Terminaison des codes convolutionnels récursifs doublement-orthogonaux

RÉSUMÉ Le travail de recherche présenté dans ce mémoire est un prolongement des travaux précédemment entamés sur les codes convolutionnels récursifs doublement orthogonaux (RCDO). Ces codes correcteurs d’erreur ont pour objectif de corriger les erreurs se produisant lors de la transmission, dans un canal bruité, de l’information entre une source et un destinataire. À ce jour, ces codes RCDO ont été seulement utilisés dans le contexte d’une transmission en continu et sont uniquement appliqués à des trames de longueurs indéfinies. Cependant, il devient indispensable de considérer le problème lié à la terminaison des codes convolutionnels RCDO lorsque nous envisageons de les utiliser au sein des systèmes de communication basés sur une transmission par paquets tels que WiMax, Ethernet, LTE, etc. Le problème de la terminaison est un problème ouvert en général pour tous les codes convolutionnels récursifs et consiste à ajouter une séquence de bits spécifique à la fin de la trame de sorte que le codeur RCDO reconverge vers l’état initial. Typiquement, cela implique que tous les éléments de délai composant les registres à décalage du codeur possèdent la valeur zéro à la fin de la transmission. Bien que la terminaison des codes RCDO entraîne une légère perte du taux de codage, des améliorations de protection contre les erreurs à la fin de la trame sont effectuées. Les objectifs de ce mémoire sont multiples. Dans un premier temps, ce travail permet d’étudier et de résoudre le problème de terminaison des codeurs RCDO. Nous avons proposé une nouvelle technique de terminaison capable de trouver les bits de terminaison qui permettent de reconduire le codeur RCDO vers l’état initial toute en assurant une complexité raisonnable associée à la génération de la séquence de terminaison. Cette technique nous a permis de réduire le plus possible la longueur de la séquence de terminaison à environ celle de la mémoire du codeur. À partir de la technique de terminaison proposée, il devient possible de définir les conditions de la terminaison qu’on doit imposer aux connexions du codeur dans le but de générer des codes RCDO terminables. En tenant compte des conditions de terminaison ainsi que des conditions de double orthogonalité, nous avons aussi proposé un nouvel algorithme de recherche capable de construire une architecture efficace des codeurs RCDO multi-registres terminables.----------Abstract This thesis presents an extension of a previous work started on Recursive Convolutional Doubly-Orthogonal (RCDO) codes. These error correcting codes are designed to detect and/or to correct errors caused by the corruptive channel noise during the transmission of data from a source to a destination. So far, RCDO codes were only decoded in a streaming fashion and were only applied to frames of indefinite lengths. However, it is essential to consider the termination for convolutional RCDO codes when used in packet-based communication systems such as WiMax, Ethernet and LTE. The termination problem remains an open and complex problem for all recursive convolutioanl codes and consists of adding a well-defined sequence of tail bits at the end of the frame so that the RCDO encoder can converge to the initial state, usually the all-zero state, which denotes that all the next information and parity bits are zero. Although the termination of RCDO codes causes a small loss of the coding rate, this can facilitate a good error performance over the last bits of a frame. This thesis has several objectives. First of all, this work studies and solves the RCDO encoder termination problem. In fact, for convolutional codes, the feedforward encoders can be terminated by simply injecting a sequence of zeros to their inputs. However, this is not a trivial problem for RCDO encoders due to their particular recursive structure. A termination sequence, whose component depends on the encoder state, can be determined by solving a system of linear equations over GF(2). For that purpose, we propose a novel termination algorithm for generating, with a reasonable complexity, the tail bits that takes the encoder back to a known state. Moreover, this algorithm allows to minimize the length of the termination sequence to approximately the code memory. From the proposed termination algorithm, it becomes possible to define a number of additional conditions, called termination conditions, which must be imposed on the encoder’s connexions in order to make the RCDO encoder terminable. Taking into consideration the terminations conditions, along with the code doubly orthogonal conditions, we also propose a new searching algorithm capable of building an efficient architecture of multi shift registers terminable RCDO encoders

