<p>Radar warning systems are life saving equipment in modern fighter aircraft. It is therefore vital that the system can tell the difference between a threat genuine frequency) and a false signal (spurious frequency). </p><p>This thesis presents a model aimed at predicting the frequencies and other parameters in the RF converter of the radar warning system. The components of the RF converter have been studied, measured, and modelled. The modelling tool has been the Simulink toolbox for Matlab. </p><p>Extreme accuracy has been sacrificed in order to make the model easy to use for the working engineer. Instead, this model presents a rough estimate of some of the most important properties of the radar warning system with just a few data sheet figures as input.</p><p>The simulation results are satisfactory as a whole. Simulink is the limiting factor in the implementation of the model. Significantly improved results can probably be obtained by working in another software environment.</p>

Maxwell's equations are a system of partial differential equations of vector fields. Imposing the constitutive relations for material properties yields equations for the curl and divergence of the electric and magnetic fields. The curl and divergence equations must be solved simultaneously, which is not the same as solving three separate scalar problems in each component of the vector field. This thesis describes a new method for solving partial differential equations of vector fields using the finite element method. New basis functions are used to solve the curl equation while allowing the divergence to be set as a constraint. The basis functions are defined on a mesh of bricks and the method is applicable for geometries that conform to a Cartesian coordinate system. The basis functions are a combination of cubic Hermite splines and second order Lagrange interpolation polynomials. The method yields a linearly independent set of constraints for the divergence, which is modelled to second order accuracy within each brick. Mesh refinement is accomplished by dividing selected bricks into $2\times 2\times 2$ smaller bricks of equal size. The change in the node pattern at an interface where mesh refinement occurs necessitates a modified implementation of the divergence constraints as well as additional constraints for hanging nodes. The mesh can be refined to an arbitrary number of levels. The basis functions can exactly model the discontinuity in the normal component of the field at a planar interface. The method is modified to solve problems with singularities at material boundaries that form $90^{\circ}$ edges and corners. The primary test problem of the new basis functions is to obtain the resonant frequencies and fields of three-dimensional cavities. The new basis functions can resolve physical solutions and non-physical, spurious modes. The eigenvalues obtained with the new method are in good agreement with exact solutions and experimental values in cases where they exist. There is also good agreement with results from second-order edge elements that are obtained with the software package HFSS. Finally, the method is modified to solve problems in cylindrical coordinates provided the domain does not contain the coordinate axis.

Maxwell's equations are a system of partial differential equations of vector fields. Imposing the constitutive relations for material properties yields equations for the curl and divergence of the electric and magnetic fields. The curl and divergence equations must be solved simultaneously, which is not the same as solving three separate scalar problems in each component of the vector field. This thesis describes a new method for solving partial differential equations of vector fields using the finite element method. New basis functions are used to solve the curl equation while allowing the divergence to be set as a constraint. The basis functions are defined on a mesh of bricks and the method is applicable for geometries that conform to a Cartesian coordinate system. The basis functions are a combination of cubic Hermite splines and second order Lagrange interpolation polynomials. The method yields a linearly independent set of constraints for the divergence, which is modelled to second order accuracy within each brick. Mesh refinement is accomplished by dividing selected bricks into $2\times 2\times 2$ smaller bricks of equal size. The change in the node pattern at an interface where mesh refinement occurs necessitates a modified implementation of the divergence constraints as well as additional constraints for hanging nodes. The mesh can be refined to an arbitrary number of levels. The basis functions can exactly model the discontinuity in the normal component of the field at a planar interface. The method is modified to solve problems with singularities at material boundaries that form $90^{\circ}$ edges and corners. The primary test problem of the new basis functions is to obtain the resonant frequencies and fields of three-dimensional cavities. The new basis functions can resolve physical solutions and non-physical, spurious modes. The eigenvalues obtained with the new method are in good agreement with exact solutions and experimental values in cases where they exist. There is also good agreement with results from second-order edge elements that are obtained with the software package HFSS. Finally, the method is modified to solve problems in cylindrical coordinates provided the domain does not contain the coordinate axis.

