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IEEE International Symposium on Future Telecommunication Technologies (SOFTT)

Extrinsic Information Transfer (EXIT) Analysis for Short Polar Codes

Fauzil Halim Mufassa, Khoirul Anwar

Published Nov 1, 2019• 0 citations
Publisher

Abstract

Ze the quality of channels into either completely noisy or noieseless channels. This paper presents extrinsic information transfer (EXIT) analysis for iterative decoding of Polar codes to reveal the mechanism of channel transformation. The purpose of understanding the transformation process are to comprehend the placement process of information bit and frozen bit and to comprehend the security standard of Polar codes. Mutual information derived based on the concept of EXIT chart for check nodes and variable nodes of low density parity check (LDPC) codes and applied to Polar codes. This paper explores the quality of the polarized channels in finite blocklength. The finite block-length is of our interest since in the fifth telecommunications generation (5G) the block length is limited. This paper reveals the EXIT curve changes of Polar codes and explores the polarization characteristics, thus, high value of mutual informations for frozen bit are needed to be detectable. If it is the other way, the error correction capability of Polar codes would be drastically decreases. These results are expected to be a reference for developments of Polar codes for 5G technologies and beyond.

DOI
10.1109/softt48120.2019.9068617

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Ze the quality of channels into either completely noisy or noieseless channels. This paper presents extrinsic information transfer (EXIT) analysis for iterative decoding of Polar codes to reveal the mechanism of channel transformation. The purpose of understanding the transformation process are to comprehend the placement process of information bit and frozen bit and to comprehend the security standard of Polar codes. Mutual information derived based on the concept of EXIT chart for check nodes and variable nodes of low density parity check (LDPC) codes and applied to Polar codes. This paper explores the quality of the polarized channels in finite blocklength. The finite block-length is of our interest since in the fifth telecommunications generation (5G) the block length is limited. This paper reveals the EXIT curve changes of Polar codes and explores the polarization characteristics, thus, high value of mutual informations for frozen bit are needed to be detectable. If it is the other way, the error correction capability of Polar codes would be drastically decreases. These results are expected to be a reference for developments of Polar codes for 5G technologies and beyond.
The extrinsic information transfer (EXIT) charts are used to analyze the sum-product decoding algorithm of low-density parity-check (LDPC) codes. The codes are optimized by performing curve fitting on EXIT charts. This technique is developed for binary erasure channels based on area property and is extended to other channels. We show that on channels where the area property is not valid, the curve fitting approach is a sub-optimal method of optimizing for the best convergence threshold. On Gaussian channels, the messages from variable nodes to check nodes are approximated by a symmetric Gaussian distribution. We prove that the check node detector mutual information transfer function derived using this approximation is an upper bound. So the EXIT chart analysis predicts an optimistic value for the convergence threshold. We give an empirical lower bound for the check node detector mutual information transfer function. For few irregular LDPC codes designed for AWGN channel using density evolution, we have examined the accuracy of convergence thresholds predicted by EXIT charts using these bounds for the check node transfer function. The analysis of EXIT chart technique presented in this paper can be extended to other channels for which the distribution of channel messages can be approximated by a symmetric Gaussian/Gaussian mixture distribution.
Low-density parity-check (LDPC) codes are channel code of fifth-generation(5G) New Radio(NR), and it is important to explore low-complexity decoding algorithms that are easy to implement on hardware. In order to realize 5G ultra-reliable low latency communication (URLLC), low-complexity decoding of short packets is necessary. For quasi-cyclic LDPC (QC-LDPC) codes in 5G NR, the layer offset min-sum (LOMS) algorithm and the layer normalized min-sum (LNMS) algorithm are two efficient algorithms with low complexity. The performance of these two simplified decoding algorithms is significantly affected by the values of the above two decoding parameters. In this paper, the 3D extrinsic information transfer (EXIT) chart is introduced as an analysis tool to indicate the performance of different values for the offset and normalization parameters. The simulation results show that with different values of the above two factors, the changing of the output information of EXIT chart is consistent with that of the block error rate (BLER) in decoding simulations. In addition, it is shown that this analysis method can be extended to the decoding with fixed-point inputs.
