Orientador: Prof. Dr. Ricardo Suyama / Dissertação (mestrado) - Universidade Federal do ABC, Programa de Pós-Graduação em Engenharia da Informação, 2017. / O metodo estatistico de Independent Component Analysis (ICA) e um dos mais amplamente utilizados para solucionar o problema de Blind Source Separation (BSS) que, junto a outros metodos de processamento de sinais, sao colocados a prova com o aumento do numero das fontes de sinais e amostras disponiveis para processamento, e sao a base de aplicacoes com requisitos de desempenho cada vez maiores. O objetivo deste trabalho e realizar o estudo do metodo ICA e analise dos algoritmos FastICA e Joint Approximate Diagonalization of Eigen-matrices (JADE) implementados em Field-Programmable Gate Array (FPGA) e seu comportamento quando variamos o numero de amostras das misturas e os numeros de iteracoes ou updates. Outros trabalhos de pesquisa ja foram realizados com o objetivo de demonstrar a viabilidade da implementacao de tais algoritmos em FPGA, mas pouco apresentam sobre o metodo utilizado para definir detalhes de implementacao como numero de amostradas utilizados, a razao da representacao numerica escolhida e sobre o thoughtput alcancado. A analise que este trabalho propos realizar, num primeiro momento, passa por demonstrar o comportamento do core dos algoritmos quando implementados utilizando diferentes representacoes numericas de ponto flutuante com precisao simples (32 bits) e ponto fixo com diferentes numeros de amostras e fontes a serem estimadas, por meio de simulacoes. Foi verificada a viabilidade desses serem utilizados para atender aplicacoes que precisam resolver o problema de BSS com boa acuracia, quando comparados com implementacoes dos mesmos algoritmos que se utilizaram de uma representacao numerica de ponto flutuante com precisao dupla (64 bits). Utilizando o Simulink R¿e a biblioteca DSP Builder R¿da Altera R¿para implementar os modelos de cada algoritmo, foi possivel analisar outros aspectos importantes, em busca de demonstrar a possibilidade da utilizacao de tais implementacoes em aplicacoes com requisitos de tempo real, que necessitam de alto desempenho, utilizando FPGA de baixo custo, como: a quantidade de recursos de FPGA necessarios na implementacao de cada algoritmo, principalmente buscando minimizar a utilizacao de blocos DSP, a latencia, e maximizar o throughput de processamento. / Independent Component Analysis (ICA) is one of the most widely used statistical method to solve the problem of Blind Source Separation (BSS), which, along with other signal processing methods, faces new challenges with the increasing the number of signal sources and samples available for processing, being the base of applications with increasing performance requirements. The aim of this work is to study the FastICA and the Joint Approximate Diagonalization of Eigen-matrices (JADE) algorithms and implement them in Field- Programmable Gate Array (FPGA). Other researches have already been carried out with the objective of demonstrating the feasibility of implementing such algorithms in FPGA, but they present little about the methodology used and implementation details such as the number of samples used, why the numerical representation was chosen and the obtained thoughtput. The analysis carried out in this work demonstrates the behavior of the core of the algorithms when implemented using different representations, such as singleprecision floating-point (32 bits) and fixed point with different numbers of samples and sources to be estimated. It was verified these immplementations are able to solve the BSS problem with good accuracy when compared with implementations of the same algorithms that exmploy a double-precision floating-point representation (64 bits). Using the Simulink R¿ and Alterafs R¿ DSP Builder R¿ library to implement the models of each algorithm, it was possible to analyze other important aspects, in order to demonstrate the possibility of using such implementations in applications with real-time requirements that require high performance, using low cost FPGA, such as: the necessary FPGA resources in the implementation of each algorithm, mainly seeking to minimize the use of DSP blocks, latency, and to maximize the processing throughput.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / This thesis will make a study of the conic, which can be defined as quadratic equations solutions with two variables, with the main objective recognition of same through a simplification of the quadratic form associated, whose procedure involves the diagonalization of symmetric matrices. Throughout this work, will address the prerequisites needed for the reader with little familiarity on the subject, can understand each stage of its development, as Euclidean spaces and matrix diagonalization. / Nesta dissertação faremos um estudo das cônicas, as quais podem ser definidas como soluções de equações do segundo grau com duas variáveis, tendo como objetivo principal o reconhecimento das mesmas por meio de uma simplificação da forma quadrática associada, cujo procedimento envolve a diagonalização de matrizes simétricas. Ao longo deste trabalho, serão abordados os pré-requisitos necessários para que o leitor, com pouca familiaridade no assunto, possa compreender cada etapa de seu desenvolvimento, como espaços euclidianos e diagonalização de matrizes.

