Mathematical optimization is applicable to nearly every scientific discipline. This thesis specifically focuses on optimization applications to digital communications and signal processing. Within the digital communications framework, the channel encoder attempts to encode a message from a source (the sender) in such a way that the channel decoder can utilize the encoding to correct errors in the message caused by the transmission over the channel. Low-density parity-check (LDPC) codes are an especially popular code for this purpose. Following the channel encoder in the digital communications framework, the modulator converts the encoded message bits to a physical waveform, which is sent over the channel and converted back to bits at the demodulator. The modulator and demodulator present special challenges for what is known as the two-antenna problem. The main results of this work are two algorithms related to the development of optimization methods for LDPC codes and the two-antenna problem. Current methods for optimization of LDPC codes analyze the degree distribution pair asymptotically as block length approaches infinity. This effectively ignores the discrete nature of the space of valid degree distribution pairs for LDPC codes of finite block length. While large codes are likely to conform reasonably well to the infinite block length analysis, shorter codes have no such guarantee. Chapter 2 more thoroughly introduces LDPC codes, and Chapter 3 presents and analyzes an algorithm for completely enumerating the space of all valid degree distribution pairs for a given block length, code rate, maximum variable node degree, and maximum check node degree. This algorithm is then demonstrated on an example LDPC code of finite block length. Finally, we discuss how the result of this algorithm can be utilized by discrete optimization routines to form novel methods for the optimization of small block length LDPC codes. In order to solve the two-antenna problem, which is introduced in greater detail in Chapter 2, it is necessary to obtain reliable estimates of the timing offset and channel gains caused by the transmission of the signal through the channel. The timing offset estimator can be formulated as an optimization problem, and an optimization method used to solve it was previously developed. However, this optimization method does not utilize gradient information, and as a result is inefficient. Chapter 4 presents and analyzes an improved gradient-based optimization method that solves the two-antenna problem much more efficiently.

With the ever-growing demand for higher data rate in next generation communication systems, researchers are pushing the limits of the existing architecture. Due to the stochastic nature of communication channels, most systems use some form of adaptive methods to adjust the transmitting parameters and allocation of resources in order to overcome channel variations and achieve optimal throughput. We will study four cases of adaptive transmission and dynamic resource allocation in collaborative systems that are practically significant. Firstly, we study hybrid automatic repeat request (HARQ) techniques that are widely used to handle transmission failures. We propose HARQ policies that improve system throughput and are suitable for point-to-point, two-hop relay, and multi-user broadcast systems. Secondly, we study the effect of having bits of mixed SNR qualities in finite length codewords. We prove that by grouping bits according to their reliability so that each codeword contains homogeneous bit qualities, the finite blocklength capacity of the system is increased. Thirdly, we study the routing and resource allocation problem in multiple collaborative networks. We propose an algorithm that enables collaboration between networks which needs little to no side information shared across networks, but rather infers necessary information from the transmissions. The collaboration between networks provides a significant gain in overall throughput compared to selfish networks. Lastly, we present an algorithm that allocates disjoint transmission channels for our cognitive radio network in the DARPA Spectrum Collaboration Challenge (SC2). This algorithm uses the real-time spectrogram knowledge perceived by the radios and allocates channels adaptively in a crowded spectrum shared with other collaborative networks.

No description available.

Dans cette thèse, nous nous sommes intéressés au problème de codage de sources avec information adjacente au décodeur seulement. Plus précisément, nous avons considéré le cas où la distribution jointe entre la source et l'information adjacente n'est pas bien connue. Dans ce contexte, pour un problème de codage sans pertes, nous avons d'abord effectué une analyse de performance à l'aide d'outils de la théorie de l'information. Nous avons ensuite proposé un schéma de codage pratique efficace malgré le manque de connaissance sur la distribution de probabilité jointe. Ce schéma de codage s'appuie sur des codes LDPC non-binaires et sur un algorithme de type Espérance-Maximisation. Le problème du schéma de codage proposé, c'est que les codes LDPC non-binaires utilisés doivent être performants. C'est à dire qu'ils doivent être construits à partir de distributions de degrés qui permettent d'atteindre un débit proche des performances théoriques. Nous avons donc proposé une méthode d'optimisation des distributions de degrés des codes LDPC. Enfin, nous nous sommes intéressés à un cas de codage avec pertes. Nous avons supposé que le modèle de corrélation entre la source et l'information adjacente était décrit par un modèle de Markov caché à émissions Gaussiennes. Pour ce modèle, nous avons également effectué une analyse de performance, puis nous avons proposé un schéma de codage pratique. Ce schéma de codage s'appuie sur des codes LDPC non-binaires et sur une reconstruction MMSE. Ces deux composantes exploitent la structure avec mémoire du modèle de Markov caché. / In this thesis, we considered the problem of source coding with side information available at the decoder only. More in details, we considered the case where the joint distribution between the source and the side information is not perfectly known. In this context, we performed a performance analysis of the lossless source coding scheme. This performance analysis was realized from information theory tools. Then, we proposed a practical coding scheme able to deal with the uncertainty on the joint probability distribution. This coding scheme is based on non-binary LDPC codes and on an Expectation-Maximization algorithm. For this problem, a key issue is to design efficient LDPC codes. In particular, good code degree distributions have to be selected. Consequently, we proposed an optimization method for the selection of good degree distributions. To finish, we considered a lossy coding scheme. In this case, we assumed that the correlation channel between the source and the side information is described by a Hidden Markov Model with Gaussian emissions. For this model, we performed again some performance analysis and proposed a practical coding scheme. The proposed scheme is based on non-binary LDPC codes and on MMSE reconstruction using an MCMC method. In our solution, these two components are able to exploit the memory induced by the Hidden Markov model.

