Spatially Coupled Low-density Parity-check Research Articles

Circular Buffer Rate-Matched Polar Codes

A practical rate-matching system for constructing rate-compatible polar codes is proposed. The proposed polar code circular buffer rate-matching is suitable for transmissions on communication channels that support hybrid automatic repeat request (HARQ) communications, as well as for flexible resource-element rate-matching on single transmission channels. Our proposed circular buffer rate matching scheme also incorporates a bit-mapping scheme for transmission on bit-interleaved coded modulation (BICM) channels using higher order modulations. An interleaver is derived from a puncturing order obtained with a low complexity progressive puncturing search algorithm on a base code of short length, and has the flexibility to achieve any desired rate at the desired code length, through puncturing or repetition. The rate-matching scheme is implied by a two-stage polarization, for transmission at any desired code length, code rate, and modulation order, and is shown to achieve the symmetric capacity of BICM channels. Numerical results on AWGN and fast fading channels show that the rate-matched polar codes have a competitive performance when compared to the spatially-coupled quasi-cyclic LDPC codes or LTE turbo codes, while having similar rate-dematching storage and computational complexities.

Spatially “Mt. Fuji” Coupled LDPC Codes

A new type of spatially coupled low density parity check (SCLDPC) codes is introduced. These codes have exponentially increasing nodes as position goes to the middle of the chain. We call the codes spatially Mt. Fuji coupled LDPC (SFCLDPC) codes because the shape constructed by the number of nodes looks like Mt. Fuji. By this structure, the SFCLDPC codes have the rate loss O(α−L), while the usual SCLDPC codes have O(L−1). In addition, the decoding processes for the SFCLDPC codes are accelerated exponentially fast as the iteration progresses. An infinite length analysis for the SFCLDPC codes with the density evolution and a finite length performance comparison between the SFCLDPC codes and the SCLDPC codes with computer simulations are made.

SC‐LDPC code design for Gaussian multiple access channel

This paper presents the design and analysis of spatially coupled low-density parity-check (SC-LDPC) codes for Gaussian multiple access channel (GMAC). For two-user GMAC, we consider that each user encodes their messages by SC-LDPC codes while joint belief propagation decoding is considered at the receiver. We present low complexity density evolution analysis of SC-LDPC GMAC codes and show simple optimisation procedure. For the asymptotic case, we observe that SC-LDPC code-based GMAC code outperforms existing irregular LDPC code-based GMAC code.

Rate‐compatible spatially coupled LDPC code ensembles based on repeat‐accumulate extensions

A family of rate-compatible capacity-approaching codes is obtained by spatially coupling multiple identical copies of rate-compatible base codes. The base code consists of a given regular low-density parity-check (LDPC) code and a parameter-adjustable repeat-accumulate (RA) extension. The RA-extension repeatedly accumulates all of the variable nodes of the given LDPC code by q + 1 times to generate q + 1 blocks of accumulated nodes, in which all of the accumulated variable nodes in the first q blocks and the α-fractional accumulated variable nodes in the last block are transmitted. The spatially coupled versions of the base codes, called RA-extended spatially coupled LDPC (SC-LDPC) codes, achieve arbitrary rates by simply adjusting parameters q and α, and thus they are rate-compatible. The potential thresholds of the base code ensembles are calculated to predict the iterative decoding performance of the proposed RA-extended SC-LDPC codes. Numerical results and simulations show that the authors’ proposed rate-compatible codes are capacity-approaching over binary erasure channels.

Threshold Saturation for Nonbinary SC-LDPC Ensembles on Discrete-Input Memoryless Channels

In this letter, we investigate the threshold saturation effect of nonbinary spatially coupled low-density parity-check (SC-LDPC) ensembles defined over GF( $ {q}$ ) when the transmission takes place over discrete-input memoryless channels. To this end, we extend the duality rule for entropy established in the analysis of binary LDPC ensembles to the nonbinary case. This enables us to derive potential functions associated with log-likelihood ratio vector densities. Finally, we prove the existence of the threshold saturation effect following the approach introduced by Yedla et al. The analysis in this letter generalizes the existing work on nonbinary SC-LDPC ensembles and implies that the proof technique based on potential functions is also applicable to nonbinary sparse graphical codes over nonbinary discrete-input channels under belief propagation decoding.

