国際会議参加報告:2014 IEEE International Symposium on Information Theory

In this paper, we construct protograph-based spatially coupled low-density parity-check (LDPC) codes by coupling together a series of $L$ disjoint, or uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying $L$ , we obtain a flexible family of code ensembles with varying rates and frame lengths that can share the same encoding and decoding architecture for arbitrary $L$ . We demonstrate that the resulting codes combine the best features of optimized irregular and regular codes in one design: capacity approaching iterative belief propagation (BP) decoding thresholds and linear growth of minimum distance with block length. In particular, we show that, for sufficiently large $L$ , the BP thresholds on both the binary erasure channel and the binary-input additive white Gaussian noise channel saturate to a particular value significantly better than the BP decoding threshold and numerically indistinguishable from the optimal maximum a posteriori decoding threshold of the uncoupled LDPC code. When all variable nodes in the coupled chain have degree greater than two, asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences of asymptotically good LDPC codes with fast convergence rates and BP thresholds close to the Shannon limit. Further, the gap to capacity decreases as the density of the graph increases, opening up a new way to construct capacity achieving codes on memoryless binary-input symmetric-output channels with low-complexity BP decoding.

https://lup.lub.lu.se/search/ws/files/4260704/7765364.pdf

Spatially Coupled LDPC Codes Constructed From Protographs

In this paper, we construct protograph-based spatially coupled low-density parity-check (SC-LDPC) codes by coupling together a series of L disjoint, or uncoupled, LDPC code Tanner graphs into a single coupled chain. By varying L, we obtain a flexible family of code ensembles with varying rates and frame lengths that can share the same encoding and decoding architecture for arbitrary L. We demonstrate that the resulting codes combine the best features of optimized irregular and regular codes in one design: capacity approaching iterative belief propagation (BP) decoding thresholds and linear growth of minimum distance with block length. In particular, we show that, for sufficiently large L, the BP thresholds on both the binary erasure channel (BEC) and the binary-input additive white Gaussian noise channel (AWGNC) saturate to a particular value significantly better than the BP decoding threshold and numerically indistinguishable from the optimal maximum a-posteriori (MAP) decoding threshold of the uncoupled LDPC code. When all variable nodes in the coupled chain have degree greater than two, asymptotically the error probability converges at least doubly exponentially with decoding iterations and we obtain sequences of asymptotically good LDPC codes with fast convergence rates and BP thresholds close to the Shannon limit. Further, the gap to capacity decreases as the density of the graph increases, opening up a new way to construct capacity achieving codes on memoryless binary-input symmetric-output (MBS) channels with low-complexity BP decoding.

Spatially Coupled LDPC Codes Constructed from Protographs

Low-density parity-check (LDPC) codes have attracted much attention over the past two decades since they can asymptotically approach the Shannon capacity in a variety of data transmission and storage scenarios. As a type of promising structured LDPC codes, the protograph LDPC codes not only inherit the advantage of conventional LDPC codes, i.e., excellent error performance, but also possess simple representations to realize fast encoding and efficient decoding. This paper provides a comprehensive survey on the state-of-the-art in protograph LDPC code design and analysis for different channel conditions, including the additive white Gaussian noise (AWGN) channels, fading channels, partial response (PR) channels, and Poisson pulse-position modulation (PPM) channels. Moreover, the applications of protograph LDPC codes to joint source-and-channel coding (JSCC) and joint channel-and-physical-layer-network coding (JCPNC) are reviewed and studied. In particular, we focus our attention on the encoding design and assume the decoder is implemented by the belief propagation (BP) algorithm. Hopefully, this survey may facilitate research in this area.

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A Survey on Protograph LDPC Codes and Their Applications

We describe coding techniques that achieve the capacity of a discrete memoryless asymmetric channel. To do so, we discuss how recent advances in coding for symmetric channels yield more efficient solutions also for the asymmetric case. In more detail, we consider three basic approaches. The first one is Gallager's scheme that concatenates a linear code with a non-linear mapper, in order to bias the input distribution. We explicitly show that both polar codes and spatially coupled codes can be employed in this scenario. Further, we derive a scaling law between the gap to capacity, the cardinality of channel input and output alphabets, and the required size of the mapper. The second one is an integrated approach in which the coding scheme is used both for source coding, in order to create codewords with the capacity-achieving 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 approach is based on an idea due to Bocherer and Mathar and separates completely the two tasks of source coding and channel coding by “chaining” together several codewords. We prove that we can combine any suitable source code with any suitable channel code in order to provide optimal schemes for asymmetric channels. In particular, polar codes and spatially coupled codes fulfill the required conditions.

