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Simplified a Posteriori Probability Calculation for Binary LDPC Codes

Latifa Mostari, Abdelmalik Taleb Ahmed

Published Oct 24, 2018• 1 citation
OpenAlex

Abstract

The LDPC (Low-Density Parity-Check) decoder must operate soft decisions calculated using: LLR (Log-Likelihood Ratio) or APP (A Posteriori Probability) according to the decoding algorithm used. The exact calculation of these decisions for high order constellations involves complicated operations. In this work, a method to simplify the APP calculation is introduced. It is programmed to adapt as perfectly as possible the transmission system to the channel type in question. This method leads to simplify the implementation of the transmission system. Simulation results show that, under the Gaussian channel, the simplified APP algorithm for 16-QAM achieves the same performance that obtained with the exact APP, while for 64-QAM we have a small performance degradation. The same simplified APP algorithm that used for the Gaussian channel can be applied, with minor operations added, for Rayleigh channel, and it shows a small performance loss with respect to the exact APP.

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The LDPC (Low-Density Parity-Check) decoder must operate soft decisions calculated using: LLR (Log-Likelihood Ratio) or APP (A Posteriori Probability) according to the decoding algorithm used. The exact calculation of these decisions for high order constellations involves complicated operations. In this work, a method to simplify the APP calculation is introduced. It is programmed to adapt as perfectly as possible the transmission system to the channel type in question. This method leads to simplify the implementation of the transmission system. Simulation results show that, under the Gaussian channel, the simplified APP algorithm for 16-QAM achieves the same performance that obtained with the exact APP, while for 64-QAM we have a small performance degradation. The same simplified APP algorithm that used for the Gaussian channel can be applied, with minor operations added, for Rayleigh channel, and it shows a small performance loss with respect to the exact APP.
Belief propagation (BP) algorithm is commonly used in soft decoding of Low Density Parity Check Code (LDPC). However, a significant disadvantage of this method is that the computational complexity is high in high order M-QAM modulation. This paper proposes an optimized algorithm to reduce the posterior probability calculation in soft decoding, which estimates an approximate decision point in the constellation map via channel equalization. Then a contractible constellation decision space is delimited by the approximate decision point and its surrounding points with the minimum European distance. Compared with the conventional soft decoding, simulation results show that the proposed algorithm can effectively reduce the computational complexity of BP decoding with tiny loss of the decoding accuracy. The proposed algorithm reduces nearly 96 percent of the computational complexity of posterior probability in the case of 256QAM modulation.
In this study, the performance of generalized low-density parity-check (GLDPC) codes under the a posteriori probability (APP) decoder is analyzed. We explore the concentration, symmetry, and monotonicity properties of GLDPC codes under the APP decoder, extending the applicability of density evolution to GLDPC codes. On the binary memoryless symmetric channels, using the BEC and BI-AWGN channels as two examples, we demonstrate that with an appropriate proportion of generalized constraint (GC) nodes, GLDPC codes can reduce the original gap to capacity compared to their original LDPC counterparts. Additionally, on the BI-AWGN channel, we apply and improve the Gaussian approximation algorithm in the density evolution of GLDPC codes. By adopting Gaussian mixture distributions to approximate the message distributions from variable nodes and Gaussian distributions for those from constraint nodes, the precision of the channel parameter threshold can be significantly enhanced while maintaining a low computational complexity similar to that of Gaussian approximations. Furthermore, we identify a class of subcodes that can greatly simplify the performance analysis and practical decoding of GLDPC codes, which we refer to as message-invariant subcodes. Using the aforementioned techniques, our simulation experiments provide empirical evidence that GLDPC codes, when decoded with the APP decoder and equipped with the right fraction of GC nodes, can achieve substantial performance improvements compared to low-density parity-check (LDPC) codes.
Aiming at the problem of large computation and high decoding complexity when updating the check node in the LLR-BP algorithm of multicode LDPC code, a simplified method of dimensionality reduction in accordance with the weights is proposed, in which the q-dimensional likelihood ratio information of a variable node is descending-ordered and a weight is set for each dimension of the node, and the dimensional information of two variable nodes will be summed up according to the weight value size set in the process of updating check node. The size of the weight value will be added according to the set rule, which effectively reduces the wasted addition operation resources on the low likelihood ratio information. The simplified algorithm reduces about 90% of computation compared to the LLR-BP algorithm during one iteration of the algorithm. The BER performance simulation of the algorithm shows that the simplified decoding algorithm has only a very small difference in performance compared to the LLR-BP algorithm, and the overall decoding iteration speed is greatly improved.
LDPC (Low Density Parity Check Codes) is a low -density puppet test code applied to channel codes. Among the existing LDPC decoding algorithms, the LLR-BP decoding arithmetic is too complicated to calculate the calculation due to the containing a large number of bilateral positive cut functions. Therefore, this article proposes a LDPC decoding algorithm that combines the Modified Weighted Bit Flipping Decoding Algorithm (MWBF) with the Logarithmic Likelihood Ratio Based Confidence Propagation Algorithm (LLR-BP). In the process of the LLR-BP decoding iteration, the introduction of the MWBF algorithm not only does it reduce the complexity of iteration and decoding, but also avoids the traditional weighted bit-flipping decoding algorithm (WBF). Excessive effects of code performance.
An effective log-likelihood-ratio-based belief propagation (LLR-BP) algorithm is proposed. It can reduce computational complexity of decoding algorithm for Low Density Parity Check (LDPC) codes. By using the Taylor series and least squares, high order multiplication based on the hyperbolic tangent (tanh) rule is converted to a first-order multiplication and addition after simplification. Moreover, all the logarithmic and exponential operations disappear without significant loss of the decoding performance. The simulation results show that the performance of the proposed scheme is similar to the general LLR-BP. In particular, we show that the modified algorithm with low complexity can achieve better BER than the other decoding algorithm in high signal-to-noise ratio region.
Low-density parity-check (LDPC) codes are widely used in communication systems. When user messages are encoded by binary LDPC codes and transmitted with higher-order modulations, bits composing the same modulated symbol are generally correlated. Traditional soft-decision decoding algorithms for binary LDPC codes like sum-product algorithm (SPA) and min-sum algorithm (MSA) update messages only based on log-likelihood ratio (LLR) of bits, and correlations introduced by modulation are not utilized. In this paper, symbol-based SPA and MSA (S-SPA and S-MSA) which incorporate symbol-level information in the decoding procedures are proposed to provide an alternative low-cost way to better utilize available information. By narrowing the search space of symbols based on bit-level information, the complexities of the proposed algorithms are independent of modulation order. S-SPA and S-MSA achieve better error-correction performances than SPA and MSA with little additional complexities and S-MSA is friendly to hardware design of LDPC decoder.