Der Einsatz von neuronalen Netzen in Edge-Geräten und deren Einbindung in unser tägliches Leben findet immer mehr Beachtung. Ihre hohen Rechenkosten machen jedoch viele eingebettete Anwendungen zu einer Herausforderung. Das Hauptziel meiner Doktorarbeit ist es, einen Beitrag zur Lösung dieses Dilemmas zu leisten: die Optimierung neuronaler Netze und die Entwicklung entsprechender neuronaler Verarbeitungseinheiten für Endgeräte. Diese Arbeit nahm die algorithmische Forschung als Ausgangspunkt und wandte dann deren Ergebnisse an, um das Architekturdesign von Neural Processing Units (NPUs) zu verbessern. Die Optimierung neuronaler Netzwerkmodelle begann mit der Quantisierung neuronaler Netzwerke mit einfacher Präzision und entwickelte sich zu gemischter Präzision. Die Entwicklung der NPU-Architektur folgte den Erkenntnissen der Algorithmusforschung, um ein Hardware/Software Co-Design zu erreichen. Darüber hinaus wurde ein neuartiger Ansatz zur gemeinsamen Entwicklung von Hardware und Software vorgeschlagen, um das Prototyping und die Leistungsbewertung von NPUs zu beschleunigen. Dieser Ansatz zielt auf die frühe Entwicklungsphase ab. Er hilft Entwicklern, sich auf das Design und die Optimierung von NPUs zu konzentrieren und verkürzt den Entwicklungszyklus erheblich. Im Abschlussprojekt wurde ein auf maschinellem Lernen basierender Ansatz angewendet, um die Rechen- und Speicherressourcen der NPU zu erkunden and optimieren. Die gesamte Arbeit umfasst mehrere verschiedene Bereiche, von der Algorithmusforschung bis zum Hardwaredesign. Sie alle arbeiten jedoch an der Verbesserung der Inferenz-Effizienz neuronaler Netze. Die Optimierung der Algorithmen zielt insbesondere darauf ab, den Speicherbedarf und die Rechenkosten von neuronalen Netzen zu verringern. Das NPU-Design hingegen konzentriert sich auf die Verbesserung der Nutzung von Hardwareressourcen. Der vorgeschlagene Ansatz zur gemeinsamen Entwicklung von Software und Hardware verkürzt den Entwurfszyklus und beschleunigt die Entwurfsiterationen. Die oben dargestellte Reihenfolge entspricht dem Aufbau dieser Dissertation. Jedes Kapitel ist einem Thema gewidmet und umfasst relevante Forschungsarbeiten, Methodik und Versuchsergebnisse.:1 Introduction 2 Convolutional Neural Networks 2.1 Convolutional layer 2.1.1 Padding 2.1.2 Convolution 2.1.3 Batch Normalization 2.1.4 Nonlinearity 2.2 Pooling Layer 2.3 Fully Connected Layer 2.4 Characterization 2.4.1 Composition of Operations and Parameters 2.4.2 Arithmetic Intensity 2.5 Optimization 3 Quantization with Double-Stage Squeeze-and-Threshold 19 3.1 Overview 3.1.1 Binarization 3.1.2 Multi-bit Quantization 3.2 Quantization of Convolutional Neural Networks 3.2.1 Quantization Scheme 3.2.2 Operator fusion of Conv2D 3.3 Activation Quantization with Squeeze-and-Threshold 3.3.1 Double-Stage Squeeze-and-Threshold 3.3.2 Inference Optimization 3.4 Experiment 3.4.1 Ablation Study of Squeeze-and-Threshold 3.4.2 Comparison with State-of-the-art Methods 3.5 Summary 4 Low-Precision Neural Architecture Search 39 4.1 Overview 4.2 Differentiable Architecture Search 4.2.1 Gumbel Softmax 4.2.2 Disadvantage and Solution 4.3 Low-Precision Differentiable Architecture Search 4.3.1 Convolution Sharing 4.3.2 Forward-and-Backward Scaling 4.3.3 Power Estimation 4.3.4 Architecture of Supernet 4.4 Experiment 4.4.1 Effectiveness of solutions to the dominance problem 4.4.2 Softmax and Gumbel Softmax 4.4.3 Optimizer and Inverted Learning Rate Scheduler 4.4.4 NAS Method Evaluation 4.4.5 Searched Model Analysis 4.4.6 NAS Cost Analysis 4.4.7 NAS Training Analysis 4.5 Summary 5 Configurable Sparse Neural Processing Unit 65 5.1 Overview 5.2 NPU Architecture 5.2.1 Buffer 5.2.2 Reshapeable Mixed-Precision MAC Array 5.2.3 Sparsity 5.2.4 Post Process Unit 5.3 Mapping 5.3.1 Mixed-Precision MAC 5.3.2 MAC Array 5.3.3 Support of Other Operation 5.3.4 Configurability 5.4 Experiment 5.4.1 Performance Analysis of Runtime Configuration 5.4.2 Roofline Performance Analysis 5.4.3 Mixed-Precision 5.4.4 Comparison with Cortex-M7 5.5 Summary 6 Agile Development and Rapid Design Space Exploration 91 6.1 Overview 6.1.1 Agile Development 6.1.2 Design Space Exploration 6.2 Agile Development Infrastructure 6.2.1 Chisel Backend 6.2.2 NPU