Mathematical approach to channel codes with a diagonal matrix structure

Digital communications have now become a fundamental part of modern society. In communications, channel coding is an effective way to reduce the information rate down to channel capacity so that the information can be transmitted reliably through the channel. This thesis is devoted to studying the mathematical theory and analysis of channel codes that possess a useful diagonal structure in the parity-check and generator matrices. The first aspect of these codes that is studied is the ability to describe the parity-check matrix of a code with sliding diagonal structure using polynomials. Using this framework, an efficient new method is proposed to obtain a generator matrix G from certain types of parity-check matrices with a so-called defective cyclic block structure. By the nature of this method, G can also be completely described by a polynomial, which leads to efficient encoder design using shift registers. In addition, there is no need for the matrices to be in systematic form, thus avoiding the need for Gaussian elimination. Following this work, we proceed to explore some of the properties of diagonally structured lowdensity parity-check (LDPC) convolutional codes. LDPC convolutional codes have been shown to be capable of achieving the same capacity-approaching performance as LDPC block codes with iterative message-passing decoding. The first crucial property studied is the minimum free distance of LDPC convolutional code ensembles, an important parameter contributing to the error-correcting capability of the code. Here, asymptotic methods are used to form lower bounds on the ratio of the free distance to constraint length for several ensembles of asymptotically good, protograph-based LDPC convolutional codes. Further, it is shown that this ratio of free distance to constraint length for such LDPC convolutional codes exceeds the ratio of minimum distance to block length for corresponding LDPC block codes. Another interesting property of these codes is the way in which the structure affects the performance in the infamous error floor (which occurs at high signal to noise ratio) of the bit error rate curve. It has been suggested that “near-codewords” may be a significant factor affecting decoding failures of LDPC codes over an additive white Gaussian noise (AWGN) channel. A near-codeword is a sequence that satisfies almost all of the check equations. These nearcodewords can be associated with so-called ‘trapping sets’ that exist in the Tanner graph of a code. In the final major contribution of the thesis, trapping sets of protograph-based LDPC convolutional codes are analysed. Here, asymptotic methods are used to calculate a lower bound for the trapping set growth rates for several ensembles of asymptotically good protograph-based LDPC convolutional codes. This value can be used to predict where the error floor will occur for these codes under iterative message-passing decoding

On Lowering the Error Floor of Short-to-Medium Block Length Irregular Low Density Parity Check Codes

Edited version embargoed until 22.03.2019 Full version: Access restricted permanently due to 3rd party copyright restrictions. Restriction set on 22.03.2018 by SE, Doctoral CollegeGallager 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