In computer graphics, the complex phenomenon of color appearance, involving the interaction of light, matter and the human visual system, is modeled by the multiplication of RGB triplets assigned to lights and materials. This efficient heuristic produces plausible images because the triplets assigned to materials usually function as color specifications. To predict color, spectral rendering is required, but the O(n) cost of computing reflections with n-dimensional point-sampled spectra is prohibitive for real-time rendering. Typical spectra are well approximated by m-dimensional linear models, where m << n, but computing reflections with this representation requires O(m^2) matrix-vector multiplication. A method by Drew and Finlayson [JOSA A 20, 7 (2003), 1181-1193], reduces this cost to O(m) by “sharpening” an n x m orthonormal basis with a linear transformation, so that the new basis vectors are approximately disjoint. If successful, this transformation allows approximated reflections to be computed as the products of coefficients of lights and materials. Finding the m x m change of basis matrix requires solving m eigenvector problems, each needing a choice of wavelengths in which to sharpen the corresponding basis vector. These choices, however, are themselves an optimization problem left unaddressed by the method's authors. Instead, we pose a single problem, expressing the total approximation error incurred across all wavelengths as the sum of dm^2 squares for some number d, where, depending on the inherent dimensionality of the rendered reflectance spectra, m <= d << n, a number that is independent of the number of approximated reflections. This problem may be solved in real time, or nearly, using standard nonlinear optimization algorithms. Results using a variety of reflectance spectra and three standard illuminants yield errors at or close to the best lower bound attained by projection onto the leading m characteristic vectors of the approximated reflections. Measured as CIEDE2000 color differences, a heuristic proxy for image difference, these errors can be made small enough to be likely imperceptible using values of 4 <= m <= 9. An examination of this problem reveals a hierarchy of simpler, more quickly solved subproblems whose solutions yield, in the typical case, increasingly inaccurate approximations. Analysis of this hierarchy explains why, in general, the lowest approximation error is not attained by simple spectral sharpening, the smallest of these subproblems, unless the spectral power distributions of all light sources in a scene are sufficiently close to constant functions. Using the methods described in this dissertation, spectra can be rendered in real time as the products of m-dimensional vectors of sharp basis coefficients at a cost that is, in a typical application, a negligible fraction above the cost of RGB rendering. / Graduate

An extended Hubbard model with large short and long-ranged Coulomb repulsions and distant transfer is numerically investigated by use of the Lanczos exact diagonalization (ED) method to study the charge order and unconditional dimerization of a chain at density rho (&#961;)= 0.4. From the analysis of the spin and charge correlation functions, a picture consistent with the formation of a dimer insulating state, which is of Wigner lattice-type (WL) charge order (CO), is obtained. The next-nearest neighbour (NNN) hopping t2 enhances the intradimer correlations and weakens the interdimer correlations. Implications for the CuO2 chains in Sr14Cu24O41 are discussed.We have also introduced a Heisenberg model which parametrically depends on hole positions. If the electrostatic hole-hole repulsion is included such a model allows to evaluate all energy eigenvalues and eigenstates (for small system size) and thus enables us to evaluate thermodynamic properties as function of temperature,magnetic field, and doping. Assuming certain exchange constants we can investigate the influence of the electrostatic hole-hole repulsion on ground state properties as well as on thermal averages like the magnetization which include contributions of low-lying spin-hole configurations.