Depuis deux décennies, la révolution technologique est avant tout numérique entrainant une forte croissance de la quantité de données à stocker. Le rythme de cette croissance est trop importante pour les solutions de stockage matérielles, provoquant une augmentation du coût de l'octet. Il est donc nécessaire d'apporter une amélioration des solutions de stockage ce qui passera par une augmentation de la taille des réseaux et par la diminution des copies de sauvegarde dans les centres de stockage de données. L'objet de cette thèse est d'étudier l'utilisation des codes en graphe dans les réseaux de stockage de donnée. Nous proposons un nouvel algorithme combinant construction de codes en graphe et allocation des noeuds de ce code sur le réseau. Cet algorithme permet d'atteindre les hautes performances des codes MDS en termes de rapport entre le nombre de disques de parité et le nombre de défaillances simultanées pouvant être corrigées sans pertes (noté R). Il bénéficie également des propriétés de faible complexité des codes en graphe pour l'encodage et la reconstruction des données. De plus, nous présentons une étude des codes LDPC Spatiallement-Couplés permettant d'anticiper le comportement de leur décodage pour les applications de stockage de données.Il est généralement nécessaire de faire des compromis entre différents paramètres lors du choix du code correcteur d'effacement. Afin que ce choix se fasse avec un maximum de connaissances, nous avons réalisé deux études théoriques comparatives pour compléter l'état de l'art. La première étude s'intéresse à la complexité de la mise à jour des données dans un réseau dynamique établi et déterminons si les codes linéaires utilisés ont une complexité de mise à jour optimale. Dans notre seconde étude, nous nous sommes intéressés à l'impact sur la charge du réseau de la modification des paramètres du code correcteur utilisé. Cette opération peut être réalisée lors d'un changement du statut du fichier (passage d'un caractère hot à cold par exemple) ou lors de la modification de la taille du réseau. L'ensemble de ces études, associé au nouvel algorithme de construction et d'allocation des codes en graphe, pourrait mener à la construction de réseaux de stockage dynamiques, flexibles avec des algorithmes d'encodage et de décodage peu complexes. / For two decades, the numerical revolution has been amplified. The spread of digital solutions associated with the improvement of the quality of these products tends to create a growth of the amount of data stored. The cost per Byte reveals that the evolution of hardware storage solutions cannot follow this expansion. Therefore, data storage solutions need deep improvement. This is feasible by increasing the storage network size and by reducing data duplication in the data center. In this thesis, we introduce a new algorithm that combines sparse graph code construction and node allocation. This algorithm may achieve the highest performance of MDS codes in terms of the ratio R between the number of parity disks and the number of failures that can be simultaneously reconstructed. In addition, encoding and decoding with sparse graph codes helps lower the complexity. By this algorithm, we allow to generalize coding in the data center, in order to reduce the amount of copies of original data. We also study Spatially-Coupled LDPC (SC-LDPC) codes which are known to have optimal asymptotic performance over the binary erasure channel, to anticipate the behavior of these codes decoding for distributed storage applications. It is usually necessary to compromise between different parameters for a distributed storage system. To complete the state of the art, we include two theoretical studies. The first study deals with the computation complexity of data update and we determine whether linear code used for data storage are update efficient or not. In the second study, we examine the impact on the network load when the code parameters are changed. This can be done when the file status changes (from a hot status to a cold status for example) or when the size of the network is modified by adding disks. All these studies, combined with the new algorithm for sparse graph codes, could lead to the construction of new flexible and dynamical networks with low encoding and decoding complexities.

In this work, a multilayer security solution for digital communication systems is provided by considering the joint effects of physical-layer security channel codes with application-layer cryptography. We address two problems: first, the cryptanalysis of error-prone ciphertext; second, the design of a practical physical-layer security coding scheme. To our knowledge, the cryptographic attack model of the noisy-ciphertext attack is a novel concept. The more traditional assumption that the attacker has the ciphertext is generally assumed when performing cryptanalysis. However, with the ever-increasing amount of viable research in physical-layer security, it now becomes essential to perform the analysis when ciphertext is unreliable. We do so for the simple substitution cipher using an information-theoretic framework, and for stream ciphers by characterizing the success or failure of fast-correlation attacks when the ciphertext contains errors. We then present a practical coding scheme that can be used in conjunction with cryptography to ensure positive error rates in an eavesdropper's observed ciphertext, while guaranteeing error-free communications for legitimate receivers. Our codes are called stopping set codes, and provide a blanket of security that covers nearly all possible system configurations and channel parameters. The codes require a public authenticated feedback channel. The solutions to these two problems indicate the inherent strengthening of security that can be obtained by confusing an attacker about the ciphertext, and then give a practical method for providing the confusion. The aggregate result is a multilayer security solution for transmitting secret data that showcases security enhancements over standalone cryptography.

