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Information Theory and Applications Workshop (ITA)

Asymptotic trapping set analysis of regular protograph-based LDPC convolutional code ensembles

David G. M. Mitchell, Alí Emre Pusane, Daniel J. Costello

Published Feb 1, 2009• 5 citations
Publisher

Abstract

It has been suggested that ldquonear-codewordsrdquo may be a significant factor affecting decoding failures of LDPC codes over the AWGN channel. A near-codeword is a sequence that satisfies almost all of the check equations. These near-codewords can be associated with so-called dasiatrapping setspsila that exist in the Tanner graph of a code. In this paper, we analyse the trapping sets of protograph-based LDPC convolutional codes. LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message-passing decoding. Further, it has been shown that some ensembles of LDPC convolutional codes are asymptotically good, in the sense that the average free distance grows linearly with constraint length. Here, asymptotic methods are used to calculate a lower bound on the trapping set growth rates for several ensembles of regular asymptotically good protograph-based LDPC convolutional codes. This provides us with an estimate of where the error floor will occur for these codes under iterative message-passing decoding.

DOI
10.1109/ita.2009.5044956

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Trapping set analysis of protograph-based LDPC convolutional codes

David G. M. Mitchell, Alí Emre Pusane, Daniel J. Costello

It has been suggested that ¿near-codewords¿ may be a significant factor affecting decoding failures of LDPC codes over the AWGN channel. A near-codeword is a sequence that satisfies almost all of the check equations. These near-codewords can be associated with so-called `trapping sets' that exist in the Tanner graph of a code. In this paper, we analyse the trapping sets of protograph-based LDPC convolutional codes. LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message-passing decoding. Further, it has been shown that some ensembles of LDPC convolutional codes are asymptotically good, in the sense that the average free distance grows linearly with constraint length. Here, asymptotic methods are used to calculate a lower bound on the trapping set growth rates for two ensembles of asymptotically good protograph-based LDPC convolutional codes. This can be used to predict where the error floor will occur for these codes under iterative message-passing decoding.

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•Sep 15, 2022•1 citation

Exact free distance and trapping set growth rates for LDPC convolutional codes

David G. M. Mitchell, Alí Emre Pusane, Michael Lentmaier +1 more

Ensembles of (J,K)-regular low-density parity-check convolutional (LDPCC) codes are known to be asymptotically good, in the sense that the minimum free distance grows linearly with the constraint length. In this paper, we use a protograph-based analysis of terminated LDPCC codes to obtain an upper bound on the free distance growth rate of ensembles of periodically time-varying LDPCC codes. This bound is compared to a lower bound and evaluated numerically. It is found that, for a sufficiently large period, the bounds coincide. This approach is then extended to obtain bounds on the trapping set numbers, which define the size of the smallest, non-empty trapping sets, for these asymptotically good, periodically time-varying LDPCC code ensembles.

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It has previously been shown that ensembles of terminated protograph-based low-density parity-check (LDPC) convolutional codes have a typical minimum distance that grows linearly with block length and that they are capable of achieving capacity approaching iterative decoding thresholds on the binary erasure channel (BEC). In this paper, we review a recent result that the dramatic threshold improvement obtained by terminating LDPC convolutional codes extends to the additive white Gaussian noise (AWGN) channel. Also, using a (3,6)-regular protograph-based LDPC convolutional code ensemble as an example, we perform an asymptotic trapping set analysis of terminated LDPC convolutional code ensembles. In addition to capacity approaching iterative decoding thresholds and linearly growing minimum distance, we find that the smallest non-empty trapping set of a terminated ensemble grows linearly with block length.