Software Stack 6.3 Modeling and Exploration 6.3.1 Area Modeling 6.3.2 Performance Modeling 6.3.3 Layered Exploration Framework 6.4 Experiment 6.4.1 Efficiency of Agile Development Infrastructure 6.4.2 Effectiveness of Agile Development Infrastructure 6.4.3 Area Modeling 6.4.4 Performance Modeling 6.4.5 Rapid Exploration and Pareto Front 6.5 Summary 7 Summary and Outlook 123 7.1 Summary 7.2 Outlook A Appendix of Double-Stage ST Quantization 127 A.1 Training setting of ResNet-18 in Table 3.3 A.2 Training setting of ReActNet in Table 3.4 A.3 Training setting of ResNet-18 in Table 3.4 A.4 Pseudocode Implementation of Double-Stage ST B Appendix of Low-Precision Neural Architecture Search 131 B.1 Low-Precision NAS on CIFAR-10 B.2 Low-Precision NAS on Tiny-ImageNet B.3 Low-Precision NAS on ImageNet Bibliography 137 / Deploying neural networks on edge devices and bringing them into our daily lives is attracting more and more attention. However, its expensive computational cost makes many embedded applications daunting. The primary objective of my doctoral studies is to make contributions towards resolving this predicament: optimizing neural networks and designing corresponding efficient neural processing units for edge devices. This work took algorithmic research, specifically the optimization of deep neural networks, as a starting point and then applied its findings to steer the architecture design of Neural Processing Units (NPUs). The optimization of neural network models started with single precision neural network quantization and progressed to mixed precision. The NPU architecture development followed the algorithmic research findings to achieve hardware/software co-design. Furthermore, a new approach to hardware and software co-development was introduced, aimed at expediting the prototyping and performance assessment of NPUs. This approach targets early-stage development. It helps developers to focus on the design and optimization of NPUs and significantly shortens the development cycle. In the final project, a machine learning-based approach was applied to explore and optimize the computational and memory resources of the NPU. The entire work covers several different areas, from algorithmic research to hardware design. But they all work on improving the inference efficiency of neural networks. Specifically, algorithm optimization aims to reduce the memory footprint and computational cost of neural networks. The NPU design, on the other hand, focuses on improving the utilization of hardware resources. The proposed software and hardware co-development approach shortens the design cycle and speeds up the design iteration. The order presented above corresponds to the structure of this dissertation. Each chapter corresponds to a topic and covers relevant research, methodology, and experimental results.:1 Introduction 2 Convolutional Neural Networks 2.1 Convolutional layer 2.1.1 Padding 2.1.2 Convolution 2.1.3 Batch Normalization 2.1.4 Nonlinearity 2.2 Pooling Layer 2.3 Fully Connected Layer 2.4 Characterization 2.4.1 Composition of Operations and Parameters 2.4.2 Arithmetic Intensity 2.5 Optimization 3 Quantization with Double-Stage Squeeze-and-Threshold 19 3.1 Overview 3.1.1 Binarization 3.1.2 Multi-bit Quantization 3.2 Quantization of Convolutional Neural Networks 3.2.1 Quantization Scheme 3.2.2 Operator fusion of Conv2D 3.3 Activation Quantization with Squeeze-and-Threshold 3.3.1 Double-Stage Squeeze-and-Threshold 3.3.2 Inference Optimization 3.4 Experiment 3.4.1 Ablation Study of Squeeze-and-Threshold 3.4.2 Comparison with State-of-the-art Methods 3.5 Summary 4 Low-Precision Neural Architecture Search 39 4.1 Overview 4.2 Differentiable Architecture Search 4.2.1 Gumbel Softmax 4.2.2 Disadvantage and Solution 4.3 Low-Precision Differentiable Architecture Search 4.3.1 Convolution Sharing 4.3.2 Forward-and-Backward Scaling 4.3.3 Power Estimation 4.3.4 Architecture of Supernet 4.4 Experiment 4.4.1 Effectiveness of solutions to the dominance problem 4.4.2 Softmax and Gumbel Softmax 4.4.3 Optimizer and Inverted Learning Rate Scheduler 4.4.4 NAS Method Evaluation 4.4.5 Searched Model Analysis 4.4.6 NAS Cost Analysis 4.4.7 NAS Training Analysis 4.5 Summary 5 Configurable Sparse