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IEEE Transactions on Communications

Robust Rate-Compatible Punctured LDPC Convolutional Codes

Hua Ren Zhou, David G. M. Mitchell, Norbert Goertz, Daniel J. Costello

Published Nov 1, 2013• 26 citations
Publisher

Abstract

A family of robust rate-compatible (RC) punctured low-density parity-check convolutional codes (LDPC-CCs) is derived from a time-invariant LDPC-CC mother code by periodically puncturing encoded bits (variable nodes) with respect to several criteria: (1) ensuring the recoverability of punctured variable nodes, (2) minimizing the number of completely punctured cycle trapping sets (CPCTSs), and (3) minimizing the number of punctured variable nodes involved in short cycles. The influence of (1) and (3) on iterative decoding performance is felt most strongly in the waterfall region of the bit-error-rate (BER) curve, while (2) has a larger effect in the error floor, or high signal-to-noise ratio (SNR), region. We show that the length of the puncturing period is an important parameter when designing high rate punctured codes and, moreover, that extending the puncturing period can improve the decoding performance and extend the range of compatible rates. As examples, we obtain families of RC LDPC-CCs from several time-invariant LDPC-CC mother codes with monomial and binomial entries in their polynomial syndrome former matrices.

DOI
10.1109/tcomm.2013.101813.120864

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IEEE Communications Letters•Feb 16, 2015•1 citation
A family of robust rate-compatible (RC) punctured low-density parity-check convolutional codes (LDPC-CCs) is derived from a time-invariant LDPC-CC mother code by periodically puncturing encoded bits (variable nodes) with respect to several criteria: (1) ensuring the recoverability of punctured variable nodes, (2) minimizing the number of completely punctured cycle trapping sets (CPCTSs), and (3) minimizing the number of punctured variable nodes involved in short cycles. The influence of (1) and (3) on iterative decoding performance is felt most strongly in the waterfall region of the bit-error-rate (BER) curve, while (2) has a larger effect in the error floor, or high signal-to-noise ratio (SNR), region. We show that the length of the puncturing period is an important parameter when designing high rate punctured codes and, moreover, that extending the puncturing period can improve the decoding performance and extend the range of compatible rates. As examples, we obtain families of RC LDPC-CCs from several time-invariant LDPC-CC mother codes with monomial and binomial entries in their polynomial syndrome former matrices.
A family of rate-compatible (RC) punctured low-density parity-check convolutional codes (LDPC-CCs) is derived from an LDPC-CC by periodically puncturing encoded bits (variable nodes) with respect to several criteria: (1) ensuring the recoverability of punctured variable nodes, (2) minimizing the number of completely punctured cycle trapping sets (CPCTSs), and (3) minimizing the number of punctured variable nodes involved in short cycles. As an example, a family of RC punctured LDPC-CCs with rates 4/9, 4/8, 4/7, 4/6, and 4/5 are obtained from the (21,3, 5) Tanner LDPC-CC.
In this paper, we propose a rate-compatible puncturing scheme for finite-length low-density parity-check (LDPC) codes over the additive white Gaussian noise (AWGN) channel. The proposed method is applicable to any LDPC mother code, both regular and irregular, and constructs punctured codes which perform well in both the waterfall and the error floor regions for a wide range of code rates. The scheme selects code bits to be punctured one at a time and based on a sequence of criteria. An important selection criterion is the number of short cycles with low approximate cycle extrinsic message degree (ACE) in which a candidate bit node participates. Simulation results demonstrate that the ACE measure, which is most often the determining criterion in the final selection of the puncturing candidates, plays an important role in improving the performance of the codes in both the waterfall and the error-floor regions. These results also demonstrate that the proposed scheme is superior to the existing puncturing methods, particularly when a wide range of code rates is desirable.
In this paper, we present an effective puncturing scheme for rate-compatible low-density parity-check (RC-LDPC) codes. Our scheme is based on the criterion of maximum average girth which is an important performance parameter for the construction of RC-LDPC codes. We select a pattern wherein the punctured bits are far apart from each other in the Tanner graph of the codes by sorting algorithm, and then puncture the bits with the maximum average girth. In the process, there is no need to do any optimization. By simulations, the proposed codes show good performance under puncturing over a wide range of rates. The introduced puncturing criterion results in lower bit error rate and error floor when compared with random puncturing. Moreover, the performance of proposed method is similar to the direct structured LDPC codes by progressive edge growth (PEG). But the scheme is very simple to implement and lowers the complexity of encoding and decoding.
We study rate-compatible (RC) puncturing algorithms for efficiently-encodable (EE) generalized LDPC (GLDPC) codes of short-to-moderate blocklength. EE-GLDPC codes are systematic codes and well-suited for puncturing, thus they are able to address a rate constraint issue, which is an obvious disadvantage of traditional GLDPC codes. In this work, we propose systematic puncturing algorithms for EE-GLDPC codes. Simulation results show that the systematic puncturing has considerable performance improvement over random puncturing. In addition, the punctured codes, like their mother codes, have the advantage of good error floor performance, which can be a very important feature in practical applications. Finally, it is also demonstrated that, compared with LDPC codes, GLDPC codes have a better error floor performance, but suffer from a decoding threshold loss at high puncturing rates; nonetheless the tradeoff between error floor and decoding threshold loss can be conveniently implemented via the method of doping.
Time-invariant low-density parity-check convolutional codes (TI LDPC-CCs) can be represented by a polynomial-domain parity-check matrix derived from the corresponding quasi-cyclic (QC) LDPC block codes (LDPC-BCs), while time-varying (TV) LDPC-CCs can be obtained by unwrapping the parity-check matrices of LDPC-BCs. The cycle enumerators for TI and TV LDPC-CCs are compared. Based on the analysis of the graphical structures of short cycles in HT(D), we introduce a method of designing the polynomial syndrome former matrix HCRT(D) for LDPC-CCs. It eliminates short cycles and shows improved decoding performance on an additive white Gaussian noise (AWGN) channel with lower bit error ratio (BER) curves.
The concept of transformed codes is introduced to algebraically evaluate the recoverability of punctured nodes for periodically punctured low-density parity-check convolutional codes (LDPC-CCs) by a simple matrix-based approach. The proposed very efficient algorithm allows to identify and exclude those candidate puncturing patterns in a computer search that contain unrecoverable nodes that would lead to bad error-correction performance. Moreover, a novel upper bound on the maximum punctured code rate is obtained, above which any punctured code must contain unrecoverable punctured nodes.