Partially Repeated SC-LDPC Codes for Multiple-Access Channel

A simple partially repeated spatially coupled low-density parity-check (SC-LDPC) code is proposed for a multiple-access channel (MAC) with a large number of users. This code is constructed by selecting part of bits of an SC-LDPC code to be repeated. Since the decoding increment is only a log-likelihood ratio (LLR) combination, compared with the original SC-LDPC code, the decoding complexity scarcely increases. It is shown that this LLR combination is effective to overcome the multi-user interference. By optimizing the ratio of coded bits for repetition, the decoding threshold of the partially repeated SC-LDPC code can be close to the MAC capacity. Experimental results show that the partially repeated SC-LDPC codes have better performance than the graph-extended and parallelly connected SC-LDPC codes over the MAC.

SC-LDPC Code Design for Half-Duplex Relay Channels

This paper studies code design for the half-duplex relay channel when the transmissions take place over binary input additive white Gaussian noise (BIAWGN) channels. Using the decode-forward relay protocol, we design the relay code based on a spatially coupled low-density parity-check (SC-LDPC) code. We show a low complexity density evolution analysis for the proposed relay code. From the density evolution results, we observe that the proposed spatially coupled relay code achieves a capacity approaching performance. We also observe that the proposed code outperforms existing optimized LDPC relay codes. Through simulation results, we evaluate the finite-length performance of the proposed code.

Design of Spatially Coupled LDPC Codes Over GF $(q)$ for Windowed Decoding

In this paper we consider the generalization of binary spatially coupled low-density parity-check (SC-LDPC) codes to finite fields GF$(q)$, $q\geq 2$, and develop design rules for $q$-ary SC-LDPC code ensembles based on their iterative belief propagation (BP) decoding thresholds, with particular emphasis on low-latency windowed decoding (WD). We consider transmission over both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (BIAWGNC) and present results for a variety of $(J,K)$-regular SC-LDPC code ensembles constructed over GF$(q)$ using protographs. Thresholds are calculated using protograph versions of $q$-ary density evolution (for the BEC) and $q$-ary extrinsic information transfer analysis (for the BIAWGNC). We show that WD of $q$-ary SC-LDPC codes provides significant threshold gains compared to corresponding (uncoupled) $q$-ary LDPC block code (LDPC-BC) ensembles when the window size $W$ is large enough and that these gains increase as the finite field size $q=2^m$ increases. Moreover, we demonstrate that the new design rules provide WD thresholds that are close to capacity, even when both $m$ and $W$ are relatively small (thereby reducing decoding complexity and latency). The analysis further shows that, compared to standard flooding-schedule decoding, WD of $q$-ary SC-LDPC code ensembles results in significant reductions in both decoding complexity and decoding latency, and that these reductions increase as $m$ increases. For applications with a near-threshold performance requirement and a constraint on decoding latency, we show that using $q$-ary SC-LDPC code ensembles, with moderate $q>2$, instead of their binary counterparts results in reduced decoding complexity.

Euclidean Geometry-Based Spatially Coupled LDPC Codes for Storage

In this paper, we construct binary spatially coupled (SC) low-density parity-check (LDPC) codes based on Euclidean geometry (EG) LDPC codes for storage applications, where high error correction capability, extremely low uncorrectable bit error rate (UBER), and low decoding complexity are required. We propose a systematic way to construct the families of SC LDPC codes from $(m,2^{s})$ EG LDPC codes, which are termed EG-SC LDPC codes. In the construction method, we propose a 2-D edge-spreading process to construct the base matrix of EG-SC LDPC codes, which consists of matrix unwrapping and periodically time-varying of a protograph. A lower bound on the rank of the parity-check matrix of an EG-SC LDPC code is derived. We evaluate the error rate performance of the constructed EG-SC LDPC codes by using a weighted bit-flipping decoding algorithm for its low decoding complexity. Numerical results show that the UBER performance of the constructed EG-SC LDPC codes is superior to that of their EG LDPC code counterparts, and show no error floor compared with the constructed protograph SC LDPC codes and regular LDPC codes.

