Spatially-coupled Low-density Parity-check Code Ensembles Research Articles

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.

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.

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.

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.

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.

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.