The first chapter considers the asymptotic validity of bootstrap methods in a linear trend model with a change in slope at an unknown time. Perron and Zhu (2005) analyzed the consistency, rate of convergence, and limiting distributions of the parameter estimates in this model. I provide theoretical results for the asymptotic validity of bootstrap methods related to forming confidence intervals for the break date. I consider two bootstrap schemes, the residual (for white noise errors) and the sieve bootstrap (for correlated errors). Simulation experiments confirm that confidence intervals obtained using bootstrap methods perform well in terms of exact coverage rate. The second chapter extends Perron and Zhu's (2005) analysis to cover more general fractionally integrated errors with memory parameter d in the interval (-0.5,1.5). My theoretical results uncover some interesting features. For example, with a concurrent level shift allowed, the rate of convergence of the estimate of the break date is the same for all values of d in the interval (-0.5,0.5), a feature linked to the contamination induced by allowing a level shift. In all other cases, the rate of convergence is decreasing as d increases. I also provide results about the spurious break issue. The third chapter considers constructing confidence intervals for the break date in linear regressions. I compare the performance of various procedures in terms of the exact coverage rates and lengths: Bai's (1997) based on the asymptotic distribution with shrinking shifts, Elliott and Müller's (EM) (2007) based on inverting a test locally invariant to the magnitude of the change, Eo and Morley's (2013) based on inverting a likelihood ratio test, and various bootstrap procedures. In terms of coverage rates, EM's approach is the best but with a high cost in terms of length. With serially correlated errors and a change in intercept or in the coefficient of a regressor with a high signal-to-noise ratio, or when a lagged dependent variable is present, the length approaches the whole sample as the magnitude of the change increases. This drawback is not present for the other methods. Theoretical results are provided to explain the drawbacks of EM's method.

Radar warning systems are life saving equipment in modern fighter aircraft. It is therefore vital that the system can tell the difference between a threat genuine frequency) and a false signal (spurious frequency). This thesis presents a model aimed at predicting the frequencies and other parameters in the RF converter of the radar warning system. The components of the RF converter have been studied, measured, and modelled. The modelling tool has been the Simulink toolbox for Matlab. Extreme accuracy has been sacrificed in order to make the model easy to use for the working engineer. Instead, this model presents a rough estimate of some of the most important properties of the radar warning system with just a few data sheet figures as input. The simulation results are satisfactory as a whole. Simulink is the limiting factor in the implementation of the model. Significantly improved results can probably be obtained by working in another software environment.