How to achieve the capacity of asymmetric channels

In this paper, we compare the finite-length performance of protograph-based spatially coupled low-density paritycheck (SC-LDPC) codes and LDPC block codes (LDPC-BCs) over GF(q). To reduce computational complexity and latency, a sliding window decoder with a stopping rule based on a soft belief propagation (BP) estimate is used for the q-ary SC-LDPC codes. Two regimes are considered: one when the constraint length of q-ary SC-LDPC codes is equal to the block length of q-ary LDPC-BCs and the other when the two decoding latencies are equal. Simulation results confirm that, in both regimes, (3,6)-, (3,9)-, and (3,12)-regular non-binary SC-LDPC codes can significantly outperform both binary and non-binary LDPC-BCs and binary SC-LDPC codes. Finally, we present a computational complexity comparison of q-ary SC-LDPC codes and q-ary LDPC-BCs under equal decoding latency and equal decoding performance assumptions.

Performance Comparison of LDPC Block and Spatially Coupled Codes over GF(q)

In this paper, we compare the finite-length performance of protograph-based spatially coupled low-density parity-check (SC-LDPC) codes and LDPC block codes (LDPC-BCs) over GF(q). In order to reduce computational complexity and latency, a sliding window decoder with a stopping rule based on a soft bit-error-rate (BER) estimate is used for the q-ary SC-LDPC codes. Two regimes are considered: one when the constraint length of q-ary SC-LDPC codes is equal to the block length of q-ary LDPC-BCs and the other when the two decoding latencies are equal. Simulation results confirm that, in both regimes, (3,6)-, (3,9)-, and (3,12)-regular non-binary SC-LDPC codes can significantly outperform both binary and non-binary LDPC-BCs and binary SC-LDPC codes. Finally, we present a computational complexity comparison of q-ary SC-LDPC codes and q-ary LDPC-BCs under equal decoding latency and equal decoding performance assumptions.

In this paper, we study spatially coupled low-density parity-check (SC-LDPC) codes over finite fields GF $(q)$ , $q\geq 2$ , and develop design rules for $q$ -ary SC-LDPC code ensembles based on their iterative belief propagation 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 the protograph versions of $q$ -ary density evolution (for the BEC) and the $q$ -ary extrinsic information transfer analysis (for the BIAWGNC). We show that the WD of $q$ -ary SC-LDPC codes provides significant threshold gains compared with 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 with standard flooding-schedule decoding, the WD of $q$ -ary SC-LDPC code ensembles results in significant reductions in both the decoding complexity and the decoding latency and that these reductions increase as $m$ increases. For the 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.

Design of Spatially Coupled LDPC Codes Over GF $(q)$ for 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 nearcapacity performance for a sufficiently large window size W on both the BEC and the BIAWGNC. Also, we show that NBSC-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 .

Threshold analysis of non-binary spatially-coupled LDPC codes with windowed decoding

In this work, we consider two possible ways of improving the performance of binary low-density parity-check (LDPC) block codes. The first uses finite fields of size q > 2 to generate non-binary LDPC (NB-LDPC) block codes [1]. The performance gain introduced in this way comes at the cost of an increase in decoding complexity, and a variety of low-complexity algorithms have been proposed [2] [3] [4] to make NB-LDPC block codes practically feasible. The second modifies a binary LDPC block code by using an edge spreading technique [5] to introduce memory into the encoding process, thereby producing an LDPC convolutional code [6]. Terminated LDPC convolutional codes, which are also known as spatially-coupled LDPC (SC-LDPC) codes, have been shown to exhibit a phenomenon called “threshold saturation” [7] [8], which allows them to perform at the maximum a posteriori (MAP) threshold of the corresponding LDPC block codes, and thus to achieve capacity for increasing graph density, with belief propagation (BP) decoding. As a result, a natural way to design good LDPC codes is to combine the two features discussed above to construct nonbinary spatially-coupled LDPC (NB-SC-LDPC) codes. BP decoding thresholds of NB-SC-LDPC code ensembles on the binary erasure channel (BEC) were reported in [9] and [10], and threshold saturation was proved in [11]. In all these papers, flooding-schedule (FS) decoding was assumed, i.e., BP decoding was carried out across the entire parity-check matrix (graph) of the code simultaneously for a fixed number of iterations. Here, we briefly summarize recent results obtained concerning (1) the BP threshold performance of NB-SC-LDPC code ensembles defined by protographs on the BEC and the binaryinput additive white Gaussian noise channel (BIAWGNC), and (2) the bit-error-rate (BER) performance of finite-length NBSC-LDPC codes based on protographs. In both cases, we assume that q = 2, where m is a positive integer, and we focus on the scenario when windowed decoding is used. Details of the results can be found in [12] and [13]. A brief introduction to key concepts and terminology now follows: 1) A protograph is a small Tanner graph that defines a structured LDPC code ensemble by placing constraints on both the degree distribution and the edge connec-

Lai Wei

Papers

Performance Comparison of LDPC Block and Spatially Coupled Codes over GF(q)

Performance Comparison of LDPC Block and Spatially Coupled Codes over GF(q)

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

Threshold analysis of non-binary spatially-coupled LDPC codes with windowed decoding

Performance comparison of non-binary LDPC block and spatially coupled codes.