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It has been suggested that ldquonear-codewordsrdquo may be a significant factor affecting decoding failures of LDPC codes over the AWGN channel. A near-codeword is a sequence that satisfies almost all of the check equations. These near-codewords can be associated with so-called dasiatrapping setspsila that exist in the Tanner graph of a code. In this paper, we analyse the trapping sets of protograph-based LDPC convolutional codes. LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message-passing decoding. Further, it has been shown that some ensembles of LDPC convolutional codes are asymptotically good, in the sense that the average free distance grows linearly with constraint length. Here, asymptotic methods are used to calculate a lower bound on the trapping set growth rates for several ensembles of regular asymptotically good protograph-based LDPC convolutional codes. This provides us with an estimate of where the error floor will occur for these codes under iterative message-passing decoding.
It has been suggested that ¿near-codewords¿ may be a significant factor affecting decoding failures of LDPC codes over the AWGN channel. A near-codeword is a sequence that satisfies almost all of the check equations. These near-codewords can be associated with so-called `trapping sets' that exist in the Tanner graph of a code. In this paper, we analyse the trapping sets of protograph-based LDPC convolutional codes. LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message-passing decoding. Further, it has been shown that some ensembles of LDPC convolutional codes are asymptotically good, in the sense that the average free distance grows linearly with constraint length. Here, asymptotic methods are used to calculate a lower bound on the trapping set growth rates for two ensembles of asymptotically good protograph-based LDPC convolutional codes. This can be used to predict where the error floor will occur for these codes under iterative message-passing decoding.
Low-density parity-check (LDPC) convolutional codes have been shown to be capable of achieving capacity-approaching performance with iterative message-passing decoding. In the first part of this paper, using asymptotic methods to obtain lower bounds on the free distance to constraint length ratio, we show that several ensembles of regular and irregular LDPC convolutional codes derived from protograph-based LDPC block codes have the property that the free distance grows linearly with respect to the constraint length, i.e., the ensembles are asymptotically good. In particular, we show that the free distance to constraint length ratio of the LDPC convolutional code ensembles exceeds the minimum distance to block length ratio of the corresponding LDPC block code ensembles. A large free distance growth rate indicates that codes drawn from the ensemble should perform well at high signal-to-noise ratios under maximum-likelihood decoding. When suboptimal decoding methods are employed, there are many factors that affect the performance of a code. Recently, it has been shown that so-called trapping sets are a significant factor affecting decoding failures of LDPC codes over the additive white Gaussian noise channel with iterative message-passing decoding. In the second part of this paper, we study the trapping sets of the asymptotically good protograph-based LDPC convolutional codes considered earlier. By extending the theory presented in part one and using similar bounding techniques, we show that the size of the smallest non-empty trapping set grows linearly with the constraint length for these ensembles.
LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message-passing decoding. In this paper, asymptotic methods are used to calculate a lower bound on the free distance for several ensembles of asymptotically good protograph-based LDPC convolutional codes. Further, we show that the free distance to constraint length ratio of the LDPC convolutional codes exceeds the minimum distance to block length ratio of corresponding LDPC block codes.
LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message passing decoding. In this dissertation, we present several methods of deriving families of time-varying and time-invariant LDPC convolutional codes from LDPC block codes. We demonstrate that the derived LDPC convolutional codes significantly outperform the underlying LDPC block codes, and we investigate the reasons for these 'convolutional gains'. It is well known that cycles in the Tanner graph representation of a sparse code affect the iterative decoding algorithm, with short cycles generally pushing its performance further away from optimum. Hence it is common practice to design codes that do not contain short cycles, so as to obtain independent messages in at least the initial iterations of the decoding process. We show that the derived LDPC convolutional codes have better graph-cycle properties than their block code counterparts. In particular, we show that the unwrapping process that is used to derive LDPC convolutional codes from LDPC block codes can 'break' some cycles of the underlying LDPC block code, but cannot create any shorter cycles. We prove that any cycle in an LDPC convolutional code always maps to a cycle of the same or smaller length in the underlying LDPC block code. Minimum distance is an important code design parameter when the channel quality is high, since codewords that are minimum distance apart are the most likely to cause errors with ML or near-ML decoding. In this case, the minimum distance determines the so-called error floor behavior in the performance curve corresponding to high channel signal-to-noise ratios. Thus studying the minimum distance properties of a code family gives insight into its error floor behavior. We use asymptotic methods to calculate a lower bound on the free distance of several ensembles of asymptotically good LDPC convolutional codes derived from protograph-based LDPC block codes. Further, we show that the free distance to constraint length ratio of the LDPC convolutional codes exceeds the minimum distance to block length ratio of the corresponding LDPC block codes. Message-passing iterative decoders for LDPC block codes are known to be subject to decoding failures due to so-called pseudo-codewords. These failures can cause the large signal-to-noise ratio performance of message-passing iterative decoding to be worse than that predicted by the maximum-likelihood decoding union bound. We address the pseudo-codeword problem from the convolutional code perspective. In particular, we show that the minimum pseudo-weight of an LDPC convolutional code is at least as large as the minimum pseudo-weight of an underlying LDPC block code. This result, which parallels a well-known relationship between the minimum Hamming weight of convolutional codes and the minimum Hamming weight of their quasi-cyclic counterparts, is due to the fact that every pseudo-codeword in the LDPC convolutional code induces a pseudo-codeword in the LDPC block code with pseudo-weight no larger than that of the convolutional pseudo-codeword. More generally, we demonstrate a difference in the weight spectra of LDPC block and convolutional codes that leads to improved performance at low-to-moderate signal-to-noise ratios for the convolutional codes, a conclusion supported by simulation results.
Ensembles of (J,K)-regular low-density parity-check convolutional (LDPCC) codes are known to be asymptotically good, in the sense that the minimum free distance grows linearly with the constraint length. In this paper, we use a protograph-based analysis of terminated LDPCC codes to obtain an upper bound on the free distance growth rate of ensembles of periodically time-varying LDPCC codes. This bound is compared to a lower bound and evaluated numerically. It is found that, for a sufficiently large period, the bounds coincide. This approach is then extended to obtain bounds on the trapping set numbers, which define the size of the smallest, non-empty trapping sets, for these asymptotically good, periodically time-varying LDPCC code ensembles.
It has previously been shown that ensembles of terminated protograph-based low-density parity-check (LDPC) convolutional codes have a typical minimum distance that grows linearly with block length and that they are capable of achieving capacity approaching iterative decoding thresholds on the binary erasure channel (BEC). In this paper, we review a recent result that the dramatic threshold improvement obtained by terminating LDPC convolutional codes extends to the additive white Gaussian noise (AWGN) channel. Also, using a (3,6)-regular protograph-based LDPC convolutional code ensemble as an example, we perform an asymptotic trapping set analysis of terminated LDPC convolutional code ensembles. In addition to capacity approaching iterative decoding thresholds and linearly growing minimum distance, we find that the smallest non-empty trapping set of a terminated ensemble grows linearly with block length.