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The purpose of this thesis is to formulate and proof some theorems about convergences of Fourier series. In essence, we shall formulate and proof Wiener's lemma and Wiener-Lévy theorem which give us weaker conditions for absolute convergence of Fourier series. This thesis follows the classical Fourier analysis approach in a straightforward and detailed way suitable for undergraduate science students.

The research presented in this dissertation is on the development and application of advanced reduced rank adaptive signal processing techniques for high data rate wireless code division multiple access (CDMA) communications systems. This is an important area of research in the field of wireless communications. Current systems are moving towards the use of multiple simultaneous users in a given channel to increase system capacity as well as spatial and/or temporal diversity for improved performance in the presence of multipath and fading channels. Furthermore, to accommodate the demand for higher data rates, fast signal processing algorithms are required, which often translate into blind signal detection and estimation and the desire for optimal, low complexity detection techniques. The research presented here shows how minimum mean square error (MMSE) receivers implemented via the multistage Wiener filter (MWF) can be employed at the receiving end of a CDMA system to perform multiuser detection (MUD) or interference suppression (IS) with no loss in performance and significant signal subspace compression better than any previous reduced rank techniques have shown. This is important for optimizing performance because it implies a reduction in the number of required samples, so it lessens the requirement that the channel be stationary for a time duration long enough to obtain enough samples for an accurate MMSE estimate. The structure of these receivers is derived for synchronous and asynchronous systems for a multipath environment, and then it is shown that implementation of the receiver in a reduced rank subspace results in no loss in performance over full rank methods. It is also shown in some instances that reduced rank exceeds full rank performance. Multiuser detectors are also studied, and the optimal reduced rank detector is shown to be equivalent to a bank of parallel single user detectors performing interference suppression (IS). The performance as a function of rank for parallel and joint multiuser detectors are compared. The research is then extended to include joint space-code (i.e. a joint multiuser detector) and joint space-time processing algorithms which employ receiver diversity for low complexity diversity gain. Non-linear techniques, namely serial interference cancellation (SIC) and parallel interference cancellation (PIC), will also be studied. The conventional matched filter correlator will be replaced by the MWF, thereby incorporating IS at each stage of the interference canceller for improved performance. A closed form expression is derived for the probability of error, and performance gains are evaluated. It will be further shown how the receiver structure can be extended when space-time codes are employed at the transmitter for additional diversity gain with minimal impact on complexity. The MMSE solution is derived and implemented via the MWF with some examples. It is believed that these new techniques will have a significant impact on the design of fourth generation (4G) and beyond cellular CDMA systems. / Ph. D.

The research presented in this dissertation is on the development of advanced reduced rank adaptive signal processing for airborne radar space-time adaptive processing (STAP) and steering vector mismatch robustness. This is an important area of research in the field of airborne radar signal processing since practical STAP algorithms should be robust against various kinds of mismatch errors. The clutter return in an airborne radar has widely spread Doppler frequencies; therefore STAP, a two-dimensional adaptive filtering algorithm is required for effective clutter and jamming cancellation. Real-world effects in nonhomogeneous environments increase the number of adaptive degrees of freedom required to adequately suppress interference. The increasing computational complexity and the need to estimate the interference from a limited sample support make full rank STAP impractical. The research presented here shows that the reduced rank multistage Wiener filter (MWF) provides significant subspace compression better than any previous techniques in a nonhomogeneous environment. In addition, the impact of steering vector mismatch will also be examined on the MWF. In an airborne radar environment, it is well known that calibration errors and steering vector mismatch can seriously degrade adaptive array performance and result in signal cancellation. These errors can be caused by many non-ideal factors such as beam steering angle errors, multipath propagation, and phase errors due to array imperfections. Since the MWF centrally features the steering vector on its formulation, it is important to assess the impact of steering vector mismatch. In this dissertation, several novel techniques for increasing robustness are examined and applied to the MWF. These include derivative constraints, quiescent pattern control (QPC) techniques, and covariance matrix tapers (CMT). This research illustrates that a combination of CMT and QPC, denoted CMTQ, is very effective at mitigating the impact of steering vector mismatch. Use of CMTQ augmentation provides the steering vector mismatch robustness that we desire while improving the reduced-rank and reduced sample characteristics of the MWF. Results using Monte Carlo simulations and experimental Multichannel Airborne Radar Measurements (MCARM) data confirm that the use of CMTQ gives superior performance to steering vector errors at a much reduced rank and sample support as compared to conventional techniques. / Ph. D.

