-LDPC codes with small period may exhibit significantly better girth properties than the corresponding time-invariant codes by exploiting a larger number of degrees of freedom in the code design, which however scale at most linearly with the product of the code period and the size of the considered base matrix.

Optimizing Code Parameters of Finite-Length SC-LDPC Codes Using the Scaling Law

In this paper, we optimize code parameters of finite-length spatially coupled low-density parity-check (SC-LDPC) codes, represented by a set of code parameters $(l,r,w,L,M)$ . Although the finite-length scaling behavior of SC-LDPC codes was studied in the existing literature, the previous works impose a constraint such that the coupling width $w$ is equal to the variable node degree $l$ and they do not focus on optimizing the code parameters for given code and decoder specifications such as the code rate, frame size, and decoding complexity. In order to optimize the code parameters with the target specifications, we first extend the scaling law of SC-LDPC codes without the constraint $w=l$ . Using the scaling law formulated with a new variable $w$ , we show that the coupling width $w$ directly affects the slope of the performance curve and performance comparisons are given to investigate trade-offs inherent in the code parameters. It is shown that there are trade-offs for the code parameters in the perspective of the asymptotic performance limit, code rate, and scaling behaviors. In addition, the scaling law allows us to find the optimal code parameter set showing the best finite-length performance. Interestingly, the optimal code parameter set $(l,r,w)$ varies depending on the coupling length $L$ and uncoupled code length $M$ that determine the code and decoder specifications, which means there is no specific code parameter set prevailing over different kinds of applications. Finally, we illustrate this result using the investigated trade-offs on the code parameters, which gives us useful insight on how to choose the code parameters.

一种基于有限域的二进制和多进制空间耦合LDPC码构造方法

一种基于有限域的二进制和多进制空间耦合LDPC码构造方法

Finite-Length Scaling of Spatially Coupled LDPC Codes Under Window Decoding Over the BEC

We analyze the finite-length performance of spatially coupled low-density parity-check (SC-LDPC) codes under window decoding over the binary erasure channel. In particular, we propose a refinement of the scaling law by Olmos and Urbanke for the frame error rate (FER) of terminated SC-LDPC ensembles under full belief propagation (BP) decoding. The refined scaling law models the decoding process as two independent Ornstein-Uhlenbeck processes, in correspondence to the two decoding waves that propagate toward the center of the coupled chain for terminated SC-LDPC codes. We then extend the proposed scaling law to predict the performance of (terminated) SC-LDPC code ensembles under the more practical sliding window decoding. Finally, we extend this framework to predict the bit error rate (BER) and block error rate (BLER) of SC-LDPC code ensembles. The proposed scaling law yields very accurate predictions of the FER, BLER, and BER for both full BP and window decoding.

Asymptotic Behavior of Spatial Coupling LDPC Coding for Compute-and-Forward Two-Way Relaying

Compute-and-forward (CAF) relaying is an effective way to increase bandwidth efficiency of wireless two-way relay channels. Design of error-correcting codes and their decoding algorithms suitable for CAF relaying schemes remains an important issue to be studied. In this paper, we will analyze an asymptotic behavior of LDPC codes over two-way relay channels based on density evolution (DE). Because of the asymmetric characteristics of the channel, we use the population dynamics DE combined with DE formulas for asymmetric channels to obtain the belief propagation (BP) thresholds. Additionally, we also evaluate the asymptotic performance of spatially coupled LDPC codes for two-way relay channels. The results indicate that the spatially coupled codes yield improvements in the BP threshold compared with corresponding uncoupled codes for two-way relay channels.

Rate-Loss Mitigation of SC-LDPC Codes Without Performance Degradation

In research on spatially-coupled low-density parity-check (SC-LDPC) codes, rate-loss of SC-LDPC codes is one of the main issues to be addressed. One way to mitigate the rate-loss is to attach additional variable nodes with an irregular degree distribution, where the degree distribution is optimized with a constraint that the belief propagation (BP) threshold should not be degraded by attaching variable nodes. However, it is observed that the degree distribution obtained with the BP threshold constraint induces degradation of the finite-length performance. In order to address the problem, we propose new optimization methods to attach additional variable nodes while minimizing performance degradation. The proposed optimization methods are based on several design techniques including the scaling law, local threshold, expected graph evolution, differential evolution algorithms, the use of a protograph structure, and puncturing codewords. Using the optimized structure for additional variable nodes, the rate-loss of SC-LDPC codes can be reduced by more than 53% without sacrificing the finite-length performance. It is also shown that the rate-loss mitigation can be translated into a performance improvement if the proposed and the conventional SC-LDPC codes are compared at the same code rate.

Code Design Based on Connecting Spatially Coupled Graph Chains

A novel code construction based on spatially coupled low-density parity-check (SC-LDPC) codes is presented. The proposed code ensembles are comprised of several protograph-based chains characterizing individual SC-LDPC codes. We demonstrate that the code ensembles obtained by connecting appropriately chosen individual SC-LDPC code chains at specific points have improved iterative decoding thresholds. In addition, the connected chain ensembles have a smaller decoding complexity required to achieve a specific bit error probability compared to the individual code chains. Moreover, we demonstrate that, like the individual component chains, the proposed constructions have a typical minimum distance that grows linearly with block length. Finally, we show that the improved asymptotic properties of the connected chain ensembles also translate into improved finite length performance.

Design of Irregular SC-LDPC Codes With Non-Uniform Degree Distributions by Linear Programming

In this paper, we propose new design algorithms of irregular spatially-coupled low-density parity-check (SC-LDPC) codes with non-uniform degree distributions using linear programming (LP). In general, irregular SC-LDPC codes with non-uniform degree distributions are difficult to design with low complexity because their density evolution equations are multi-dimensional. To overcome this problem, proposed design algorithms are based on three main ideas: a local design of degree distributions, pre-computation of the input/output message relationship, and selection of a proper objective function. These ideas make it possible to design degree distributions of irregular SC-LDPC codes by solving low-complexity LP problems over the binary erasure channel (BEC). It is shown that the proposed irregular SC-LDPC codes designed by the proposed algorithms are superior to regular SC-LDPC codes in terms of both asymptotic and finite-length performances over the BEC. We also confirm that the proposed irregular SC-LDPC code achieves better performance compared with an optimized irregular block LDPC code in the same blocklength, which implies that the proposed design algorithms also provide a new way to construct capacity-approaching block LDPC codes.