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Decoding low-density parity check codes with normalized APP-based algorithm

Jinghu Chen, M.P.C. Fossorier

Published Nov 13, 2002• 54 citations
Publisher

Abstract

We propose a normalized a posteriori probability (APP) based algorithm for the decoding of low-density parity check (LDPC) codes. The normalized APP-based algorithm utilizes normalization to improve the accuracy of the soft values delivered by the simplified APP-based algorithm from one iteration to another during the iterative decoding, and can achieve very good tradeoff between decoding complexity and performance.

DOI
10.1109/glocom.2001.965573

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International Symposium on Communications, Control and Signal Processing•Mar 1, 2008•4 citations
We propose a normalized a posteriori probability (APP) based algorithm for the decoding of low-density parity check (LDPC) codes. The normalized APP-based algorithm utilizes normalization to improve the accuracy of the soft values delivered by the simplified APP-based algorithm from one iteration to another during the iterative decoding, and can achieve very good tradeoff between decoding complexity and performance.
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.
In this paper, we propose a belief-propagation (BP)-based decoding algorithm which utilizes normalization to improve the accuracy of the soft values delivered by a previously proposed simplified BP-based algorithm. The normalization factors can be obtained not only by simulation, but also, importantly, theoretically. This new BP-based algorithm is much simpler to implement than BP decoding as it requires only additions of the normalized received values and is universal, i.e., the decoding is independent of the channel characteristics. Some simulation results are given, which show this new decoding approach can achieve an error performance very close to that of BP on the additive white Gaussian noise channel, especially for low-density parity check (LDPC) codes whose check sums have large weights. The principle of normalization can also be used to improve the performance of the max-log-MAP algorithm in turbo decoding, and some coding gain can be achieved if the code length is long enough.
Iterative decoding techniques have been receiving more and more attentions with the invention of turbo codes and the rediscovery of low-density parity-check (LDPC) codes. An important aspect in the study of iterative decoding is the tradeoff between decoding performance and complexities. For both LDPC codes and turbo codes, optimum decoding algorithms can provide very good performance. However, complicated operations are involved in the optimum decoding, and prohibit the wide applications of LDPC codes and turbo codes in the next generation digital communication and storage systems. Although there exist sub-optimum decoding algorithms for both LDPC codes and turbo codes, the decoding performance is degraded with the sub-optimum algorithms, and under some circumstances, the gap is very large. This research investigates the reduced complexity decoding algorithms of LDPC codes and turbo codes. For decoding LDPC codes, new algorithms, namely the normalized BP-based algorithm and the offset BP-based algorithm, are proposed. For these two reduced complexity algorithms, density evolution algorithms are derived, and are used to determine the best decoder parameters associated with each of the algorithms. Numerical results show that the new algorithms can achieve near optimum decoding performances for infinite code lengths, and simulation results reveal the same conclusion for short to medium code lengths. In addition to the advantage of low computational complexities, the two new algorithms are less subject to quantization errors and correlation effects than the optimum BP algorithm, and consequently are more suitable for hardware implementation. For a special kind of LDPC codes - the geometric LDPC codes, we propose the normalized APP-based algorithm, which is even more simplified yet still can achieve the near optimum performance. For decoding turbo codes, two new sub-optimum decoding algorithms are proposed. The first is the bi-directional soft-output Viterbi algorithm (bi-SOVA), which is based on utilizing a backward SOYA decoding in addition to the conventional forward one, and can achieve better performance than the uni-directional SOYA. The second is the normalized Max-Log-MAP algorithm, which improves the performance of the Max-Log-MAP decoding by scaling the soft outputs with some predetermined factors.
Low-density parity-check (LDPC) codes deliver very good performance when decoded with the BP algorithm. Unfortunately, this algorithm has the disadvantage to be complex for implementation. In this paper, we investigate the BP algorithm and its simplifications (BP-Based, normalized BP-Based, offset BP-Based). Previous works on normalized and offset BP-Based proposed theoretical values for the offset and normalization factors, we perform simulations which allow to validate these theoretical results.