Submitted by Erika Demachki (erikademachki@gmail.com) on 2014-10-01T10:43:24Z No. of bitstreams: 2 TCC 30_06_2013 Marco Aurélio PROFMAT.pdf: 3494241 bytes, checksum: 199c5af10fd068461af3db98f96eaf49 (MD5) license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) / Approved for entry into archive by Cláudia Bueno (claudiamoura18@gmail.com) on 2014-10-31T19:45:56Z (GMT) No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) TCC 30_06_2013 Marco Aurélio PROFMAT.pdf: 3494241 bytes, checksum: 199c5af10fd068461af3db98f96eaf49 (MD5) / Made available in DSpace on 2014-10-31T19:45:56Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) TCC 30_06_2013 Marco Aurélio PROFMAT.pdf: 3494241 bytes, checksum: 199c5af10fd068461af3db98f96eaf49 (MD5) Previous issue date: 2013-04-12 / In this thesis we study linear transformations, eigenvalues and eigenvectors with the objective of solve a system of linear ordinary differential equations with constant coefficients. / Nesta dissertação estudamos transformações lineares, autovalores e autovetores com o intuito de resolvermos um sistema de equações diferenciais ordinárias lineares com coeficientes constantes.

Cette thèse présente de nouveaux algorithmes de diagonalisation conjointe par similitude. Cesalgorithmes permettent, entre autres, de résoudre le problème de décomposition canonique polyadiquede tenseurs. Cette décomposition est particulièrement utilisée dans les problèmes deséparation de sources. L’utilisation de la diagonalisation conjointe par similitude permet de paliercertains problèmes dont les autres types de méthode de décomposition canonique polyadiquesouffrent, tels que le taux de convergence, la sensibilité à la surestimation du nombre de facteurset la sensibilité aux facteurs corrélés. Les algorithmes de diagonalisation conjointe par similitudetraitant des données complexes donnent soit de bons résultats lorsque le niveau de bruit est faible,soit sont plus robustes au bruit mais ont un coût calcul élevé. Nous proposons donc en premierlieu des algorithmes de diagonalisation conjointe par similitude traitant les données réelles etcomplexes de la même manière. Par ailleurs, dans plusieurs applications, les matrices facteursde la décomposition canonique polyadique contiennent des éléments exclusivement non-négatifs.Prendre en compte cette contrainte de non-négativité permet de rendre les algorithmes de décompositioncanonique polyadique plus robustes à la surestimation du nombre de facteurs ou lorsqueces derniers ont un haut degré de corrélation. Nous proposons donc aussi des algorithmes dediagonalisation conjointe par similitude exploitant cette contrainte. Les simulations numériquesproposées montrent que le premier type d’algorithmes développés améliore l’estimation des paramètresinconnus et diminue le coût de calcul. Les simulations numériques montrent aussi queles algorithmes avec contrainte de non-négativité améliorent l’estimation des matrices facteurslorsque leurs colonnes ont un haut degré de corrélation. Enfin, nos résultats sont validés à traversdeux applications de séparation de sources en télécommunications numériques et en spectroscopiede fluorescence. / This thesis introduces new joint eigenvalue decomposition algorithms. These algorithms allowamongst others to solve the canonical polyadic decomposition problem. This decomposition iswidely used for blind source separation. Using the joint eigenvalue decomposition to solve thecanonical polyadic decomposition problem allows to avoid some problems whose the others canonicalpolyadic decomposition algorithms generally suffer, such as the convergence rate, theoverfactoring sensibility and the correlated factors sensibility. The joint eigenvalue decompositionalgorithms dealing with complex data give either good results when the noise power is low, orthey are robust to the noise power but have a high numerical cost. Therefore, we first proposealgorithms equally dealing with real and complex. Moreover, in some applications, factor matricesof the canonical polyadic decomposition contain only nonnegative values. Taking this constraintinto account makes the algorithms more robust to the overfactoring and to the correlated factors.Therefore, we also offer joint eigenvalue decomposition algorithms taking advantage of thisnonnegativity constraint. Suggested numerical simulations show that the first developed algorithmsimprove the estimation accuracy and reduce the numerical cost in the case of complexdata. Our numerical simulations also highlight the fact that our nonnegative joint eigenvaluedecomposition algorithms improve the factor matrices estimation when their columns have ahigh correlation degree. Eventually, we successfully applied our algorithms to two blind sourceseparation problems : one concerning numerical telecommunications and the other concerningfluorescence spectroscopy.