Low density parity check (LDPC) codes are known to achieve a performance very close to the Shannon capacity limit on Additive White Gaussian Noise (AWGN) and erasure channels. The extrinsic information transfer (EXIT) chart is a powerful method for analyzing iteratively decoded codes such as Turbo codes. EXIT charts were introduced as a method for representing how the mutual information between the decoder output and the transmitted bits changes over turbo decoding iterations. It is possible to apply the EXIT chart method to LDPC codes by treating the LDPC decoder as a concatenation of variable and check nodes. However the achieved convergence threshold values obtained with the EXIT charts are (plusmn0.1) dB away from the more precise results derived from the density evolution (DE) method. Subsequently, the EXIT trajectories method was proposed as an improved performance-analysis method for LDPC codes under belief propagation (BP) decoding and achieved more accurate convergence threshold values than the EXIT chart method. In this paper, the EXIT chart and the EXIT trajectories methods are proposed as analysis tools to compare between the BP, log likelihood and max log likelihood decoding algorithms for LDPC codes in terms of convergence thresholds. Furthermore, a comparison between EXIT chart, EXIT trajectories and DE methods is introduced for the mentioned LDPC codes. Simulations and numerical calculations on convergence thresholds for various ensembles of (dv, dc) regular LDPC codes for binary input AWGN channels are performed which results that the BP is the best decoding algorithm for LDPC code followed by the log likelihood and the max log likelihood algorithms, respectively. Also, the EXIT trajectory method yields more accurate results than EXIT charts which come very close to the DE method, while being more convenient numerically as a code research tool than the latter which require many Fourier transform operations.
Low-density parity-check (LDPC) codes are widely recognized for their excellent forward error correction, near-Shannon-limit performance, and support for high data rates with effective hardware parallelization. Their convolutional counterpart, LDPC convolutional codes (LDPC-CCs), offer additional advantages such as variable codeword lengths, unlimited parity-check matrices, and simpler encoding and decoding. These features make LDPC-CCs particularly suitable for practical implementations with varying channel conditions and data frame sizes. This paper investigates the performance of LDPC-CCs using the extrinsic information transfer (EXIT) chart, a graphical tool for analyzing iterative decoding. EXIT charts visualize mutual information exchange and help predict convergence behavior, estimate performance thresholds, and optimize code design. Starting with the EXIT chart principles for LDPC codes, we derived the mutual information functions for variable and check nodes in regular and irregular LDPC-CC tanner graphs. This involved adapting existing EXIT functions to the periodic parity-check matrix of LDPC-CCs. We compare regular and irregular LDPC-CC constructions, examining the impact of degree distributions and the number of periods in the parity-check matrix on convergence behavior. Our simulations show that irregular LDPC-CCs consistently outperform regular ones, and the EXIT chart analysis confirms that LDPC-CCs demonstrate superior bit error rate (BER) performance compared to equivalent LDPC block codes.
The most popular methods for the analysis of LDPC codes, e.g. density evolution and EXIT charts, apply to infinite length ensembles. Finite length analysis is less common, among the notable examples are scaling law analysis and the EXIT band chart. The scaling law is limited to the waterfall region and furthermore is limited to the final error rates and does not give insight into the progression and dynamics of belief propagation decoding. The EXIT band chart was based on empirical considerations and does not enjoy a strong underlying theory. The developments of this paper are motivated by the recent results by Polyanskiy, Poor and Verdu showing that the rate penalty due to finite blocklength depends on the channel dispersion. We propose tracking the dispersion between the transmitted bits and the log-likelihood ratio across the decoder iterations to capture the effect of the codeword length. Based on the proposed analysis, a simple lower bound on the block error probability as a function of the blocklength is derived. The bound is corroborated by simulation.
In this paper extrinsic information transfer (EXIT) charts are proposed to design non-binary low-density parity-check (LDPC) codes for the AWGN channel. A new metric is presented to describe the mutual information of the non-binary messages. The a priori information is modelled using a Gaussian mixture distribution. Analytical expressions are given for the EXIT curves of the variable and check node decoders for both regular and irregular LDPC codes. The analytical expressions are shown to agree well with simulation results. Finally, by matching the variable and check node EXIT curves, it is shown that good nonbinary LDPC can be designed for the AWGN channel.