The VLSI implementation complexity of a low density parity check (LDPC) decoder is largely influenced by the interconnect and the storage requirements. This dissertation presents the decoder architectures for regular and irregular LDPC codes that provide substantial gains over existing academic and commercial implementations. Several structured properties of LDPC codes and decoding algorithms are observed and are used to construct hardware implementation with reduced processing complexity. The proposed architectures utilize an on-the-fly computation paradigm which permits scheduling of the computations in a way that the memory requirements and re-computations are reduced. Using this paradigm, the run-time configurable and multi-rate VLSI architectures for the rate compatible array LDPC codes and irregular block LDPC codes are designed. Rate compatible array codes are considered for DSL applications. Irregular block LDPC codes are proposed for IEEE 802.16e, IEEE 802.11n, and IEEE 802.20. When compared with a recent implementation of an 802.11n LDPC decoder, the proposed decoder reduces the logic complexity by 6.45x and memory complexity by 2x for a given data throughput. When compared to the latest reported multi-rate decoders, this decoder design has an area efficiency of around 5.5x and energy efficiency of 2.6x for a given data throughput. The numbers are normalized for a 180nm CMOS process. Properly designed array codes have low error floors and meet the requirements of magnetic channel and other applications which need several Gbps of data throughput. A high throughput and fixed code architecture for array LDPC codes has been designed. No modification to the code is performed as this can result in high error floors. This parallel decoder architecture has no routing congestion and is scalable for longer block lengths. When compared to the latest fixed code parallel decoders in the literature, this design has an area efficiency of around 36x and an energy efficiency of 3x for a given data throughput. Again, the numbers are normalized for a 180nm CMOS process. In summary, the design and analysis details of the proposed architectures are described in this dissertation. The results from the extensive simulation and VHDL verification on FPGA and ASIC design platforms are also presented.

The importance of quantum error correction in paving the way to build a practical quantum computer is no longer in doubt. Despite the large body of literature in quantum coding theory, many important questions, especially those centering on the issue of "good codes" are unresolved. In this dissertation the dominant underlying theme is that of constructing good quantum codes. It approaches this problem from three rather different but not exclusive strategies. Broadly, its contribution to the theory of quantum error correction is threefold. Firstly, it extends the framework of an important class of quantum codes - nonbinary stabilizer codes. It clarifies the connections of stabilizer codes to classical codes over quadratic extension fields, provides many new constructions of quantum codes, and develops further the theory of optimal quantum codes and punctured quantum codes. In particular it provides many explicit constructions of stabilizer codes, most notably it simplifies the criteria by which quantum BCH codes can be constructed from classical codes. Secondly, it contributes to the theory of operator quantum error correcting codes also called as subsystem codes. These codes are expected to have efficient error recovery schemes than stabilizer codes. Prior to our work however, systematic methods to construct these codes were few and it was not clear how to fairly compare them with other classes of quantum codes. This dissertation develops a framework for study and analysis of subsystem codes using character theoretic methods. In particular, this work established a close link between subsystem codes and classical codes and it became clear that the subsystem codes can be constructed from arbitrary classical codes. Thirdly, it seeks to exploit the knowledge of noise to design efficient quantum codes and considers more realistic channels than the commonly studied depolarizing channel. It gives systematic constructions of asymmetric quantum stabilizer codes that exploit the asymmetry of errors in certain quantum channels. This approach is based on a Calderbank- Shor-Steane construction that combines BCH and finite geometry LDPC codes.

Traditional solutions to information security in communication systems act in the application layer and are oblivious to the effects in the physical layer. Physical-layer security methods, of which information-theoretic security is a special case, try to extract security from the random effects in the physical layer. In information-theoretic security, there are two asymptotic notions of secrecy---weak and strong secrecy This dissertation investigates the problem of information-theoretic strong secrecy on the binary erasure wiretap channel (BEWC) with a specific focus on designing practical codes. The codes designed in this work are based on analysis and techniques from error-correcting codes. In particular, the dual codes of certain low-density parity-check (LDPC) codes are shown to achieve strong secrecy in a coset coding scheme. First, we analyze the asymptotic block-error rate of short-cycle-free LDPC codes when they are transmitted over a binary erasure channel (BEC) and decoded using the belief propagation (BP) decoder. Under certain conditions, we show that the asymptotic block-error rate falls according to an inverse square law in block length, which is shown to be a sufficient condition for the dual codes to achieve strong secrecy. Next, we construct large-girth LDPC codes using algorithms from graph theory and show that the asymptotic bit-error rate of these codes follow a sub-exponential decay as the block length increases, which is a sufficient condition for strong secrecy. The secrecy rates achieved by the duals of large-girth LDPC codes are shown to be an improvement over that of the duals of short-cycle-free LDPC codes.

No description available.