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International Conference on Internet of Things, Embedded Systems and Communications (IINTEC)

An LP-based algorithm for decoding terminated LDPC convolutional codes

Hayfa Ben Thameur, Nadia Khouja, Fethi Tlili, Bertrand Le Gal, Christophe Jégo

Published Oct 1, 2017• 1 citation
Publisher

Abstract

Linear programming (LP) technique was demonstrated as efficient for binary and non-binary LDPC block decoding. Indeed, it improves the error correction performance of codes and provides the Maximum-likelihood (ML) certificate. In this paper, the LP algorithm is applied for decoding LDPC convolutional codes (LDPC-CC). The proposed algorithm is built on the decomposition method based on the alternating direction method of multipliers (ADMM). Motivated by the strong convergence guarantees of an ADMM based LDPC block code (LDPC-BC) decoder and its comparable complexity to a Belief Propagation (BP) based LDPC-BC decoder, the efficiency of decoding LDPC-CCs using the ADMM algorithm is empirically analyzed in this paper. Simulations are carried over LDPC-CCs derived from LDPC-BCs and the LDPC-CC proposed by the IEEE 1901 standard. Experimental results show that the LP decoding approach is a promising technique for improving LDPC-CC error correction performance.

DOI
10.1109/iintec.2017.8325920

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Linear programming (LP) technique was demonstrated as efficient for binary and non-binary LDPC block decoding. Indeed, it improves the error correction performance of codes and provides the Maximum-likelihood (ML) certificate. In this paper, the LP algorithm is applied for decoding LDPC convolutional codes (LDPC-CC). The proposed algorithm is built on the decomposition method based on the alternating direction method of multipliers (ADMM). Motivated by the strong convergence guarantees of an ADMM based LDPC block code (LDPC-BC) decoder and its comparable complexity to a Belief Propagation (BP) based LDPC-BC decoder, the efficiency of decoding LDPC-CCs using the ADMM algorithm is empirically analyzed in this paper. Simulations are carried over LDPC-CCs derived from LDPC-BCs and the LDPC-CC proposed by the IEEE 1901 standard. Experimental results show that the LP decoding approach is a promising technique for improving LDPC-CC error correction performance.
The ADMM based linear programming (LP) technique shows interesting error correction performance when decoding binary LDPC block codes. Nonetheless, it's applicability to decode LDPC convolutional codes (LDPC-CC) has not been yet investigated. In this paper, a first flooding based formulation of the ADMM-LP for decoding LDPC-CCs is described. In addition, reduced complexity decoding schedules to lessen the storage requirements and improve the convergence speed of an ADMM-LP based LDPC-CC decoder without significant loss in error correction performances are proposed and assessed from an algorithmic and computational/memory complexity perspectives.
The enforcement of the linear programming (LP) decoding was recently extended to LDPC convolutional codes (LDPC-CC). It was demonstrated that their convolutional structure suites the message-passing based representation of the LP problem, thanks to the application of the alternating directions method of multipliers (ADMM). In this paper, a modified formulation of the ADMM-LP problem is introduced for pipeline decoding of LDPC-CCs based on the layered schedule. Moreover, an assessment of the fixed-point format required for hardware implementation is provided to evaluate the complexity of the underlying decoding algorithm. Simulations show that an 8-bit quantization scheme yields negligible error correction performances loss of about 0.05 dB regarding the floating-precision ADMM based LDPC-CC decoder. Further, an algorithmic optimization of the Euclidean projection skipping technique, described in [1] [2], is proposed. This improved version enables to reduce the computational complexity of the decoding process without memory penalty contrary to the original formulation.
A node-wise (NS) schedule has been recently proposed for decoding LDPC codes with the linear programming (LP) decoding approach, based on the alternate direction method of multipliers (ADMM). It improves the error correction performances as well as the convergence speed of the ADMM-LP decoder. However, it suffers from a high computational complexity resulted from residuals calculation. In this paper, a new formulation of the ADMM-LP algorithm with a modified computational scheduling is proposed for decoding LDPC convolutional codes. Simulations show that it reduces the computational complexity while improving the error correction performances and speeding up the decoding process. The decoding complexity is further reduced when considering the NS-ADMM approximation which allows savings up to 70% of the residuals calculation.