Zugl.: Darmstadt, Techn. Univ., Diss., 2009

Title: On Selected Geometric Properties of Brownian Motion Paths Author: Mgr. Ondřej Honzl E-mail Address: honzl@karlin.mff.cuni.cz Department: Department of Probability and Mathematical Statistics Supervisor: Prof. RNDr. Jan Rataj, CSc. E-mail Address: rataj@karlin.mff.cuni.cz Department: Mathematical Institute, Charles University Abstract: Our thesis is focused on certain geometric properties of Brownian motion paths. Firstly, it deals with cone points of Brownian motion in the plane and we show some connections between cone points and critical points of Brownian motion. The motivation of the study of critical points is provided by a pleasant behavior of the distance function outside of the set of these points. We prove the theorem on a non-existence of π+ cone points on fixed line. This statement leads us to the conjecture that there are only countably many critical points of the Brownian motion path in the plane. Next, the thesis discusses an asymptotic behavior of the surface area of r-neigh- bourhood of Brownian motion, which is called Wiener sausage. Using the proper- ties of a Kneser function, we prove the claim about the relation of the Minkowski content and S-content. As the consequence, we obtain a limit behavior of the surface area of the Wiener sausage almost surely in dimension d ≥ 3. Finally,...

Dans cette thèse, nous donnerons tout d'abord des conditions sur les paramètres d’une EDSR à générateur lipschitzien ou à croissance quadratique telles que les processus solutions de l’EDSR admettent des densités par rapport à la mesure de Lebesgue. Puis, nous donnerons des conditions sur les paramètres d’une EDSR non-markovienne à générateur lipschitzien ou quadratique telles que les processus solutions de l’EDSR admettent une dérivée de Malliavin, à l’aide d’une nouvelle caractérisation de cette dérivée. Ce résultat nous fournira une nouvelle structure interne des espaces de Malliavin que nous étudierons. Nous donnerons ensuite des conditions nous assurant que des solutions d’EDSR non-markoviennes à générateurs lipschitziens stochastiques sont différentiables au sens de Malliavin en utilisant cette caractérisation. Nous ferons ensuite une analyse de densités pour les lois des solutions de telles EDSR et nous appliquerons nos résultats à la biologie. Enfin, nous étudierons deux exemples d’utilisations des EDSR en finance. On s’intéressera tout d’abord à un problème de maximisation d’utilité avec un horizon aléatoire que nous réduirons à l’analyse d’un nouveau type d’EDSR à coefficients singuliers et nous illustrerons nos résultats par des simulations numériques. Puis, nous résoudrons un problème de type Principal/Agent sous volatilité incertaine. / In the first part of this PhD thesis, we give conditions on the parameters of Lipschitz and quadratic growth BSDEs such that the laws of the components Y and Z of the solutions to such BSDEs admit densities with respect to the Lebesgue measure. We then provide conditions on the parameters of non-Markovian Lipschitz or quadratic growth BSDEs such that the components Y and Z of their solutions are Malliavin differentiable. We obtain these conditions by applying a new characterization of the Malliavin differentiability, as an Lp convergence criterion of difference quotients. This result provide also a new characterization of the Malliavin-Sobolev spaces that we study in detail. To finish this first theoretical part, we provide conditions ensuring that solutions of non-Markovian stochastic-Lipschitz BSDEs are Malliavin differentiable by applying the characterization of the Malliavin differentiability obtained. We then analyse the existence of densities for the laws of the components of solutions to such BSDEs and we apply our result to a model of gene expression. In the second part of this thesis, we investigate financial problems dealing with BSDEs. We first solve a utility maximization problem with a random horizon, characterized by an exogenous default time. We reduce it to the analysis of a specific BSDE, which we call BSDE with singular coefficients, when the default time is assumed to be bounded. We give conditions ensuring the existence and the uniqueness of solutions to such BSDE and we illustrate our results by numerical simulations. Then, we solve a Principal/Agent problem with ambiguity, in which the "Nature" impacts both the utilities of the Agent and the Principal, charaterized by sets of probability measures which modify the volatility.

