Математическая модель сигнала с низкоплотностным кодированием, позволяющая выявить существенные свойства кода и использовать их при декодировании с целью минимизации вероятности битовой ошибки
Today the steady tendency of growth of popularity of a LDPC code was outlined in systems of satellite communication, it is caused first of all thanks to its good correcting opportunities. For decoding of this code broad application was received by algorithm of IBP. Use of algorithm IBP allows to receive exact a posteriori estimates of probabilities of symbols of rather accepted code word only in case of lack of cycles in the Tanner graph of code that in practice, at synthesis of a test matrix of a code, is impracticable. Existence of cycles in the Tanner graph of code is a major factor of decrease in correcting ability of algorithm of decoding as cycles result in dependence of decoded value on the current iteration from values of decoding of the same symbol on the previous iterations. In article the model, allowing to reduce negative influence of cycles of the minimum length is offered and to achieve more independence of estimates of decoding and as, it is offered to use statistics of decoding of symbols of the code word for additional updating of decisions for the purpose of reduction probability of an error of decoding in output sequence of bits in comparison with known decisions.
Today the steady tendency of growth of popularity of a LDPC code was outlined in systems of satellite communication, it is caused first of all thanks to its good correcting opportunities. For decoding of this code broad application was received by algorithm of IBP. Use of algorithm IBP allows to receive exact a posteriori estimates of probabilities of symbols of rather accepted code word only in case of lack of cycles in the Tanner graph of code that in practice, at synthesis of a test matrix of a code, is impracticable. Existence of cycles in the Tanner graph of code is a major factor of decrease in correcting ability of algorithm of decoding as cycles result in dependence of decoded value on the current iteration from values of decoding of the same symbol on the previous iterations. In article the model, allowing to reduce negative influence of cycles of the minimum length is offered and to achieve more independence of estimates of decoding and as, it is offered to use statistics of decoding of symbols of the code word for additional updating of decisions for the purpose of reduction probability of an error of decoding in output sequence of bits in comparison with known decisions.
Now the active junction of vendors of telecommunication equipment of the backbone fiber optic networks on use the low-density parity check (LDPC) codes for correction of errors in communication links is watched.This tendency is connected to that LDPC codes possess the best noise immunity on comparing with all noiseproof codes used earlier in practice (Read Solomon, BCH, block turbocodes). For LDPC decoding in satellite communication systems broad application was received by algorithm of iterative believe propagation and its different modifications. However, the LDPC codes in case of implementation of decoders at speeds of 10, 40 and 100 Gbit\s leads use of these algorithms of decoding to the significant increase in resource intensity, or decoding time.The modern programmable logic devices of the maximum integration scale can''t provide required resource intensity for implementation of decoders in real time. In article the modified algorithm of decoding LDPC codes, allowing considerably to reduce resource intensity of developed decoders is offered.
В статье рассматриваются способы упрощения расчета корректирующей способности LDPC-кодов и вероятности битовой ошибки при их использовании для случая применения порождающей матрицы канонического вида. В первой части статьи дано представление о кодировании LDPC, проанализированы узловые требования к порождающим матрицам, определены основные достоинства их канонического вида, отмечены способы обнаружения битовых ошибок на приемной стороне. Предложен алгоритм оценки корректирующей способности LDPC на основе статистических розыгрышей, приведена его схема, позволяющая смягчить имеющиеся в математических пакетах ограничения размерности матриц. Представлены результаты оценки максимального количества восстанавливаемых бит при использовании LDPC с разной кодовой скоростью. The article discusses ways to simplify the calculation of the correcting ability of LDPC codes and the probability of bit error when using them for the case of using a generator matrix of canonical form. In the first part of the article, an idea of LDPC coding is provided, nodal requirements for generator matrices are analyzed, the main advantages of their canonicalform are determined, and methods for detecting bit errors on the receiving side are noted. An algorithm for estimating the correcting ability of LDPC based on statistical drawings is proposed, and its scheme is given, which allows mitigating the limitations of the matrix dimension existing in mathematical packages. The results of estimating the maximum number of recoverable bits when using LDPC with different code rates are presented.
В статье рассмотрены особенности декодирования низкоплотностного кода. Приведены основные факторы, снижающие эффективность кодека. Описаны примеры стоп-множеств и предложены способы борьбы с ранним проявлением «полки» при декодировании. The article discusses features of decoding a low-density parity-check (LDPC) code. The main factors that reduce the efficiency of the codec are given. Examples of “stopping sets” are considered, and methods for mitigating the early appearance of “error floor” are suggested.
