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Register a new userThreshold Saturation via Spatial Coupling: Why Convolutional LDPC Ensembles Perform so well over the BEC
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Added by
Shrinivas Kudekar Mr.
Publication date
2010
fields
Informatics Engineering
and research's language is
English
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Convolutional LDPC ensembles, introduced by Felstrom and Zigangirov, have excellent thresholds and these thresholds are rapidly increasing as a function of the average degree. Several variations on the basic theme have been proposed to date, all of which share the good performance characteristics of convolutional LDPC ensembles. We describe the fundamental mechanism which explains why convolutional-like or spatially coupled codes perform so well. In essence, the spatial coupling of the individual code structure has the effect of increasing the belief-propagation (BP) threshold of the new ensemble to its maximum possible value, namely the maximum-a-posteriori (MAP) threshold of the underlying ensemble. For this reason we call this phenomenon threshold saturation. This gives an entirely new way of approaching capacity. One significant advantage of such a construction is that one can create capacity-approaching ensembles with an error correcting radius which is increasing in the blocklength. Our proof makes use of the area theorem of the BP-EXIT curve and the connection between the MAP and BP threshold recently pointed out by Measson, Montanari, Richardson, and Urbanke. Although we prove the connection between the MAP and the BP threshold only for a very specific ensemble and only for the binary erasure channel, empirically a threshold saturation phenomenon occurs for a wide class of ensembles and channels. More generally, we conjecture that for a large range of graphical systems a similar saturation of the dynamical threshold occurs once individual components are coupled sufficiently strongly. This might give rise to improved algorithms as well as to new techniques for analysis.
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In this paper we are concerned with the asymptotic analysis of nonbinary spatially-coupled low-density parity-check (SC-LDPC) ensembles defined over GL$left(2^{m}right)$ (the general linear group of degree $m$ over GF$left(2right)$). Our purpose is to prove threshold saturation when the transmission takes place on the binary erasure channel (BEC). To this end, we establish the duality rule for entropy for nonbinary variable-node (VN) and check-node (CN) convolutional operators to accommodate the nonbinary density evolution (DE) analysis. Based on this, we construct the explicit forms of the potential functions for uncoupled and coupled DE recursions. In addition, we show that these functions exhibit similar monotonicity properties as those for binary LDPC and SC-LDPC ensembles over general binary memoryless symmetric (BMS) channels. This leads to the threshold saturation theorem and its converse for nonbinary SC-LDPC ensembles on the BEC, following the proof technique developed by S. Kumar et al.
We consider spatially coupled code ensembles. A particular instance are convolutional LDPC ensembles. It was recently shown that, for transmission over the binary erasure channel, this coupling increases the belief propagation threshold of the ensemble to the maximum a-priori threshold of the underlying component ensemble. We report on empirical evidence which suggest that the same phenomenon also occurs when transmission takes place over a general binary memoryless symmetric channel. This is confirmed both by simulations as well as by computing EBP GEXIT curves and by comparing the empirical BP thresholds of coupled ensembles to the empirically determined MAP thresholds of the underlying regular ensembles. We further consider ways of reducing the rate-loss incurred by such constructions.
In this paper, we analyze the tradeoff between coding rate and asymptotic performance of a class of generalized low-density parity-check (GLDPC) codes constructed by including a certain fraction of generalized constraint (GC) nodes in the graph. The rate of the GLDPC ensemble is bounded using classical results on linear block codes, namely Hamming bound and Varshamov bound. We also study the impact of the decoding method used at GC nodes. To incorporate both bounded-distance (BD) and Maximum Likelihood (ML) decoding at GC nodes into our analysis without resorting on multi-edge type of degree distributions (DDs), we propose the probabilistic peeling decoding (P-PD) algorithm, which models the decoding step at every GC node as an instance of a Bernoulli random variable with a successful decoding probability that depends on both the GC block code as well as its decoding algorithm. The P-PD asymptotic performance over the BEC can be efficiently predicted using standard techniques for LDPC codes such as density evolution (DE) or the differential equation method. Furthermore, for a class of GLDPC ensembles, we demonstrate that the simulated P-PD performance accurately predicts the actual performance of the GLPDC code under ML decoding at GC nodes. We illustrate our analysis for GLDPC code ensembles with regular and irregular DDs. In all cases, we show that a large fraction of GC nodes is required to reduce the original gap to capacity, but the optimal fraction is strictly smaller than one. We then consider techniques to further reduce the gap to capacity by means of random puncturing, and the inclusion of a certain fraction of generalized variable nodes in the graph.
Spatially coupled codes have been shown to universally achieve the capacity for a large class of channels. Many variants of such codes have been introduced to date. We discuss a further such variant that is particularly simple and is determined by a very small number of parameters. More precisely, we consider time-invariant low-density convolutional codes with very large constraint lengths. We show via simulations that, despite their extreme simplicity, such codes still show the threshold saturation behavior known from the spatially coupled codes discussed in the literature. Further, we show how the size of the typical minimum stopping set is related to basic parameters of the code. Due to their simplicity and good performance, these codes might be attractive from an implementation perspective.
This paper is focused on the derivation of some universal properties of capacity-approaching low-density parity-check (LDPC) code ensembles whose transmission takes place over memoryless binary-input output-symmetric (MBIOS) channels. Properties of the degree distributions, graphical complexity and the number of fundamental cycles in the bipartite graphs are considered via the derivation of information-theoretic bounds. These bounds are expressed in terms of the target block/ bit error probability and the gap (in rate) to capacity. Most of the bounds are general for any decoding algorithm, and some others are proved under belief propagation (BP) decoding. Proving these bounds under a certain decoding algorithm, validates them automatically also under any sub-optimal decoding algorithm. A proper modification of these bounds makes them universal for the set of all MBIOS channels which exhibit a given capacity. Bounds on the degree distributions and graphical complexity apply to finite-length LDPC codes and to the asymptotic case of an infinite block length. The bounds are compared with capacity-approaching LDPC code ensembles under BP decoding, and they are shown to be informative and are easy to calculate. Finally, some interesting open problems are considered.
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