Charge excitations in different 3d transition metal compounds are studied. In particular, the influence of the lattice geometry on the character of these excitations is investigated. For this purpose, the momentum dependent loss function of electron energy-loss spectroscopy (EELS) as well as the optical conductivity are calculated and compared with the experimental data of NaV$_{2}$O$_{5}$, LiV$_{2}$O$_{5}$, Sr$_{2}$CuO$_{3}$, and CuGeO$_{3}$ . A quarter-filled extended Hubbard model on a system of coupled ladders provides a qualitative explanation for the highly anisotropic charge excitations of NaV$_{2}$O$_{5}$ and LiV$_{2}$O$_{5}$. These ladder compounds do not only differ from the charge ordering pattern but also from the coupling between different ladders: In LiV$_{2}$O$_{5}$ one finds a strong inter-ladder hopping which is very small in NaV$_{2}$O$_{5}$. On the other hand, in NaV$_{2}$O$_{5}$ the ladders are coupled by a strong inter-ladder Coulomb interaction. The charge excitations of quasi one-dimensional cuprates reflect both the properties of the CuO$_{4}$ plaquettes and the character of the coupling between different plaquettes. Independently from the geometry of the cuprat chains, the local excitation of the copper hole onto the adjacent oxygen orbitals is always found. Further transitions with an excitation energy below the local excitation of a single plaquette result from a hole transfer to another plaquette. These excitations with hole delocalization dominate the spectra of the corner-shared Sr$_{2}$CuO$_{3}$. In contrast to this, the hole transfer leads only to a pre-peak in the spectra of the edge-shared CuGeO$_{3}$. Furthermore, it is shown that the hole transfer is determined by the geometry of the edge-shared CO. / Gegenstand dieser Arbeit ist die theoretische Analyse von Ladungsanregungen in verschiedenen niedrigdimensionalen 3d-Übergangsmetallverbindungen, wobei insbesondere der Einfluß der Gittergeometrie auf die Charakteristik der Anregungen untersucht wurde. Mit Hilfe des Lanczos-Algorithmus' wurden dazu sowohl die impulsabhängige Verlustfunktion der Elektron-Energie-Verlust-Spektroskopie (EELS) als auch die optische Leitfähigkeit für NaV$_{2}$O$_{5}$, LiV$_{2}$O$_{5}$, Sr$_{2}$CuO$_{3}$ und CuGeO$_{3}$ berechnet und mit den experimentellen Ergebnissen verglichen. Unter der Verwendung eines Modells viertelgefüllter Leitern kann man die Ladungsanregungen sowohl für NaV$_{2}$O$_{5}$ als auch LiV$_{2}$O$_{5}$ sehr gut beschreiben. In diesen Materialien findet man nicht nur unterschiedliche Ladungsordnungen sondern vor allem auch verschiedene Kopplungsarten zwischen den Leitern. Während die Leitern im NaV$_{2}$O$_{5}$ durch die Coulomb-Wechselwirkung miteinander gekoppelt sind, existiert im LiV$_{2}$O$_{5}$ ein Austausch aufgrund einer starken Hybridisierung zwischen den Leitern. Die Ladungsanregungen von quasi eindimensionalen Kupratketten spiegeln sowohl die Plaketteneigenschaften als auch die Plakettenkopplung wider. Unabhängig von der Geometrie der Ketten findet man stets die lokale Anregung des Kupferloches auf die umliegenden Sauerstofforbitale. Aus einem möglichen Lochtransfer zu benachbarten Plaketten resultieren außerdem noch Anregungen, die energetisch unterhalb der Plakettenanregung liegen und unmittelbar von der Kettengeometrie abhängen. Während im eckenvernetzten Sr$_{2}$CuO$_{3}$ diese Anregungen die Spektren dominieren, spielt der Lochtransfer im kantenvernetzten CuGeO$_{3}$ nur eine untergeordnete Rolle.