Continuous Transmission of Spatially Coupled LDPC Code Chains

We propose a novel encoding/transmission scheme called continuous chain (CC) transmission that is able to improve the finite-length performance of a system using spatially-coupled low-density parity-check (SC-LDPC) codes. In CC transmission, instead of transmitting a sequence of independent codewords from a terminated SC-LDPC code chain, we connect multiple chains in a layered format, where encoding, transmission, and decoding are now performed in a continuous fashion. The connections between chains are created at specific points, chosen to improve the finite-length performance of the code structure under iterative decoding. We describe the design of CC schemes for different SC-LDPC code ensembles constructed from protographs: a (J,K)-regular SC-LDPC code chain, a spatially-coupled repeat-accumulate (SC-RA) code, and a spatially-coupled accumulate-repeat-jagged-accumulate (SC- ARJA) code. In all cases, significant performance improvements are reported and, in addition, it is shown that using CC transmission only requires a small increase in decoding complexity and decoding delay with respect to a system employing a single SC-LDPC code chain for transmission.

SC-LDPC Code With Nonuniform Degree Distribution Optimized by Using Genetic Algorithm

We propose a class of spatially-coupled low-density parity-check (SC-LDPC) codes consisting of a finite-length chain of component LDPC matrices which are assigned with different degree distributions, i.e., the strength of spatial coupling is nonuniform within an SC-LDPC code. For this class of codes, we use a modified density evolution algorithm to calculate the belief-propagation (BP) threshold on the binary erasure channel (BEC). We then optimize the degree distributions by a genetic algorithm where the BP-threshold is considered as the fitness value. The resultant codes were confirmed to exhibit improved decoding performance compared with SC-LDPC codes having uniform degree distributions.

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.

Spatially coupled LDPC coding in cooperative wireless networks

This paper proposes a novel technique of spatially coupled low-density parity-check (SC-LDPC) code-based soft forwarding relaying scheme for a two-way relay system. We introduce an array-based optimized SC-LDPC codes in relay channels. A more precise model is proposed to characterize the residual noise on the soft symbols, using a pre-calculated look-up table at the destination. This requires less signaling overhead compared to previous soft noise modeling techniques. We also introduce a variance correction factor to provide a rectification to the equivalent total noise variance at the destination. Finally, we propose an appropriate log likelihood ratio (LLR) former at the destination which is tailored to the proposed soft parity generation technique. Simulation results demonstrate that the proposed relay protocol yields an improved bit error rate (BER) performance compared to competing schemes proposed in the literature.

On the Waterfall Performance of Finite-Length SC-LDPC Codes Constructed From Protographs

An analysis of spatially coupled low-density parity-check (SC-LDPC) codes constructed from protographs is proposed. Given the protograph used to generate the SC-LDPC code ensemble, a set of scaling parameters to characterize the average finite-length performance in the waterfall region is computed. The error performance of structured SC-LDPC code ensembles is shown to follow a scaling law similar to that of unstructured randomly constructed SC-LDPC codes. Under a finite-length perspective, some of the most relevant SC-LDPC protograph structures proposed to date are compared. The analysis reveals significant differences in their finite-length scaling behavior, which is corroborated by simulation. Spatially coupled repeat-accumulate codes present excellent finite-length performance, as they outperform in the waterfall region SC-LDPC codes of the same rate and better asymptotic thresholds.

Performance Analysis of Block Markov Superposition Transmission of Short Codes

In this paper, we consider the asymptotic and finite-length performance of block Markov superposition transmission (BMST) of short codes, which can be viewed as a new class of spatially coupled (SC) codes where the generator matrices of short codes (referred to as basic codes ) are coupled. A modified extrinsic information transfer (EXIT) chart analysis that takes into account the relation between mutual information (MI) and bit-error-rate (BER) is presented to study the convergence behavior of BMST codes. Using the modified EXIT chart analysis, we investigate the impact of various parameters on BMST code performance, thereby providing theoretical guidance for designing and implementing practical BMST codes suitable for window decoding. Then, we present a performance comparison of BMST codes and SC low-density parity-check (SC-LDPC) codes on the basis of equal decoding latency. Also presented is a comparison of computational complexity. Simulation results show that, under the equal decoding latency constraint, BMST codes using the repetition code as the basic code can outperform both (3,6)-regular SC-LDPC codes and (4,8)-regular SC-LDPC codes in the waterfall region but have a higher computational complexity.