Aналитический подход к определению оптимальной функции постобработки для минимальной операции в алгоритме MIN-SUM, ранее полученный для обычных кодов проверки на четность с низкой плотностью (LDPC-коды), распространяется на нерегулярные коды LDPC. Оптимальное выражение постобработки для нестандартного случая варьируется от одного контрольного узла к другому, а также от одной итерации к следующей. Для практического использования необходимо аппроксимировать эту оптимальную функцию. В отличие от обычного случая, когда можно было бы использовать уникальную функцию постобработки на протяжении всего процесса декодирования без потери производительности битовых ошибок, для нерегулярных кодов критически важно варьировать постобработку от одной итерации к следующей, чтобы добиться хорошей производительности. С использованием этого подхода было выявлено, что качество битовых ошибок от алгоритма распространения доверия соответствует улучшению на 1 дБ по сравнению с MIN-SUM алгоритмом без постобработки. Сначала будет представлен обзор подхода и представлена аналитическая основа для оптимальной постобработки. Далее будет представлена оптимальная функция постобработки для нерегулярных кодов и обсуждены возможные упрощения. И наконец, показаны результаты моделирования и преимущества аппроксимации We extended an analytical approach to determining the optimal post-processing function for the minimum operation in the MIN-SUM algorithm, previously obtained for conventional low density parity check codes (LDPC codes), to irregular LDPC codes. The optimal post-processing expression for the non-standard case varies from one control node to another, as well as from one iteration to the next. For practical use, it is necessary to approximate this optimal function. Unlike the usual case where one could use a unique post-processing function throughout the entire decoding process without sacrificing bit code performance, it is critical for irregular codes to distinguish post-processing from one iteration to the next in order to achieve good performance. Using this approach, we found that the quality of bit errors from the trust algorithm corresponds to an improvement of 1 level compared to the MIN-SUM algorithm without post-processing. First, we provide an overview and analytical framework for optimal post-processing. Then, we present the optimal post-processing function for irregular codes and discuss possible simplifications. Finally, we show the simulation results and the benefits of the approximation
Gallager proposed and developed low density parity check (LDPC) codes in the early 1960s. LDPC codes were rediscovered in the early 1990s and shown to be capacity approaching over the additive white Gaussian noise (AWGN) channel. Subsequently, density evolution (DE) optimized symbol node degree distributions were used to significantly improve the decoding performance of short to medium length irregular LDPC codes. Currently, the short to medium length LDPC codes with the lowest error floor are DE optimized irregular LDPC codes constructed using progressive edge growth (PEG) algorithm modifications which are designed to increase the approximate cycle extrinsic message degrees (ACE) in the LDPC code graphs constructed. The aim of the present work is to find efficient means to improve on the error floor performance published for short to medium length irregular LDPC codes over AWGN channels in the literature. An efficient algorithm for determining the girth and ACE distributions in short to medium length LDPC code Tanner graphs has been proposed. A cyclic PEG (CPEG) algorithm which uses an edge connections sequence that results in LDPC codes with improved girth and ACE distributions is presented. LDPC codes with DE optimized/’good’ degree distributions which have larger minimum distances and stopping distances than previously published for LDPC codes of similar length and rate have been found. It is shown that increasing the minimum distance of LDPC codes lowers their error floor performance over AWGN channels; however, there are threshold minimum distances values above which there is no further lowering of the error floor performance. A minimum local girth (edge skipping) (MLG (ES)) PEG algorithm is presented; the algorithm controls the minimum local girth (global girth) connected in the Tanner graphs of LDPC codes constructed by forfeiting some edge connections. A technique for constructing optimal low correlated edge density (OED) LDPC codes based on modified DE optimized symbol node degree distributions and the MLG (ES) PEG algorithm modification is presented. OED rate-½ (n, k)=(512, 256) LDPC codes have been shown to have lower error floor over the AWGN channel than previously published for LDPC codes of similar length and rate. Similarly, consequent to an improved symbol node degree distribution, rate ½ (n, k)=(1024, 512) LDPC codes have been shown to have lower error floor over the AWGN channel than previously published for LDPC codes of similar length and rate. An improved BP/SPA (IBP/SPA) decoder, obtained by making two simple modifications to the standard BP/SPA decoder, has been shown to result in an unprecedented generalized improvement in the performance of short to medium length irregular LDPC codes under iterative message passing decoding. The superiority of the Slepian Wolf distributed source coding model over other distributed source coding models based on LDPC codes has been shown.
Для итеративного декодирования на графах используется новый альтернативный подход - это стохастическое декодирование. Возможность стохастического декодирования была недавно предложена для декодирования LDPC-кодов. Эта статья расширяет применение стохастического подхода для декодирования линейных блочных кодов с помощью проверочных матриц (PCM), таких как коды Боуза - Чоудхури - Хоквингема (BCH), коды Рида - Соломона (RS) и блочные турбокоды на основе компонентов кодов BCH. Показано, как стохастический подход способен генерировать информацию мягкого выхода для итеративного декодирования с мягким входом и мягким выходом Soft - Input Soft - Output (SISO). Описывается структура стохастических переменных узлов высокой степени, используемых в кодах с помощью проверочных матриц PCM. Результаты моделирования для кода BCH (128, 120), кода RS (31, 25) и RS (63, 55) и турбокода блока BCH (256, 121) и (1024, 676) демонстрируют эффективность декодирования при закрытии к итеративному декодеру SISO с реализацией с плавающей запятой. Эти результаты показывают производительность декодирования, близкую к адаптивному алгоритму распространения доверия и/или турбо-ориентированному адаптированному алгоритму распространения доверия Stochastic decoding capability has recently been proposed for decoding LDPC codes. This paper expands on the application of the stochastic approach to decoding linear block codes using parity check matrices (PCMs) such as Bose-Chowdhury-Hawkingham (BCH) codes, Reed-Solomon (RS) codes, and BCH component-based block turbo codes. We show how the stochastic approach is able to generate soft-output information for iterative decoding with soft-input and soft-output Soft-Input Soft-Output (SISO). We describe the structure of high degree stochastic node variables used in codes using PCM parity check matrices. Simulation results for BCH code (128, 120), RS code (31, 25) and RS (63, 55), and BCH block turbo code (256, 121) and (1024, 676) demonstrate the decoding efficiency on close to SISO iterative decoder with floating point implementation. These results show decoding performance close to the adaptive trust propagation algorithm and / or turbo-oriented adapted trust propagation algorithm