Neural Processing Unit 65 5.1 Overview 5.2 NPU Architecture 5.2.1 Buffer 5.2.2 Reshapeable Mixed-Precision MAC Array 5.2.3 Sparsity 5.2.4 Post Process Unit 5.3 Mapping 5.3.1 Mixed-Precision MAC 5.3.2 MAC Array 5.3.3 Support of Other Operation 5.3.4 Configurability 5.4 Experiment 5.4.1 Performance Analysis of Runtime Configuration 5.4.2 Roofline Performance Analysis 5.4.3 Mixed-Precision 5.4.4 Comparison with Cortex-M7 5.5 Summary 6 Agile Development and Rapid Design Space Exploration 91 6.1 Overview 6.1.1 Agile Development 6.1.2 Design Space Exploration 6.2 Agile Development Infrastructure 6.2.1 Chisel Backend 6.2.2 NPU Software Stack 6.3 Modeling and Exploration 6.3.1 Area Modeling 6.3.2 Performance Modeling 6.3.3 Layered Exploration Framework 6.4 Experiment 6.4.1 Efficiency of Agile Development Infrastructure 6.4.2 Effectiveness of Agile Development Infrastructure 6.4.3 Area Modeling 6.4.4 Performance Modeling 6.4.5 Rapid Exploration and Pareto Front 6.5 Summary 7 Summary and Outlook 123 7.1 Summary 7.2 Outlook A Appendix of Double-Stage ST Quantization 127 A.1 Training setting of ResNet-18 in Table 3.3 A.2 Training setting of ReActNet in Table 3.4 A.3 Training setting of ResNet-18 in Table 3.4 A.4 Pseudocode Implementation of Double-Stage ST B Appendix of Low-Precision Neural Architecture Search 131 B.1 Low-Precision NAS on CIFAR-10 B.2 Low-Precision NAS on Tiny-ImageNet B.3 Low-Precision NAS on ImageNet Bibliography 137

[pt] Eficiência energética e espectral são características importantes para comunicações militares e internet das coisas (IoT). Nesta tese, métodos e sistemas de quantização de 1-bit com modulação de fase contínua (CPM) são estudados e propostos para resolver as necessidades de sistemas de comunicações modernos com baixo consumo energético. Nesse contexto, o método de superamostragem em relação a duração de um símbolo é promissor, pois a informação está contida ao longo da transição de fase de sinais CPM, que não são estritamente limitados em banda. Consequentemente, a perda de taxa alcançável causada pela quantização de 1-bit pode ser reduzida consideravelmente, até mesmo para esquemas com maior ordem de modulação. Este estudo investiga diferentes abordagens para melhorar o desempenho do modelo de sistema proposto. Um esquema de codificação de canal é projetado com mapeamento de bits adaptado ao problema de quantização grosseira, fazendo uso de um soft-in soft-out (SISO) turbo receiver. Formas de onda CPM com duração de símbolo significamente menor que o inverso da banda do sinal são propostas, nomeadas de faster-than-Nyquist CPM. Um fator maior de superamostragem é aplicado com uma estratégia de seleção de amostras em um modelo de amostragem adaptativa. Finalmente, resultados numéricos confirmam melhor desempenho em taxa de erro de bit, eficiência espectral e taxa alcançável para os métodos propostos, em comparação às técnicas recentemente utilizadas. / [en] Energy and spectral efficiency are appealing features for military communications and internet of things (IoT). On this thesis, systems and schemes with 1-bit quantization and continuous phase modulation (CPM) are studied and proposed to address the needs for modern and power efficient communications. In this context, oversampling with respect to the symbol duration is promising because the information is conveyed in the phase transitions of the CPM signals, which are not strictly bandlimited. With this, the loss in achievable rate caused by the coarse quantization can be greatly reduced, even for higher order modulation schemes. This study investigates different approaches to enhancing the performance of the proposed system model. A channel coding scheme is designed with a tailored bit mapping, by means of employing a soft-in soft-out (SISO) turbo receiver. CPM waveforms with symbol durations significantly shorter than the inverse of the signal bandwidth are proposed, termed faster-than-Nyquist CPM. Higher oversampling is applied with a sample selection strategy for a nonuniform adaptive oversampling model. Finally, numerical results confirm better performance on bit error rate, spectral efficiency and achievable rate for the proposed methods in comparison with state of the art techniques.

