LISA Pathfinder (LPF) has been a space mission led by ESA with NASA contributions, operating between March 2016 and July 2017. LPF demonstrated the feasibility of setting bodies in space along freely-falling geodetic trajectories, complying with the residual acceleration requirements of the future gravitational-wave observatory LISA. After operations, the LPF Collaboration pointed out that two phenomena, affecting the sub-mHz performance, were not completely understood and needed deeper analyses. This, despite performing better than requirements. Such phenomena are, namely, the low-frequency acceleration noise, and the sub-pN transient acceleration glitches. This thesis work focuses entirely on analyzing these observations, in view of the future mission LISA. Regarding the low-frequency sub-mHz noise, first, we make a preliminary analysis. We investigate its evolution in time, its properties, its stability, and its nature. We find that the low-frequency noise has had a remarkably stable behavior for nearly two years, but noise fluctuations are not compatible with an overall unique noise. We develop results on multivariate spectral estimation. Implementing results from complex-variable statistics, we show that cross-power spectral density matrices follow complex-Wishart probability distributions; we develop a Bayesian tool for the posterior inference of spectral parameters. We develop decorrelation tools to understand the measured noise's physical origin. In particular, we aim at finding, if any, correlations between the main acceleration measurement and synchronously measured time series. Then, we summarize the most recent understanding of the LPF acceleration performance. We expand previous analyses about the LPF outgassing environment, through the analysis of the white "Brownian" noise, and the long-term quasi-static acceleration drift observed on LPF, proposing a physical model. We extensively analyze the second phenomenon impacting low-frequency performances, the acceleration transient glitches. We show that LPF glitches spanned a wide range of amplitudes, transferring impulses between a few fN s, to some nN s, and showing durations ranging from a few seconds to hours. We show that LPF glitches fall into two rather distinct categories: fast transients in the interferometric motion readout and long-lasting sub-pN force transient events, acting on the test masses. We present an analysis of the physical and statistical properties of both, including a cross-investigation with other time series and other dynamical variables, and examine the possible sources of glitches, identifying the most likely ones.

This thesis consists of six papers. The main topic of all these papers is modeling a class of linear time-invariant systems. The system class is parameterized in the context of interpolation theory with a degree constraint. In the papers included in the thesis, this parameterization is the key tool for the design of dynamical system models in fields such as spectral estimation and model reduction. A problem in spectral estimation amounts to estimating a spectral density function that captures characteristics of the stochastic process, such as covariance, cepstrum, Markov parameters and the frequency response of the process. A  model reduction problem consists in finding a small order system which replaces the original one so that the behavior of both systems is similar in an appropriately defined sense.  In Paper A a new spectral estimation technique based on the rational covariance extension theory is proposed. The novelty of this approach is in the design of a spectral density that optimally matches covariances and approximates the frequency response of a given process simultaneously.In Paper B  a model reduction problem is considered. In the literature there are several methods to perform model reduction. Our attention is focused on methods which preserve, in the model reduction phase, the stability and the positive real properties of the original system. A reduced-order model is computed employing the analytic interpolation theory with a degree constraint. We observe that in this theory there is a freedom in the placement of the spectral zeros and interpolation points. This freedom can be utilized for the computation of a rational positive real function of low degree which approximates the best a given system. A problem left open in Paper B is how to select spectral zeros and interpolation points in a systematic way in order to obtain the best approximation of a given system. This problem is the main topic in Paper C. Here, the problem is investigated in the analytic interpolation context and spectral zeros and interpolation points are obtained as solution of a optimization problem.In Paper D, the problem of modeling a floating body by a positive real function is investigated. The main focus is  on modeling the radiation forces and moment. The radiation forces are described as the forces that make a floating body oscillate in calm water. These forces are passive and usually they are modeled with system of high degree. Thus, for efficient computer simulation it is necessary to obtain a low order system which approximates the original one. In this paper, the procedure developed in Paper C is employed. Thus, this paper demonstrates the usefulness of the methodology described in Paper C for a real world application.In Paper E, an algorithm to compute the steady-state solution of a discrete-type Riccati equation, the Covariance Extension Equation, is considered. The algorithm is based on a homotopy continuation method with predictor-corrector steps. Although this approach does not seem to offer particular advantage to previous solvers, it provides insights into issues such as positive degree and model reduction, since the rank of the solution of the covariance extension problem coincides with the degree of the shaping filter. In Paper F a new algorithm for the computation of the analytic interpolant of a bounded degree is proposed. It applies to the class of non-strictly positive real interpolants and it is capable of treating the case with boundary spectral zeros. Thus, in Paper~F, we deal with a class of interpolation problems which could not be treated by the optimization-based algorithm proposed by Byrnes, Georgiou and Lindquist. The new procedure computes interpolants by solving a system of nonlinear equations. The solution of the system of nonlinear equations is obtained by a homotopy continuation method. / QC 20100721