Iterative decoding algorithms based on the alternating direction method of multipliers (ADMM) decoding of low density parity check (LDPC) codes has emerged as an alternating decoding method and bringed a boom of research on drawing upon mathematical optimization to LDPC decoding. Improving error-correcting performance is a key issue to enhance the superiority of ADMM decoding. In this letter, we investigate an efficient ADMM-based iterative decoder for binary LDPC codes. First, we build an mathematical programming equivalence of the maximum likelihood (ML) decoding problem by transforming parity-check constraints to multiple equivalent linear constraints and eliminating check-polytope projection (CPP). Then, an iterative algorithm based on ADMM technique is developed to solve this free-of-CPP (FCPP) equivalence and each ADMM update can be computed efficiently. Moreover, the proposed ADMM-FCPP decoding algorithm is analyzed to display a linear complexity to the length of the LDPC code at each iteration. Finally, simulation results demonstrate the superiority of the proposed decoder in error-correcting performance compared with the state-of-the-art ADMM-based decoders.
In this dissertation, we study linear programming (LP) decoding of low-density parity-check (LDPC) codes based on the alternating direction method of multipliers (ADMM) technique, or ADMM decoding for short.The decoding of LDPC codes is formulated as an LP model and solved efficiently by the ADMM technique.However, compared with traditional belief-propagation decoding, ADMM decoding suffers from higher complexity and worse error correction performance, which prevents the employment of ADMM decoding in reality.To reduce the complexity of ADMM decoding, we first focus on the simplification of the check polytope projection, the most complex operation in ADMM decoding.We propose an iterative check polytope projection algorithm without the sorting operation.The proposed algorithm converges with the increase of iterations.Moreover, for a fixed number of iterations, its average complexity and the worst-case complexity are linear in the input dimension.Another direction we propose to simplify ADMM decoding is to devise a better scheduling scheme than the standard flooding scheme.We start from the node-wise scheduling scheme which only updates the check node messages with the maximum message residual.Then we simplify the calculation of the message residual and propose a reduced-complexity node-wise scheduling scheme.To improve the error correction performance of ADMM decoding, we propose a novel ADMM penalized decoding algorithm whose penalty term is based on check nodes (CN).We investigate the properties of the CN penalty functions and show several examples.We make conclusions on its convergence properties and prove its failure probability is independent of the transmitted codewords for symmetric channels.Monte-Carlo simulation and the instanton analysis show its better error correction performance in the low and high SNR regions.Finally, we propose a post-processing technique to lower the error floor of LDPC codes.The output of the first stage decoder can be revised if the syndrome is found This is a journey with joys and tears.I encountered many amazing people.They instructed me, helped me, and accompanied me.First of all, I would like to thank Prof. Amir H. Banihashemi for his guidance.His attitudes towards the research and devotion to one specific area are impressive.All these will have a life-long influence on me.It is also a great time working with people in his coding research group.I have a great memory of learning from their presentations.Next, I should thank Prof.
In this paper, we develop an efficient quadratic programming (QP) decoding algorithm via the alternating direction method of multipliers (ADMM) technique for binary low density parity check (LDPC) codes. Its main content is as follows: first, through transforming the three-variables parity check equation to its equivalent expression, we relax the maximum likelihood decoding problem to a quadratic program. Second, the ADMM technique is exploited to design the solving algorithm of the resulting QP decoding model. Compared with the existing ADMM-based mathematical programming (MP) decoding algorithms, our proposed algorithm eliminates complex Euclidean projection onto the check polytope. Third, we prove that the proposed algorithm satisfies the favorable property of all-zeros assumption. Moreover, by exploiting the inside structure of the QP model, we show that the decoding complexity of our proposed algorithm in each iteration is linear in terms of LDPC code length. Simulation results demonstrate that the proposed QP decoder attains better error-correction performance than the sum-product BP decoder and costs the least amount of decoding time amongst the state-of-the-art ADMM-based MP decoding algorithms.