Die AdS/CFT-Dualität zwischen N=4 SYM und dem AdS_5 × S^5 Superstring zeigt Quanten-Integrabilität im planaren Limes und erlaubte die Konstruktion mächtiger Methoden, welche das Spektrale Problem zu lösen scheinen. Unser Verständnis der direkten Quantisierung des AdS_5 × S^5 Superstrings ist jedoch weiterhin unbefriedigend und besonders das Spektrum kurzer Stringzustände war bisher nur in führender Ordnung in starker ''t Hooft-Kopplung bekannt. In dieser Arbeit untersuchen wir verschiedene Methoden der perturbativen Quantisierung kurzer Strings über die führende Ordnung hinaus, wodurch wir uns auch einen besseres Verständnis der vorhandenen Quanten-Symmetrien erhoffen. Wir fokusieren auf die niedrigst angeregten Stringzustände, dual zum Konishi-Supermultiplet, und begutachten kritisch eine angeblichen Berechnung der Konishi anomalen Skalendimension im Pure-Spinor-Superstring-Formalismus. Als nächstes betrachten wir den bosonischen AdS_5 × S^5 String in statischer Eichung und konstruieren eine sog. Einzelmoden-Stringlösung, eine Veralgemeinerung des pulsierenden Strings durch unbeschränkte Nullmoden. Diese ist klassisch integrabel und quanteninvariant unter den Isometrien SO(2,4) × SO(6). Mögliche Korrekturen der vernachlässigten Supersymmetrie werden heuristisch berücksichtigt, wodurch die ersten Quantenkorrekturen der Konishi anomale Skalendimension reproduzieren werden. Wir implementieren statische Eichung für den AdS_5 × S^5 Superstring und finden elegante Ausdrücke für die Lagrangedichte und Superladungen. Unter Beschränkung auf das Superteilchen finden wir auf zwei unterschiedliche Arten kanonische Koordinaten in quadratischer Ordnung in Fermionen. Schließlich betrachten wir eine weitere Quantisierungsmethode: Da der Einzelmoden-String die SO(2,4) × SO(6)-Bahn des pulsierenden Strings ist, wenden wir Bahn-Methoden-Quantisierung auf das Teilchen und Spinning Strings in bosonischem AdS_3 × S^3 an und erhalten konsistente Ergebnisse für die Spektra. / The initial AdS/CFT duality pair, the duality between N=4 SYM and the AdS_5 × S^5 superstring, appears to enjoy quantum integrability in the planar limit, which allowed to devise powerful methods ostensibly solving the spectral problem. However, quantization of the AdS_5 × S^5 superstring from first principles is still an open question and especially the spectrum of short string states has previously been derived only at leading order in large ''t Hooft coupling. In this thesis we investigate possible routes to quantize short string states perturbatively beyond the leading order, where equally our aim is to gain better appreciation of the quantum symmetries at play. A prominent role is played by the lowest excited string states, dual to the Konishi supermultiplet, and we start by reviewing critically an asserted derivation of the Konishi anomalous dimension in the setup of pure spinor string theory. Next, we constrain ourselves to bosonic AdS_5 × S^5 String in static gauge, where we construct a so-called single-mode string solution, a generalization of the pulsating string allowing for unconstrained zero-modes. This solution shows classical integrability and invariance under the isometries SO(2,4) × SO(6) at the quantum level. Arguing heuristically about the effects of supersymmetry, we indeed recover the first non-trivial quantum correction to the Konishi anomalous dimension. We continue by implementing static gauge for the full AdS_5 × S^5 superstring and find elegant expressions for the Lagrangian density and the supercharges. We then constrain our interest to the superparticle and, using two different methods, find canonical coordinates at quadratic order in fermions. We conclude by exploring another quantization scheme: As the single-mode string is nothing but the SO(2,4) × SO(6) orbit of the pulsating string, we apply orbit method quantization to the particle and spinning string solutions in bosonic AdS_3 × S^3 yielding consistent results for the spectra.

1-bit analog-to-digital conversion is very attractive for low-complexity communications receivers. A major drawback is, however, the small spectral efficiency when sampling at symbol rate. This can be improved through oversampling by exploiting the signal distortion caused by the transmission channel. This paper analyzes the achievable data rate of band-limited communications channels that are subject to additive noise and inter-symbol-interference with 1-bit quantization and oversampling at the receiver. It is shown that not only the channel noise but also the inter-symbol-interference can be exploited to benefit from oversampling.