The present dissertation aims to discuss diagonalization of linear operators, so that we can explore these concepts in the solution of systems of ordinary differential equations and in the identification of conics. A linear operator in a vector space of finite dimension can be represented by a matrix. Since diagonal arrays are the simplest from the point of view of matrix operations, we will show under what conditions, given a linear operator it is possible to represent it by a diagonal matrix. Thus, this paper presents the process of operator diagonalization, introduces basic concepts about systems of ordinary differential equations and applications. / A presente dissertação tem como objetivo discorrer sobre diagonalização de operadores lineares, de modo que possamos explorar esses conceitos na solução de sistemas de equações diferenciais ordinárias e na identificação de cônicas. Um operador linear em um espaço vetorial de dimensão finita, pode ser representado por uma matriz. Sendo as matrizes diagonais as mais simples do ponto de vista das operações matriciais, mostraremos sob que condições, dado um operador linear é possível representá-lo por uma matriz diagonal. Dessa forma, este trabalho apresenta o processo de diagonalização de operadores, introduz conceitos básicos sobre sistemas de equações diferenciais ordinárias e aplicações. / São Cristóvão, SE

Charge excitations in different 3d transition metal compounds are studied. In particular, the influence of the lattice geometry on the character of these excitations is investigated. For this purpose, the momentum dependent loss function of electron energy-loss spectroscopy (EELS) as well as the optical conductivity are calculated and compared with the experimental data of NaV$_{2}$O$_{5}$, LiV$_{2}$O$_{5}$, Sr$_{2}$CuO$_{3}$, and CuGeO$_{3}$ . A quarter-filled extended Hubbard model on a system of coupled ladders provides a qualitative explanation for the highly anisotropic charge excitations of NaV$_{2}$O$_{5}$ and LiV$_{2}$O$_{5}$. These ladder compounds do not only differ from the charge ordering pattern but also from the coupling between different ladders: In LiV$_{2}$O$_{5}$ one finds a strong inter-ladder hopping which is very small in NaV$_{2}$O$_{5}$. On the other hand, in NaV$_{2}$O$_{5}$ the ladders are coupled by a strong inter-ladder Coulomb interaction. The charge excitations of quasi one-dimensional cuprates reflect both the properties of the CuO$_{4}$ plaquettes and the character of the coupling between different plaquettes. Independently from the geometry of the cuprat chains, the local excitation of the copper hole onto the adjacent oxygen orbitals is always found. Further transitions with an excitation energy below the local excitation of a single plaquette result from a hole transfer to another plaquette. These excitations with hole delocalization dominate the spectra of the corner-shared Sr$_{2}$CuO$_{3}$. In contrast to this, the hole transfer leads only to a pre-peak in the spectra of the edge-shared CuGeO$_{3}$. Furthermore, it is shown that the hole transfer is determined by the geometry of the edge-shared CO. / Gegenstand dieser Arbeit ist die theoretische Analyse von Ladungsanregungen in verschiedenen niedrigdimensionalen 3d-Übergangsmetallverbindungen, wobei insbesondere der Einfluß der Gittergeometrie auf die Charakteristik der Anregungen untersucht wurde. Mit Hilfe des Lanczos-Algorithmus' wurden dazu sowohl die impulsabhängige Verlustfunktion der Elektron-Energie-Verlust-Spektroskopie (EELS) als auch die optische Leitfähigkeit für NaV$_{2}$O$_{5}$, LiV$_{2}$O$_{5}$, Sr$_{2}$CuO$_{3}$ und CuGeO$_{3}$ berechnet und mit den experimentellen Ergebnissen verglichen. Unter der Verwendung eines Modells viertelgefüllter Leitern kann man die Ladungsanregungen sowohl für NaV$_{2}$O$_{5}$ als auch LiV$_{2}$O$_{5}$ sehr gut beschreiben. In diesen Materialien findet