Abstract Oversampling data converters based on Delta-Sigma modulation are a popular solution for modern high-resolution applications. In the design of digital-to-analog or analog-to-digital Delta-sigma converters there are common obstacles due to the difficulties on predicting and verifying their performance. Being a highly nonlinear system, a Delta-Sigma modulator’s (DSM) quantization noise and therefore the spurious tones are difficult to analyze and predict. Multi-bit DACs can be used to improve the performance and linearize the behavior of DSMs. However, this will give rise to the need for linearizing the multi-bit DAC. A popular DAC linearization method, data weighted averaging (DWA) shapes the DAC mismatch noise spectrum. There are many variants of DWA, for low-pass and band-pass DSMs. This thesis proposes a generalization which integrates a few published variants into one, broader DWA scheme. The generalization enables expanding the tone-suppression studies into a larger concept. The performance of one- or multibit DSMs is usually verified by simulations. This thesis proposes a simulation-based qualification (characterization) method that can be used to repeatedly verify and compare the performance of multibit DSM with a DAC mismatch shaping or scrambling scheme. The last contribution of this thesis is a very simple model for tonal behavior. The model enables accurate prediction of spurious tones from both DSMs and DWA-DACs. The model emulates the tone behavior by its true birth-mechanism: frequency modulation. The proposed prediction model for tone-behavior can be used for developing new tone-cancelation methods. Based on the model, a DWA linearization method is also proposed. / Tiivistelmä Delta-Sigma modulaatio on suosituin tekniikka ylinäytteistävissä datan muuntimissa. Riippumatta toteutustarkoituksesta (analogia-digitaali- tai digitaali-analogia-muunnos), Delta-Sigma (DS) modulaatiossa on yleisesti tunnettuja käyttäytymisen ennustamiseen liittyviä ongelmia. Nämä ongelmat ovat peräisin modulaattorin luontaisesta epälineaarisuudesta: DS-muunnin on nimittäin vahvasti epälineaarinen takaisinkytketty systeemi, jonka harhatoistojen ennustaminen ja analysointi on erittäin hankalaa. Yksibittisestä monibittiseen DS-muuntimeen siirryttäessä muuntimen suorituskyky paranee, ja muuntimen kohinakäyttäytyminen on lineaarisempaa. Tämä kuitenkin kostautuu tarpeena linearisoida DS-muuntimen digitaali-analogia (D/A) muunnin. Tällä hetkellä tunnetuin linearisointimenetelmä on nimeltään DWA (data weighted averaging) algoritmi. Tässä työssä DWA:lle ja sen lukuisille varianteille esitellään eräänlainen yleistys, jonka avulla algoritmia voidaan soveltaa sekä alipäästö- että kaistanpäästö-DS-muuntimelle. Kuten tunnettua, DS-modulaattorin analyyttinen tarkastelu on raskasta. Yksi- ja monibittisten DS-muuntimien suunnitellun käyttäytymisen varmistaminen tapahtuukin yleensä simulointien avulla. Työssä esitetään simulointiperiaate, jolla voidaan kvalifioida (karakterisoida) monibittinen DS-muunnin. Tarkemmin, kvalifioinnin kohteena on DWA:n kaltaiset D/A -muuntimien linearisointimentelmät. Kyseessä on pyrkimys ennen kaikkea toistettavaan menetelmään, jolla eri menetelmiä voidaan verrata nopeasti ja luotettavasti. Tämän väitöstyön viimeinen kontribuutio on matemaattinen malli harhatoistojen syntymekanismille. Mallilla sekä DS-muunnoksen että DWA-D/A -muunnokseen liittyvät harhatoistot voidaan ennustaa tarkasti. Harhatoistot mallinnetaan yksinkertaisella havaintoihin perustuvalla FM-modulaatiokaavalla. Syntymekanismin mallinnus mahdollistaa DS-muuntimien ennustettavuuden ja täten auttaa harhatoiston kumoamismenetelmien kehittämistä. Työssä esitetään yksi matemaattisen mallin avulla kehitetty DWA-D/A -muunnoksen linearisointimenetelmä.

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This work deals with the design of a hybrid time integrator that couples spatially explicit and implicit time integrators. In order to cope with the industrial solver of Ariane Group called FLUSEPA, the explicit scheme of Heun and the implicit scheme of Crank-Nicolson are hybridized using the transition parameter : the whole technique is called AION time integration. The latter is studied into details with special focus on spectral behaviour and on its ability to keep the accuracy. It is shown that the hybrid technique has interesting dissipation and dispersion properties while maintaining precision and avoiding spurious waves. Moreover, this hybrid approach is validated on several academic test cases for both convective and diffusive fluxes. And as expected the method is more interesting in term of computational time than standard time integrators. For the extension of this hybrid approach to the temporal adaptive method implemented in FLUSEPA, it was necessary to improve some treatments in order to maintain conservation and acceptable spectral properties. Finally the hybrid time integration was also applied to a RANS/LES turbulent test case with interesting computational time while capturing the flow physics.