Submitted by Maike Costa (maiksebas@gmail.com) on 2016-03-15T12:16:24Z No. of bitstreams: 1 arquivototal.pdf: 5169144 bytes, checksum: 66d534e3f0c0c59bf5d35a45290fa390 (MD5) / Made available in DSpace on 2016-03-15T12:16:24Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 5169144 bytes, checksum: 66d534e3f0c0c59bf5d35a45290fa390 (MD5) Previous issue date: 2015-02-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In this thesis, we propose an experimental setup for the study of Landau quantization and associated effects in a two-dimensional ultracold atomic gas. Gauge fields can emerge in the equation of motion for the optically addressed ultracold atoms. To this end, spatially dependent dark states are necessary for the internal states of the atoms. A tripod level scheme yields two degenerate dark states which can leads to either an Abelian U(1) U(1) gauge field or a non-Abelian SU(2) gauge field. Using a suitable laser configuration, we obtain a uniform U(1) U(1) magnetic field which causes the atoms organize themselves in Landau levels. The strength of the effective magnetic field depends on the relative intensity of the lasers beams at the atomic cloud. We estimate the degeneracy of the energy levels for an atomic gas formed by atoms of 87Rb. In addition, we establish the experimental conditions to reach the lowest Landau level regime. In the zero-temperature limit, we realize the emergence of magnetic oscillations in the atomic energy and its derivative as function of the inverse of the effective magnetic field (de Haas van Alphen effect). The period of the de Haas van Alphen oscillation allow us to determine area of the Fermi circle for the atomic gas via an Onsager-like relation. We also show that detuning the a laser from the two-photon resonance we generate a parabolic scalar potential that laterally confines the atoms. As a consequence, the Landau levels degeneracy is removed, since the energy spectrum depends explicitly on the transverse atomic momentum. We show that the Landau levels presents a reminiscent degeneracy when the boundaries conditions are considered. The residual degeneracy occurs when different energy levels overlap. We map the residual degeneracy points as a function of the effective magnetic field. Finally, we present an experimental scheme for observing the spin Hall effect for ultracold atoms in a tripod configuration. / Nesta tese, propomos um arranjo experimental para o estudo da quantização de Landau e efeitos associados em um gás atômico ultrafrio bidimensional. Campos de calibre podem surgir na equação de movimento para átomos ultrafrios oticamente vestidos. Para que isto ocorra, estados escuros espacialmente dependentes são necessários a partir dos estados internos dos átomos. Átomos numa configuração de níveis de energia do tipo tripod produzem dois estados escuros degenerados, que podem levar a campos de calibre Abelianos U(1) U(1) ou não-Abelianos SU(2). Utilizando uma configuração adequada de lasers, mostramos que é possível se produzir um campo magnético sintético uniforme U(1) U(1) que atua nos átomos neutros fazendo-os se organizarem em níveis de Landau. A intensidade do campo efetivo depende da intensidade relativa dos feixes de luz na nuvem atômica. Estimamos a degenerescência dos níveis de energia para um gás atômico formado por átomos de 87Rb e estabelecemos as condições experimentais para que seja atingido o regime em que todos os átomos populam unicamente o nível de Landau menos energético. Considerando o limite de temperatura nula, verificamos o surgimento de oscilações magnéticas na energia e em sua derivada como uma função do inverso do campo magnético efetivo (efeito de Haas van Alphen). O período da oscilação magnética nos permite determinar a área do círculo de Fermi para o gás atômico através de uma expressão similar a de Onsager para sistemas eletrônicos. Mostramos também que dessintonizando um dos lasers em relação à ressonância de dois fótons geramos um potencial escalar parabólico que faz com os átomos sejam lateralmente confinados. Isto resulta na remoção da degenerescência dos níveis de Landau, uma vez que a energia depende explicitamente do momento atômico transverso. Demonstramos que, aplicando condições periódicas de contorno ao sistema, temos o surgimento de uma degenerescência residual. A degenerescência remanescente ocorre quando diferentes níveis de energia se superpõem. Mapeamos os pontos de degenerescência como uma função do campo magnético efetivo. Por fim, apresentamos um esquema experimental para a observação do efeito spin Hall para átomos ultrafrios em uma configuração tripod.

Made available in DSpace on 2014-12-17T14:54:59Z (GMT). No. of bitstreams: 1 FabianaTS_TESE.pdf: 1364206 bytes, checksum: 5e147adc9ca5829c7a40ed214ab434d2 (MD5) Previous issue date: 2011-12-02 / Coordena??o de Aperfei?oamento de Pessoal de N?vel Superior / In this work we use Interval Mathematics to establish interval counterparts for the main tools used in digital signal processing. More specifically, the approach developed here is oriented to signals, systems, sampling, quantization, coding and Fourier transforms. A detailed study for some interval arithmetics which handle with complex numbers is provided; they are: complex interval arithmetic (or rectangular), circular complex arithmetic, and interval arithmetic for polar sectors. This lead us to investigate some properties that are relevant for the development of a theory of interval digital