Novel Code-Construction for (3, k) Regular Low Density Parity Check Codes

Communication system links that do not have the ability to retransmit generally rely on forward error correction (FEC) techniques that make use of error correcting codes (ECC) to detect and correct errors caused by the noise in the channel. There are several ECC’s in the literature that are used for the purpose. Among them, the low density parity check (LDPC) codes have become quite popular owing to the fact that they exhibit performance that is closest to the Shannon’s limit. This thesis proposes a novel code-construction method for constructing not only (3, k) regular but also irregular LDPC codes. The choice of designing (3, k) regular LDPC codes is made because it has low decoding complexity and has a Hamming distance, at least, 4. In this work, the proposed code-construction consists of information submatrix (Hinf) and an almost lower triangular parity sub-matrix (Hpar). The core design of the proposed code-construction utilizes expanded deterministic base matrices in three stages. Deterministic base matrix of parity part starts with triple diagonal matrix while deterministic base matrix of information part utilizes matrix having all elements of ones. The proposed matrix H is designed to generate various code rates (R) by maintaining the number of rows in matrix H while only changing the number of columns in matrix Hinf. All the codes designed and presented in this thesis are having no rank-deficiency, no pre-processing step of encoding, no singular nature in parity part (Hpar), no girth of 4-cycles and low encoding complexity of the order of (N + g2) where g2«N. The proposed (3, k) regular codes are shown to achieve code performance below 1.44 dB from Shannon limit at bit error rate (BER) of 10 −6 when the code rate greater than R = 0.875. They have comparable BER and block error rate (BLER) performance with other techniques such as (3, k) regular quasi-cyclic (QC) and (3, k) regular random LDPC codes when code rates are at least R = 0.7. In addition, it is also shown that the proposed (3, 42) regular LDPC code performs as close as 0.97 dB from Shannon limit at BER 10 −6 with encoding complexity (1.0225 N), for R = 0.928 and N = 14364 – a result that no other published techniques can reach

Novel Code-Construction for (3, k) Regular Low Density Parity Check Codes

Communication system links that do not have the ability to retransmit generally rely on forward error correction (FEC) techniques that make use of error correcting codes (ECC) to detect and correct errors caused by the noise in the channel. There are several ECC’s in the literature that are used for the purpose. Among them, the low density parity check (LDPC) codes have become quite popular owing to the fact that they exhibit performance that is closest to the Shannon’s limit. This thesis proposes a novel code-construction method for constructing not only (3, k) regular but also irregular LDPC codes. The choice of designing (3, k) regular LDPC codes is made because it has low decoding complexity and has a Hamming distance, at least, 4. In this work, the proposed code-construction consists of information submatrix (Hinf) and an almost lower triangular parity sub-matrix (Hpar). The core design of the proposed code-construction utilizes expanded deterministic base matrices in three stages. Deterministic base matrix of parity part starts with triple diagonal matrix while deterministic base matrix of information part utilizes matrix having all elements of ones. The proposed matrix H is designed to generate various code rates (R) by maintaining the number of rows in matrix H while only changing the number of columns in matrix Hinf. All the codes designed and presented in this thesis are having no rank-deficiency, no pre-processing step of encoding, no singular nature in parity part (Hpar), no girth of 4-cycles and low encoding complexity of the order of (N + g2) where g2«N. The proposed (3, k) regular codes are shown to achieve code performance below 1.44 dB from Shannon limit at bit error rate (BER) of 10 −6 when the code rate greater than R = 0.875. They have comparable BER and block error rate (BLER) performance with other techniques such as (3, k) regular quasi-cyclic (QC) and (3, k) regular random LDPC codes when code rates are at least R = 0.7. In addition, it is also shown that the proposed (3, 42) regular LDPC code performs as close as 0.97 dB from Shannon limit at BER 10 −6 with encoding complexity (1.0225 N), for R = 0.928 and N = 14364 – a result that no other published techniques can reach