O assunto principal deste trabalho é qualificar, quantificar e implementar o comportamento numérico de estruturas discretizadas através do método dos elementos finitos. Serão abordados apenas os elementos lineares unidimensionais dinâmicos, porém a aplicabilidade da formulação proposta pode se estender para elementos bi e tridimensionais lineares dinâmicos. Inicia-se com uma introdução ao tema. Com certo desenvolvimento matemático, pode-se isolar analiticamente a parcela relacionada ao erro numérico. Elevando a ordem do erro de truncamento, obtém-se precisão elevada na resposta numérica. Inspirado no integrador temporal de Newmark, projetam-se elementos que apresentam estabilidade incondicional para os chamados efeitos espúrios. O efeito evanescente é um fenômeno espúrio onde a onda se propaga ao longo da estrutura acompanhada de um amortecimento puramente numérico ao longo do domínio do espaço. Outro efeito analisado é a reflexão espúria. Quando dois elementos adjacentes têm comprimentos diferentes, surge uma onda de reflexão (ou duas, no caso do elemento de viga) na interface deles. Tal onda, também de origem puramente matemática, existe devido à diferença entre as massas e as rigidezes absolutas dos elementos envolvidos, independente do fato de que eles tenham as mesmas características físicas. A relação entre o incremento de tempo e o período de oscilação é convenientemente empregada como principal parâmetro para quantificar a discretização no domínio temporal. No domínio do espaço, a relação empregada é entre o comprimento do elemento e o comprimento de onda. / The main subject of this work is to qualify, quantify and implement the numerical behavior of discrete structures through the finite element method. It will be investigated only the dynamic onedimensional linear elements, but the applicability of the proposed formulation can be extended to the bi and tri-dimensional cases. It begins with an introduction to the theme. With some mathematical development, the related numerical error can be isolated analytically. Once the truncation error is isolate, a high precision numerical response is obtained. Inspired in the Newmark time integrator, unconditionally stable elements for spurious effects are idealized. The evanescent effect is a spurious phenomenon where the wave propagates along the structure subjected to a numerical damping in the spatial domain. Another effect analyzed here is the spurious wave reflection. When two adjacent elements have different lengths, a reflected wave exists (two waves for the beam element) at their interface. This wave, which meaning is purely mathematical, exists due to the difference of their absolute mass and stiffness between the finite elements involved, even when both elements have the same physical properties. The rate between the time increment and the period of oscillation is conveniently employed as the main parameter to quantify the time discretization. In the spatial domain, the used parameter is the relation between the element and the wave length.

O assunto principal deste trabalho é qualificar, quantificar e implementar o comportamento numérico de estruturas discretizadas através do método dos elementos finitos. Serão abordados apenas os elementos lineares unidimensionais dinâmicos, porém a aplicabilidade da formulação proposta pode se estender para elementos bi e tridimensionais lineares dinâmicos. Inicia-se com uma introdução ao tema. Com certo desenvolvimento matemático, pode-se isolar analiticamente a parcela relacionada ao erro numérico. Elevando a ordem do erro de truncamento, obtém-se precisão elevada na resposta numérica. Inspirado no integrador temporal de Newmark, projetam-se elementos que apresentam estabilidade incondicional para os chamados efeitos espúrios. O efeito evanescente é um fenômeno espúrio onde a onda se propaga ao longo da estrutura acompanhada de um amortecimento puramente numérico ao longo do domínio do espaço. Outro efeito analisado é a reflexão espúria. Quando dois elementos adjacentes têm comprimentos diferentes, surge uma onda de reflexão (ou duas, no caso do elemento de viga) na interface deles. Tal onda, também de origem puramente matemática, existe devido à diferença entre as massas e as rigidezes absolutas dos elementos envolvidos, independente do fato de que eles tenham as mesmas características físicas. A relação entre o incremento de tempo e o período de oscilação é convenientemente empregada como principal parâmetro para quantificar a discretização no domínio temporal. No domínio do espaço, a relação empregada é entre o comprimento do elemento e o comprimento de onda. / The main subject of this work is to qualify, quantify and implement the numerical behavior of discrete structures through the finite element method. It will be investigated only the dynamic onedimensional linear elements, but the applicability of the proposed formulation can be extended to the bi and tri-dimensional cases. It begins with an introduction to the theme. With some mathematical development, the related numerical error can be isolated analytically. Once the truncation error is isolate, a high precision numerical response is obtained. Inspired in the Newmark time integrator, unconditionally stable elements for spurious effects are idealized. The evanescent effect is a spurious phenomenon where the wave propagates along the structure subjected to a numerical damping in the spatial domain. Another effect analyzed here is the spurious wave reflection. When two adjacent elements have different lengths, a reflected wave exists (two waves for the beam element) at their interface. This wave, which meaning is purely mathematical, exists due to the difference of their absolute mass and stiffness between the finite elements involved, even when both elements have the same physical properties. The rate between the time increment and the period of oscillation is conveniently employed as the main parameter to quantify the time discretization. In the spatial domain, the used parameter is the relation between the element and the wave length.