signal processing. It is shown that the sets IR and R(C) endowed with any correct arithmetic is not an algebraic field, meaning that those sets do not behave like real and complex numbers. An alternative to the notion of interval complex width is also provided and the Kulisch- Miranker order is used in order to write complex numbers in the interval form enabling operations on endpoints. The use of interval signals and systems is possible thanks to the representation of complex values into floating point systems. That is, if a number x 2 R is not representable in a floating point system F then it is mapped to an interval [x;x], such that x is the largest number in F which is smaller than x and x is the smallest one in F which is greater than x. This interval representation is the starting point for definitions like interval signals and systems which take real or complex values. It provides the extension for notions like: causality, stability, time invariance, homogeneity, additivity and linearity to interval systems. The process of quantization is extended to its interval counterpart. Thereafter the interval versions for: quantization levels, quantization error and encoded signal are provided. It is shown that the interval levels of quantization represent complex quantization levels and the classical quantization error ranges over the interval quantization error. An estimation for the interval quantization error and an interval version for Z-transform (and hence Fourier transform) is provided. Finally, the results of an Matlab implementation is given / Neste trabalho utiliza-se a matem?tica intervalar para estabelecer os conceitos intervalares das principais ferramentas utilizadas em processamento digital de sinais. Mais especificamente, foram desenvolvidos aqui as abordagens intervalares para sinais, sistemas, amostragem, quantiza??o, codifica??o, transformada Z e transformada de Fourier. ? feito um estudo de algumas aritm?ticas que lidam com n?meros complexos sujeitos ? imprecis?es, tais como: aritm?tica complexa intervalar (ou retangular), aritm?tica complexa circular, aritm?tica setorial e aritm?tica intervalar polar. A partir da?, investiga-se algumas propriedades que ser?o relevantes para o desenvolvimento e aplica??o no processamento de sinais discretos intervalares. Mostra-se que nos conjuntos IR e R(C), seja qual for a aritm?tica correta adotada, n?o se tem um corpo, isto ?, os elementos desses conjuntos n?o se comportam como os n?meros reais ou complexos com suas aritm?ticas cl?ssicas e que isso ir? requerer uma avalia??o matem?tica dos conceitos necess?rios ? teoria de sinais e a rela??o desses com as aritm?ticas intervalares. Tamb?m tanto ? introduzido o conceito de amplitude intervalar complexa, como alternativa ? defini??o cl?ssica quanto utiliza-se a ordem de Kulisch-Miranker para n?meros complexos afim de que se escreva n?meros complexos intervalares na forma de intervalos, o que torna poss?vel as opera??es atrav?s dos extremos. Essa rela??o ? utilizada em propriedades de somas de intervalos de n?meros complexos. O uso de sinais e sistemas intervalares foi motivado pela representa??o intervalar num sistema de ponto flutuante abstrato. Isto ?, se um n?mero x 2 R n?o ? represent?vel em um sistema de ponto flutuante F, ele ? mapeado para um intervalo [x;x], tal que x ? o maior dos n?meros menores que x represent?vel em F e x ? o menor dos n?meros maiores que x represent?vel em F. A representa??o intervalar ? importante em processamento digital de sinais, pois a imprecis?o em dados ocorre tanto no momento da medi??o de determinado sinal, quanto no momento de process?-los computacionalmente. A partir da?, define-se sinais e sistemas intervalares que assumem tanto valores reais quanto complexos. Para isso, utiliza-se o estudo feito a respeito das aritm?ticas complexas intervalares e mostram-se algumas propriedades dos sistemas intervalares, tais como: causalidade, estabilidade, invari?ncia no tempo, homogeneidade, aditividade e linearidade. Al?m disso, foi definida a representa??o intervalar de fun??es complexas. Tal fun??o estende sistemas cl?ssicos a sistemas intervalares preservando as principais propriedades. Um conceito muito importante no processamento digital de sinais ? a quantiza??o, uma vez que a maioria dos sinais ? de natureza cont?nua e para process?-los ? necess?rio convert?-los em sinais discretos. Aqui, este processo ? descrito detalhadamente com o uso da matem?tica intervalar, onde se prop?em, inicialmente, uma amostragem intervalar utilizando as id?ias de representa??o no sistema de ponto flutuante. Posteriormente, s?o definidos n?veis de quantiza??o intervalares e, a partir da?, ? descrito o processo para se obter o sinal quantizado intervalar e s?o definidos o erro de quantiza??o intervalar e o sinal codificado intervalar. ? mostrado que os n?veis de quantiza??o intervalares representam os n?veis de quantiza??o cl?ssicos e o erro de quantiza??o intervalar representa o e erro de quantiza??o cl?ssico. Uma estimativa para o erro de quantiza??o intervalar ? apresentada. Utilizando a aritm?tica retangular e as defini??es e propriedades de sinais e sistemas intervalares, ? introduzida a transformada Z intervalar e s?o analisadas as condi??es de converg?ncia e as principais propriedades. Em particular, quando a vari?vel complexa z ? unit?ria, define-se a transformada de Fourier intervalar para sinais discretos no tempo, al?m de suas propriedades. Por fim, foram apresentadas as implementa??es dos resultados que foram feitas no software Matlab

