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IEEE Transactions on Circuits and Systems II Analog and Digital Signal Processing

Degree-Matched Check Node Decoding for Regular and Irregular LDPCs

Sheryl L. Howard, Christian B. Schlegel, Vincent Gaudet

Published Oct 1, 2006• 22 citations
Publisher

Abstract

This brief examines different parity-check node decoding algorithms for low-density parity-check (LDPC) codes, seeking to recoup the performance loss incurred by the min-sum approximation compared to sum-product decoding. Two degree-matched check node decoding approximations that depend on the check node degree dcare presented. Both have low complexity and can be applied to any degree distribution. Simulation results show near sum-product decoding performance for both degree-matched check node approximations for regular and irregular LDPCs

DOI
10.1109/tcsii.2006.882204

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This brief examines different parity-check node decoding algorithms for low-density parity-check (LDPC) codes, seeking to recoup the performance loss incurred by the min-sum approximation compared to sum-product decoding. Two degree-matched check node decoding approximations that depend on the check node degree dcare presented. Both have low complexity and can be applied to any degree distribution. Simulation results show near sum-product decoding performance for both degree-matched check node approximations for regular and irregular LDPCs
This paper examines ways to recoup the performance loss incurred when using the min-sum approximation instead of the exact sum-product algorithm for decoding low-density parity check codes (LDPCs). Approximations to the correction factor exactly expressing the difference between these two decoding algorithms exist for degree 3 check nodes, and can be applied to higher degree nodes by subdividing them into component degree 3 nodes. However, this results in replication of the approximation. An asymptotic expression for the correction factor at a check node of any degree is derived in this paper, and used to develop two simple approximations to the correction factor, matched to the check node degree. One has very low complexity, and both only need be applied once per check node extrinsic message. Simulation results are presented for each check node approximation when decoding a regular and an irregular LDPC. Both degree-matched check node approximations achieve sum-product decoding performance
A check node degree-based modified min-sum algorithm which classifies the codes was developed to reduce the complexity and improve the performance of decoding algorithm for low-density parity-check(LDPC) codes.The algorithm corrects both the minimum and sub-minimum absolute values of incoming messages in the check nodes.Two correction factors are obtained by calculating the offset between the belief propagation(BP) algorithm and the min-sum algorithm based on the check node degree.Simulations show that the algorithm achieves better performance than the BP algorithm at high signal noise ratio.The algorithm is also less complex than the BP algorithm and provides a good tradeoff between performance and complexity for any check node degree distribution in the LDPC codes.
This paper analyzes the performance of quantized min-sum decoding algorithms for irregular low-density parity-check (LDPC) codes. For regular LDPC codes, it is known that the normalized or offset min-sum decoding algorithm with quantization bits less than 6 bits achieves good performances over wide range of signal-to-noise ratios (SNR). However, finite precision effects in decoding irregular LDPC codes are different from that in decoding regular LDPC codes which is caused by the difference of convergence speeds between low degree nodes and high degree nodes. This paper proposes a novel method to improve the performance of the conventional normalized or offset min-sum decoding algorithm when it is approximated with finite precision for hardware implementations. The proposed method applies down-scaling factors to intrinsic information which has effects on increasing the reliability of extrinsic information at variable nodes and compensating the quantization errors caused by finite precision. Computer simulation results for irregular LDPC codes show that the proposed normalized and offset min-sum decoding algorithms achieve better performances at high SNR compared to the conventional normalized and offset min-sum algorithms under (6:2) quantization scheme.
An analytical approach to determining the optimal post-processing function for the minimum operation in the SumMin algorithm, previously derived for regular Low-Density Parity-Check (LOPe) codes, is extended to irregular LDPC codes. The optimal post-processing expression for the irregular case varies from one check node to another as well as from one iteration to the next. For practical use, it is necessary to approximate this optimal function. Unlike for the regular case, where one could use a unique post-processing function throughout the decoding process without a loss of bit error performance, for irregular codes it is crucial to vary the post-processing from one iteration to the next in order to achieve good performance. Using this approach, a bit error performance of 0.15dS from the Sum-Product decoder was demonstrated, corresponding to a IdS improvement compared to the Sum-Min algorithm without post-processing.
We construct lattices with high coding gains based on nested low-density parity-check (LDPC) codes by Construction D′.We generalize the LDPC lattices [1] to have irregular degrees and call them irregular LDPC lattices. To construct good irregular LDPC lattices, we optimize the degree distributions by using density evolution and a modified sequential quadratic programming (SQP) and show our optimized irregular LDPC lattice has a threshold 0.48dB from the capacity, which is about 0.6dB better than the regular one in [1]. To construct a Tanner graph corresponding to LDPC lattices, we generalize the progressive-edge growth (PEG) algorithm and show a lower bound on the girth of the graph. Simulation is performed using the sum-product algorithm. Also, we compare the performance of joint decoding and multi-stage decoding [2] and show the advantage of joint decoding.
In this paper a modified Offset Min-Sum decoding algorithm for Low-Density Parity Check Codes is presented. In this modified algorithm, the offset factor in the original Offset Min-Sum algorithm is adjusted iteratively with the assistance of the check-node computation outputs of the Normalized Min-Sum algorithm. As a result, the offset factor in our modified algorithm can be calculated adaptively and more efficiently according to the previous decoding procedure, which can minimize the decoding degradation compared with the Belief Propagation decoding algorithm. The simulation results show that, compared with the original Offset Min-Sum decoding algorithm, our modified algorithm can achieve noticeable performance improvement with minor extra hardware complexity. For example, when BER is 10−5, our algorithm can achieve 0.1dB and 0.2dB decoding gain over Offset Min-Sum algorithm for regular and irregular LDPC codes respectively.