Finite-Length Algebraic Spatially-Coupled Quasi-Cyclic LDPC Codes

The replicate-and-mask (R&M) construction of finite-length spatially-coupled (SC) LDPC codes is proposed in this paper. The proposed R&M construction generalizes the conventional matrix unwrapping construction and contains it as a special case. The R&M construction of a class of algebraic spatially coupled (SC) quasi-cyclic (QC) LDPC codes over arbitrary finite fields is demonstrated. The girth, rank, and time-varying periodicity of the proposed R&M SC QC LDPC codes are analyzed. The error rate performance of finite-length nonbinary algebraic SC QC LDPC codes is investigated with window decoding. Compared to the conventional unwrapping construction, it is found through numerical simulations that the R&M construction resulted in SC QC LDPC codes with better block error rate performance and lower error floors. With a flooding schedule decoder, it is shown that the proposed R&M algebraic SC QC LDPC codes have better error performance than the corresponding LDPC block codes and random SC codes. The R&M construction of irregular SC QC LDPC codes is demonstrated. It is shown that low-complexity regular puncturing schemes can be deployed on these codes to construct families of rate-compatible irregular SC QC LDPC codes with good performance.

Optimized Design of Finite-Length Separable Circulant-Based Spatially-Coupled Codes: An Absorbing Set-Based Analysis

In this paper, we characterize the finite-length performance of separable circulant-based spatially-coupled (SCB-SC) LDPC codes for transmission over the additive white Gaussian noise channel. For a general class of finite-length graph-based codes, it is known that the existence of small absorbing sets causes a performance degradation in the error floor regime. We first present the mathematical conditions for the existence of absorbing sets in binary SCB-SC codes. This analysis enables us to find the exact number of absorbing sets as a function of the design parameters. In particular, our results show that the choice of the cutting vector affects the number of absorbing sets and, therefore, the error floor performance of the code. For a fixed column weight, we find provably optimal cutting vectors that result in the least number of absorbing sets. Furthermore, we extend our analysis to nonbinary SCB-SC codes, where we show that the choice of the cutting vector is not as critical as in the binary case. We provide an algorithm which provably removes the problematic nonbinary absorbing sets from nonbinary SCB-SC codes by informed selection of edge labels. Our simulation results show the superior error floor performance of our designed binary and nonbinary SCB-SC codes compared with binary unstructured and nonbinary quasi-cyclic SC codes available in the open literature.

Spatially Coupled LDPC Codes Constructed by Parallelly Connecting Multiple Chains

A particular class of spatially coupled low-density parity-check (SC-LDPC) codes are constructed by parallelly connecting multiple different coupled chains. The connection structure is realized by edge exchanges between adjacent chains, which can help these chains support each other and improve the iterative decoding thresholds. By varying the number of the connected chains and the degree of each chain, a family of SC-LDPC codes with wide rate range can be obtained. Density evolution analysis shows that the decoding thresholds of the proposed codes are very close to Shannon limits and are slightly better than that constructed by a single chain over the binary erasure channel.

Coded Cooperation for Multiway Relaying in Wireless Sensor Networks.

Wireless sensor networks have been considered as an enabling technology for constructing smart cities. One important feature of wireless sensor networks is that the sensor nodes collaborate in some manner for communications. In this manuscript, we focus on the model of multiway relaying with full data exchange where each user wants to transmit and receive data to and from all other users in the network. We derive the capacity region for this specific model and propose a coding strategy through coset encoding. To obtain good performance with practical codes, we choose spatially-coupled LDPC (SC-LDPC) codes for the coded cooperation. In particular, for the message broadcasting from the relay, we construct multi-edge-type (MET) SC-LDPC codes by repeatedly applying coset encoding. Due to the capacity-achieving property of the SC-LDPC codes, we prove that the capacity region can theoretically be achieved by the proposed MET SC-LDPC codes. Numerical results with finite node degrees are provided, which show that the achievable rates approach the boundary of the capacity region in both binary erasure channels and additive white Gaussian channels.

A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes

Spatially-coupled low-density parity-check (SC-LDPC) codes are known to have excellent asymptotic properties. Much less is known regarding their finite-length performance. We propose a scaling law to predict the error probability of finite-length spatially coupled code ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters. The ultimate goal of this line of research is to develop analytic tools for the design of SC-LDPC codes under practical constraints.