Cette thèse comporte deux parties distinctes, toutes les deux liées à la théorie du transport optimal. Dans la première partie, nous considérons une variété riemannienne, deux mesures à densité régulière et un coût de transport, typiquement la distance géodésique quadratique et nous nous intéressons à la régularité de l’application de transport optimal. Le critère décisif à cette régularité s’avère être le signe du tenseur de Ma-Trudinger-Wang (MTW). Nous présentons tout d’abord une synthèse des travaux réalisés sur ce tenseur. Nous nous intéressons ensuite au lien entre la géométrie des lieux d’injectivité et le tenseur MTW. Nous montrons que dans de nombreux cas, la positivité du tenseur MTW implique la convexité des lieux d’injectivité. La deuxième partie de cette thèse est liée aux équations aux dérivées partielles. Certaines peuvent être considérées comme des flots gradients dans l’espace de Wasserstein W2. C’est le cas de l’équation de Keller-Segel en dimension 2. Pour cette équation nous nous intéressons au problème de quantification de la masse lors de l’explosion des solutions ; cette explosion apparaît lorsque la masse initiale est supérieure à un seuil critique Mc. Nous cherchons alors à montrer qu’elle consiste en la formation d’un Dirac de masse Mc. Nous considérons ici un modèle particulaire en dimension 1 ayant le même comportement que l’équation de Keller-Segel. Pour ce modèle nous exhibons des bassins d’attractions à l’intérieur desquels l’explosion se produit avec seulement le nombre critique de particules. Finalement nous nous intéressons au profil d’explosion : à l’aide d’un changement d’échelle parabolique nous montrons que la structure de l’explosion correspond aux points critiques d’une certaine fonctionnelle. / This thesis consists in two distinct parts both related to the optimal transport theory.The first part deals with the regularity of the optimal transport map. The key tool is the Ma-Trundinger-Wang tensor and especially its positivity. We first give a review of the known results about the MTW tensor. We then explore the geometrical consequences of the MTW tensor on the injectivity domain. We prove that in many cases the positivity of MTW implies the convexity of the injectivity domain. The second part is devoted to the behaviour of a Keller-Segel solution in the super critical case. In particular we are interested in the mass quantization problem: we wish to quantify the mass aggregated when the blow-up occurs. In order to study the behaviour of the solution we consider a particle approximation of a Keller-Segel type equation in dimension 1. We define this approximation using the gradient flow interpretation of the Keller-Segel equation and the particular structure of the Wasserstein space in dimension 1. We show two kinds of results; we first prove a stability theorem for the blow-up mechanism: we exhibit basins of attraction in which the solution blows up with only the critical number of particles. We then prove a rigidity theorem for the blow-up mechanism: thanks to a parabolic rescaling we prove that the structure of the blow-up is given by the critical points of a certain functional.

Les Processus Markoviens Déterministes par Morceaux (PDMP) ont été introduits dans la littérature par M.H.A Davis comme une classe générale de modèles stochastiques. Les PDMP forment une famille de processus markoviens qui décrivent une trajectoire déterministe ponctuée par des sauts aléatoires. Dans une première partie, les PDMP sont utilisés pour calculer des probabilités d'événements redoutés pour un cas-test de la fiabilité dynamique (le réservoir chauffé) par deux méthodes numériques différentes : la première est basée sur la résolution du système différentieldécrivant l'évolution physique du réservoir et la seconde utilise le calcul de l'espérancede la fonctionnelle d'un PDMP par un système d'équations intégro-différentielles.Dans la seconde partie, nous proposons une méthode numérique pour approcher lafonction valeur du problème d'arrêt optimal pour un PDMP. Notre approche estbasée sur la quantification de la position après saut et le temps inter-sauts de lachaîne de Markov sous-jacente au PDMP, et la discréetisation en temps adaptée à latrajectoire du processus. Ceci nous permet d'obtenir une vitesse de convergence denotre schéma numérique et de calculer un temps d'arrêt ε-optimal. / Piecewise Deterministic Markov Processes (PDMP's) have been introduced inthe literature by M.H.A. Davis as a general class of stochastics models. PDMP's area family of Markov processes involving deterministic motion punctuated by randomjumps. In a first part, PDMP's are used to compute probabilities of top eventsfor a case-study of dynamic reliability (the heated tank system) with two di#erentmethods : the first one is based on the resolution of the differential system giving thephysical evolution of the tank and the second uses the computation of the functionalof a PDMP by a system of integro-differential equations. In the second part, wepropose a numerical method to approximate the value function for the optimalstopping problem of a PDMP. Our approach is based on quantization of the post-jump location and inter-arrival time of the Markov chain naturally embedded in thePDMP, and path-adapted time discretization grids. It allows us to derive boundsfor the convergence rate of the algorithm and to provide a computable ε-optimalstopping time.

