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IEEE Communications Letters

Recoverability of Variable Nodes in Periodically Punctured LDPC Convolutional Codes

Hua Ren Zhou, Norbert Goertz

Published Feb 16, 2015• 1 citation
Publisher

Abstract

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.

DOI
10.1109/lcomm.2015.2404434

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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.
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.
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.
In this paper, we analyze the effect of puncturing patterns for rate-compatible Low-Density Parity-Check (LDPC) codes. Each punctured variable node are connected with multiple check nodes and the check nodes are categorized into two groups named as ‘survived check (SC) node’ and ‘dead check (DC) node’ according to the role of them [3]. With this categorization, we derived the node-wise error probability using density evolution and presented simulation results which were well matched with the analysis.
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.
In this paper, we propose a non-greedy punctur- ing method for constructing binary rate-compatible low-density parity-check codes. First, we show that the recovery error prob- ability for a punctured variable node can be reduced by allocat- ing a small number of unpunctured variable nodes in the recovery tree. Then, we redefine the recovery tree according to the princi- ple of iterative decoding with the sum-product algorithm. The pro- posed non-greedy puncturing algorithm lies mainly in the dynam- ical minimization of the number of unpunctured variable nodes in the redefined recovery tree for the punctured variable nodes. Fi- nally, simulation results show that the proposed puncturing algo- rithm outperforms the existing best puncturing algorithms in bit error rate performances of iterative decoding.
In this paper, a new construction of irregular low-density parity-check convolutional codes (LDPC-CCs) is presented. Both upper and lower bounds on the free distance are derived as well. Compared with previously constructed irregular LDPC-CCs, our design can not only avoid a girth less than 8 but also provide an enlarged free distance. Since some undesired cycles occur due to the binomial terms in the parity-check matrices of general LDPC-CCs, we provide a specific procedure to remove this kind of length-8 cycles. It can also be applied to our irregular construction. Simulation results show that the codes based on the new construction and the cycle-removal procedure can achieve better bit-error-rate performance and a lower error-floor.