Design and Analysis of GFDM-Based Wireless Communication Systems

Le multiplexage généralisé par répartition en fréquence (GFDM), une méthode de traitement par blocs de modulation multiporteuses non orthogonales, est une candidate prometteuse pour les technologies de forme d'onde pour les systèmes sans fil au-delà de la cinquième génération (5G). La capacité du GFDM à ajuster de manière flexible la taille du bloc et le type de filtres de mise en forme des impulsions en fait une méthode appropriée pour répondre à plusieurs exigences importantes, comme une faible latence, un faible rayonnement hors bande (OOB) et des débits de données élevés. En appliquant aux systèmes GFDM la technique des systèmes à entrées multiples et sorties multiples (MIMO), la technique de MIMO massif ou des codes de contrôle de parité à faible densité (LDPC), il est possible d'améliorer leurs performances. Par conséquent, l'étude de ces systèmes combinés sont d'une grande importance théorique et pratique. Dans cette thèse, nous étudions les systèmes de communication sans fil basés sur le GFDM en considérant trois aspects. Tout d'abord, nous dérivons une borne d'union sur le taux d'erreur sur les bits (BER) pour les systèmes MIMO-GFDM, technique qui est basée sur des probabilités d'erreur par paires exactes (PEP). La PEP exacte est calculée en utilisant la fonction génératrice de moments(MGF) pour les détecteurs à maximum de vraisemblance (ML). La corrélation spatiale entre les antennes et les erreurs d'estimation de canal sont prises en compte dans l'environnement de canal étudié. Deuxièmement, les estimateurs et les précodeurs de canal de faible complexité basés sur une expansion polynomiale sont proposés pour les systèmes MIMO-GFDM massifs. Des pilotes sans interférence sont utilisés pour l'estimation du canal basée sur l'erreur quadratique moyenne minimale(MMSE) pour lutter contre l'influence de la non-orthogonalité entre les sous-porteuses dans le GFDM. La complexité de calcul cubique peut être réduite à une complexité d'ordre au carré en utilisant la technique d'expansion polynomiale pour approximer les inverses de matrices dans l'estimation MMSE conventionnelle et le précodage. De plus, nous calculons les limites de performance en termes d'erreur quadratique moyenne (MSE) pour les estimateurs proposés, ce qui peut être un outil utile pour prédire la performance des estimateurs dans la région de Eₛ/N₀ élevé. Une borne inférieure de Cramér-Rao(CRLB) est dérivée pour notre modèle de système et agit comme une référence pour les estimateurs. La complexité de calcul des estimateurs de canal proposés et des précodeurs et les impacts du degré du polynôme sont également étudiés. Enfin, nous analysons les performances de la probabilité d'erreur des systèmes GFDM combinés aux codes LDPC. Nous dérivons d'abord les expressions du ratio de vraisemblance logarithmique (LLR) initiale qui sont utilisées dans le décodeur de l'algorithme de somme de produits (SPA). Ensuite, basé sur le seuil de décodage, nous estimons le taux d'erreur de trame (FER) dans la région de bas E[indice b]/N₀ en utilisant le BER observé pour modéliser les variations du canal. De plus, une borne inférieure du FER du système est également proposée basée sur des ensembles absorbants. Cette borne inférieure peut agir comme une estimation du FER dans la région de E[indice b]/N₀ élevé si l'ensemble absorbant utilisé est dominant et que sa multiplicité est connue. La quantification a également un impact important sur les performances du FER et du BER. Des codes LDPC basés sur un tableau et construit aléatoirement sont utilisés pour supporter les analyses de performances. Pour ces trois aspects, des simulations et des calculs informatiques sont effectués pour obtenir des résultats numériques connexes, qui vérifient les méthodes proposées.8 372162\u a Generalized frequency division multiplexing (GFDM) is a block-processing based non-orthogonal multi-carrier modulation scheme, which is a promising candidate waveform technology for beyond fifth-generation (5G) wireless systems. The ability of GFDM to flexibly adjust the block size and the type of pulse-shaping filters makes it a suitable scheme to meet several important requirements, such as low latency, low out-of-band (OOB) radiation and high data rates. Applying the multiple-input multiple-output (MIMO) technique, the massive MIMO technique, or low-density parity-check (LDPC) codes to GFDM systems can further improve the systems performance. Therefore, the investigation of such combined systems is of great theoretical and practical importance. This thesis investigates GFDM-based wireless communication systems from the following three aspects. First, we derive a union bound on the bit error rate (BER) for MIMO-GFDM systems, which is based on exact pairwise error probabilities (PEPs). The exact PEP is calculated using the moment-generating function (MGF) for maximum likelihood (ML) detectors. Both the spatial correlation between antennas and the channel estimation errors are considered in the investigated channel environment. Second, polynomial expansion-based low-complexity channel estimators and precoders are proposed for massive MIMO-GFDM systems. Interference-free pilots are used in the minimum mean square error (MMSE) channel estimation to combat the influence of non-orthogonality between subcarriers in GFDM. The cubic computational complexity can be reduced to square order by using the polynomial expansion technique to approximate the matrix inverses in the conventional MMSE estimation and precoding. In addition, we derive performance limits in terms of the mean square error (MSE) for the proposed estimators, which can be a useful tool to predict estimators performance in the high Eₛ/N₀ region. A Cramér-Rao lower bound (CRLB) is derived for our system model and acts as a benchmark for the estimators. The computational complexity of the proposed channel estimators and precoders, and the impacts of the polynomial degree are also investigated. Finally, we analyze the error probability performance of LDPC coded GFDM systems. We first derive the initial log-likelihood ratio (LLR) expressions that are used in the sum-product algorithm (SPA) decoder. Then, based on the decoding threshold, we estimate the frame error rate (FER) in the low E[subscript b]/N₀ region by using the observed BER to model the channel variations. In addition, a lower bound on the FER of the system is also proposed based on absorbing sets. This lower bound can act as an estimate of the FER in the high E[subscript b]/N₀ region if the absorbing set used is dominant and its multiplicity is known. The quantization scheme also has an important impact on the FER and BER performances. Randomly constructed and array-based LDPC codes are used to support the performance analyses. For all these three aspects, software-based simulations and calculations are carried out to obtain related numerical results, which verify our proposed methods