To meet the exponentially increasing traffic data driven by the rapidly growing mobile subscriptions, both industry and academia are exploring the potential of a new genera- tion (5G) of wireless technologies. An important 5G goal is to achieve high data rate. Small cells with spectrum sharing and multiple-input multiple-output (MIMO) techniques are one of the most promising 5G technologies, since it enables to increase the aggregate data rate by improving the spectral efficiency, nodes density and transmission bandwidth, respectively. However, the increased interference in the densified networks will in return limit the achievable rate performance if not properly managed. The considered setup can be modeled as MIMO interference networks, which can be classified into the K-user MIMO interference channel (IC) and the K-cell MIMO interfering broadcast channel/multiple access channel (MIMO-IBC/IMAC) according to the number of mobile stations (MSs) simultaneously served by each base station (BS). The thesis considers two physical layer (PHY) resource allocation problems that deal with the interference for both models: 1) Pareto boundary computation for the achiev- able rate region in a K-user single-stream MIMO IC and 2) grouping-based interference alignment (GIA) with optimized IA-Cell assignment in a MIMO-IMAC under limited feedback. In each problem, the thesis seeks to provide a deeper understanding of the system and novel mathematical results, along with supporting numerical examples. Some of the main contributions can be summarized as follows. It is an open problem to compute the Pareto boundary of the achievable rate region for a K-user single-stream MIMO IC. The K-user single-stream MIMO IC models multiple transmitter-receiver pairs which operate over the same spectrum simultaneously. Each transmitter and each receiver is equipped with multiple antennas, and a single desired data stream is communicated in each transmitter-receiver link. The individual achievable rates of the K users form a K-dimensional achievable rate region. To find efficient operating points in the achievable rate region, the Pareto boundary computation problem, which can be formulated as a multi-objective optimization problem, needs to be solved. The thesis transforms the multi-objective optimization problem to two single-objective optimization problems–single constraint rate maximization problem and alternating rate profile optimization problem, based on the formulations of the ε-constraint optimization and the weighted Chebyshev optimization, respectively. The thesis proposes two alternating optimization algorithms to solve both single-objective optimization problems. The convergence of both algorithms is guaranteed. Also, a heuristic initialization scheme is provided for each algorithm to achieve a high-quality solution. By varying the weights in each single-objective optimization problem, numerical results show that both algorithms provide an inner bound very close to the Pareto boundary. Furthermore, the thesis also computes some key points exactly on the Pareto boundary in closed-form. A framework for interference alignment (IA) under limited feedback is proposed for a MIMO-IMAC. The MIMO-IMAC well matches the uplink scenario in cellular system, where multiple cells share their spectrum and operate simultaneously. In each cell, a BS receives the desired signals from multiple MSs within its own cell and each BS and each MS is equipped with multi-antenna. By allowing the inter-cell coordination, the thesis develops a distributed IA framework under limited feedback from three aspects: the GIA, the IA-Cell assignment and dynamic feedback bit allocation (DBA), respec- tively. Firstly, the thesis provides a complete study along with some new improvements of the GIA, which enables to compute the exact IA precoders in closed-form, based on local channel state information at the receiver (CSIR). Secondly, the concept of IA-Cell assignment is introduced and its effect on the achievable rate and degrees of freedom (DoF) performance is analyzed. Two distributed matching approaches and one centralized assignment approach are proposed to find a good IA-Cell assignment in three scenrios with different backhaul overhead. Thirdly, under limited feedback, the thesis derives an upper bound of the residual interference to noise ratio (RINR), formulates and solves a corresponding DBA problem. Finally, numerical results show that the proposed GIA with optimized IA-Cell assignment and the DBA greatly outperforms the traditional GIA algorithm.

Dans cette thèse on s’est intéressé aux méthodes de génération de clés secrètes symétriques en utilisant la couche physique ultra large bande impulsionnelle (IR-UWB). Les travaux ont été réalisés selon trois axes, les deux premiers concernant la communication point-à-point et le dernier, les communications coopératives. Tout d’abord, la quantification des signaux typiques IR-UWB (soit directement échantillonnés, soit estimés) a été investiguée, principalement du point de vue du compromis entre la robustesse (ou réciprocité) des séquences binaires obtenues et leur caractère aléatoire. Différents algorithmes de quantification valorisant l’information temporelle offerte par les canaux IR-UWB pour améliorer ce compromis ont alors été proposés. Ensuite, des études concernant les échanges publics nécessaires à l’étape de réconciliation (visant la correction d’éventuels désaccords entre les séquences binaires générées de part et d’autre du lien) ont montré qu’il était possible d’être plus robuste face aux attaques passives en utilisant des informations de plus haut niveau, inhérentes à cette technologie et disponibles à moindre coût (ex. via une estimation précise du temps de vol aller-retour). Finalement, une nouvelle méthode a été développée afin d’étendre les schémas de génération de clé point-à-point à plusieurs nœuds (trois dans nos études) en utilisant directement la couche physique fournie par les liens radio entre les nœuds. / Emerging decentralized wireless systems, such as sensor or ad-hoc networks, will demand an adequate level of security in order to protect the private and often sensitive information that they carry. The main security mechanism for confidentiality in such networks is symmetric cryptography, which requires the sharing of a symmetric key between the two legitimate parties. According to the principles of physical layer security, wireless devices within the communication range can exploit the wireless channel in order to protect their communications. Due to the theoretical reciprocity of wireless channels, the spatial decorrelation property (e.g., in rich scattering environments), as well as the fine temporal resolution of the Impulse Radio - Ultra Wideband (IR-UWB) technology, directly sampled received signals or estimated channel impulse responses (CIRs) can be used for symmetric secret key extraction under the information-theoretic source model. Firstly, we are interested in the impact of quantization and channel estimation algorithms on the reciprocity and on the random aspect of the generated keys. Secondly, we investigate alternative ways of limiting public exchanges needed for the reconciliation phase. Finally, we develop a new signal-based method that extends the point-to-point source model to cooperative contexts with several nodes intending to establish a group key.