man nicht nur unterschiedliche Ladungsordnungen sondern vor allem auch verschiedene Kopplungsarten zwischen den Leitern. Während die Leitern im NaV$_{2}$O$_{5}$ durch die Coulomb-Wechselwirkung miteinander gekoppelt sind, existiert im LiV$_{2}$O$_{5}$ ein Austausch aufgrund einer starken Hybridisierung zwischen den Leitern. Die Ladungsanregungen von quasi eindimensionalen Kupratketten spiegeln sowohl die Plaketteneigenschaften als auch die Plakettenkopplung wider. Unabhängig von der Geometrie der Ketten findet man stets die lokale Anregung des Kupferloches auf die umliegenden Sauerstofforbitale. Aus einem möglichen Lochtransfer zu benachbarten Plaketten resultieren außerdem noch Anregungen, die energetisch unterhalb der Plakettenanregung liegen und unmittelbar von der Kettengeometrie abhängen. Während im eckenvernetzten Sr$_{2}$CuO$_{3}$ diese Anregungen die Spektren dominieren, spielt der Lochtransfer im kantenvernetzten CuGeO$_{3}$ nur eine untergeordnete Rolle.

La diagonalisation conjointe approximée d’un ensemble de matrices permet de résoudre le problème de séparation aveugle de sources et trouve de nombreuses applications, notamment pour l’électroencéphalographie, une technique de mesure de l’activité cérébrale.La diagonalisation conjointe se formule comme un problème d’optimisation avec trois composantes : le choix du critère à minimiser, la contrainte de non-dégénérescence de la solution et l’algorithme de résolution.Les approches existantes considèrent principalement deux critères, les moindres carrés et la log-vraissemblance.Elles sont spécifiques à une contrainte et se restreignent à un seul type d’algorithme de résolution.Dans ce travail de thèse, nous proposons de formuler le problème de diagonalisation conjointe selon un modèle géométrique, qui généralise les travaux précédents et permet de définir des critères inédits, notamment liés à la théorie de l’information.Nous proposons également d’exploiter l’optimisation riemannienne et nousdéfinissons un ensemble d’outils qui permet de faire varier les trois composantes indépendamment, créant ainsi de nouvelles méthodes et révélant l’influence des choix de modélisation.Des expériences numériques sur des données simulées et sur des enregistrements électroencéphalographiques montrent que notre approche par optimisation riemannienne donne des résultats compétitifs par rapport aux méthodes existantes.Elles indiquent aussi que les deux critères traditionnels ne sont pas les meilleurs dans toutes les situations. / The approximate joint diagonalisation of a set of matrices allows the solution of the blind source separation problem and finds several applications, for instance in electroencephalography, a technique for measuring brain activity.The approximate joint diagonalisation is formulated as an optimization problem with three components: the choice of the criterion to be minimized, the non-degeneracy constraint on the solution and the solving algorithm.Existing approaches mainly consider two criteria, the least-squares and the log-likelihood.They are specific to a constraint and are limited to only one type of solving algorithms.In this thesis, we propose to formulate the approximate joint diagonalisation problem in a geometrical fashion, which generalizes previous works and allows the definition of new criteria, particularly those linked to information theory.We also propose to exploit Riemannian optimisation and we define tools that allow to have the three components varying independently, creating in this way new methods and revealing the influence of the choice of the model.Numerical experiments on simulated data as well as on electroencephalographic recordings show that our approach by means of Riemannian optimisation gives results that are competitive as compared to existing methods.They also indicate that the two traditional criteria do not perform best in all situations.