Made available in DSpace on 2016-08-17T14:53:17Z (GMT). No. of bitstreams: 1 Ulysses Santos Sousa.pdf: 1410915 bytes, checksum: 88235f7f4a3bc07a4da1b27c23dc71ca (MD5) Previous issue date: 2011-05-13 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / Cancer is one of the biggest health problems worldwide, and the breast cancer is the one that causes more deaths among women. Also it is the second most frequent type in the world. The chances of survival for a patient with breast cancer increases the sooner this disease is discovered. Several Computer Aided Detection/Diagnosis Systems has been used to assist health professionals. This work presents a methodology to discriminate and classify mammographic tissues regions in mass and non-mass. For this purpose the Shannon-Wiener‟s Diversity Index, which is applied to measure the biodiversity in ecosystem, is used to describe pattern of breast image region with four approaches: global, in circles, in rings and directional. After, a Support Vector Machine is used to classify the regions in mass and non-mass. The methodology presents promising results for classification of mammographic tissues regions in mass and non-mass, achieving 99.85% maximum accuracy. / O câncer é um dos maiores problemas de saúde mundial, sendo o câncer de mama o que mais causa óbito entre as mulheres e o segundo tipo mais freqüente no mundo. As chances de uma paciente sobreviver ao câncer de mama aumentam à medida que a doença é descoberta mais cedo. Diversos Sistemas de Detecção e Diagnóstico auxiliados por computador (Computer Aided Detection/Diagnosis) têm sido utilizados para auxiliar profissionais de saúde. Este trabalho apresenta uma metodologia de discriminação e classificação de regiões de tecidos de mamografias em massa e não massa. Para este propósito utiliza-se o Índice de Diversidade de Shannon-Wiener, comumente aplicado para medir a biodiversidade em um ecossistema, para descrever padrões de regiões de imagens de mama com quatro abordagens: global, em círculos, em anéis e direcional. Em seguida, utiliza-se o classificador Support Vector Machine para classificar estas regiões em massa e não massa. A metodologia apresenta resultados promissores para a classificação de regiões de tecidos de mamografia em massa e não massa, obtendo uma acurácia máxima de 99,85%.

L'objet de cette thèse est d'étudier la théorie du transport optimal sur un espace de Wiener abstrait. Les résultats qui se trouvent dans quatre principales parties, portent :Sur la convexité de l'entropie relative. On prolongera des résultats connus en dimension finie, sur l'espace de Wiener muni d'une norme uniforme, à savoir que l'entropie relative est (au moins faiblement) 1-convexe le long des géodésiques induites par un transport optimal sur l'espace de Wiener.Sur les mesures à densité logarithmiquement concaves. Le premier des résultats importants consiste à montrer qu'une inégalité de type Harnack est vraie pour le semi-groupe induit par une telle mesure sur l'espace de Wiener. Le second des résultats obtenus nous fournit une inégalité en dimension finie (mais indépendante de la dimension), contrôlant la différence de deux applications de transport optimal.Sur le problème de Monge. On s'intéressera au problème de Monge sur l'espace de Wiener, muni de plusieurs normes : des normes à valeurs finies, ou encore la pseudo-norme de Cameron-Martin.Sur l'équation de Monge-Ampère. Grâce aux inégalités obtenues précédemment, nous serons en mesure de construire des solutions fortes de l'équation de Monge-Ampère (induite par le coût quadratique) sur l'espace de Wiener, sous de faibles hypothèses sur les densités des mesures considérées / The aim of this PhD is to study the optimal transportation theory in some abstract Wiener space. You can find the results in four main parts and they are aboutThe convexity of the relative entropy. We will extend the well known results in finite dimension to the Wiener space, endowed with the uniform norm. To be precise the relative entropy is (at least weakly) geodesically 1-convex in the sense of the optimal transportation in the Wiener space.The measures with logarithmic concave density. The first important result consists in showing that the Harnack inequality holds for the semi-group induced by such a measure in the Wiener space. The second one provides us a finite dimensional and dimension-free inequality which gives estimate on the difference between two optimal maps.The Monge Problem. We will be interested in the Monge Problem on the Wiener endowed with different norms: either some finite valued norms or the pseudo-norm of Cameron-Martin.The Monge-Ampère equation. Thanks to the inequalities obtained above, we will be able to build strong solutions of the Monge-Ampère (those which are induced by the quadratic cost) equation on the Wiener space, provided the considered measures satisfy weak conditions

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