This thesis is divided into two parts. The first part deals with theNevanlinna-Pick interpolation problem, a problem which occursnaturally in several applications such as robust control, signalprocessing and circuit theory. We consider the problem of shaping andapproximating solutions to the Nevanlinna-Pick problem in a systematicway. In the second part, we study distance measures between powerspectra for spectral estimation. We postulate a situation where wewant to quantify robustness based on a finite set of covariances, andthis leads naturally to considering the weak*-topology. Severalweak*-continuous metrics are proposed and studied in this context.In the first paper we consider the correspondence between weighted entropyfunctionals and minimizing interpolants in order to find appropriateinterpolants for, e.g., control synthesis. There are two basic issues that weaddress: we first characterize admissible shapes of minimizers bystudying the corresponding inverse problem, and then we developeffective ways of shaping minimizers via suitable choices of weights.These results are used in order to systematize feedback controlsynthesis to obtain frequency dependent robustness bounds with aconstraint on the controller degree.The second paper studies contractive interpolants obtained as minimizersof a weighted entropy functional and analyzes the role of weights andinterpolation conditions as design parameters for shaping theinterpolants. We first show that, if, for a sequence of interpolants,the values of the corresponding entropy gains converge to theoptimum, then the interpolants converge in H_2, but not necessarily inH-infinity. This result is then used to describe the asymptoticbehaviour of the interpolant as an interpolation point approaches theboundary of the domain of analyticity.A quite comprehensive theory of analytic interpolation with degreeconstraint, dealing with rational analytic interpolants with an apriori bound, has been developed in recent years. In the third paper,we consider the limit case when this bound is removed, and only stableinterpolants with a prescribed maximum degree are sought. This leadsto weighted H_2 minimization, where the interpolants areparameterized by the weights. The inverse problem of determining theweight given a desired interpolant profile is considered, and arational approximation procedure based on the theory is proposed. Thisprovides a tool for tuning the solution for attaining designspecifications. The purpose of the fourth paper is to study the topology and develop metricsthat allow for localization of power spectra, based on second-orderstatistics. We show that the appropriate topology is theweak*-topology and give several examples on how to construct suchmetrics. This allows us to quantify uncertainty of spectra in anatural way and to calculate a priori bounds on spectral uncertainty,based on second-order statistics. Finally, we study identification ofspectral densities and relate this to the trade-off between resolutionand variance of spectral estimates.In the fifth paper, we present an axiomatic framework for seekingdistances between power spectra. The axioms requirethat the sought metric respects the effects of additive andmultiplicative noise in reducing our ability to discriminate spectra.They also require continuity of statistical quantities withrespect to perturbations measured in the metric. We then present aparticular metric which abides by these requirements. The metric isbased on the Monge-Kantorovich transportation problem and iscontrasted to an earlier Riemannian metric based on theminimum-variance prediction geometry of the underlying time-series. Itis also being compared with the more traditional Itakura-Saitodistance measure, as well as the aforementioned prediction metric, ontwo representative examples. / QC 20100817

Analytical interpolation theory has several applications in systems and control. In particular, solutions of low degree, or more generally of low complexity, are of special interest since they allow for synthesis of simpler systems. The study of degree constrained analytic interpolation was initialized in the early 80's and during the past decade it has had significant progress. This thesis contributes in three different aspects to complexity constrained analytic interpolation: theory, numerical algorithms, and design paradigms. The contributions are closely related; shortcomings of previous design paradigms motivate development of the theory, which in turn calls for new robust and efficient numerical algorithms. Mainly two theoretical developments are studied in the thesis. Firstly, the spectral Kullback-Leibler approximation formulation is merged with simultaneous cepstral and covariance interpolation. For this formulation, both uniqueness of the solution, as well as smoothness with respect to data, is proven. Secondly, the theory is generalized to matrix-valued interpolation, but then only allowing for covariance-type interpolation conditions. Again, uniqueness and smoothness with respect to data is proven. Three algorithms are presented. Firstly, a refinement of a previous algorithm allowing for multiple as well as matrix-valued interpolation in an optimization framework is presented. Secondly, an algorithm capable of solving the boundary case, that is, with spectral zeros on the unit circle, is given. This also yields an inherent numerical robustness. Thirdly, a new algorithm treating the problem with both cepstral and covariance conditions is presented. Two design paradigms have sprung out of the complexity constrained analytical interpolation theory. Firstly, in robust control it enables low degree Hinf controller design. This is illustrated by a low degree controller design for a benchmark problem in MIMO sensitivity shaping. Also, a user support for the tuning of controllers within the design paradigm for the SISO case is presented. Secondly, in ARMA estimation it provides unique model estimates, which depend smoothly on the data as well as enables frequency weighting. For AR estimation, a covariance extension approach to frequency weighting is discussed, and an example is given as an illustration. For ARMA estimation, simultaneous cepstral and covariance matching is generalized to include prefiltering. An example indicates that this might yield asymptotically efficient estimates. / QC 20100928

The performance of radar detection and imaging systems strongly depends on the characteristics of radar clutter. In order to improve the radar signal processing algorithms, successful analysis and modeling of radar clutter are required. For a successful model of radar clutter, both the spectral and statistical characteristics of the clutter should be revealed. Within the scope of this study, an experimental radar data acquisition system is established to analyze radar clutter. The hardware and the data processing system are first verified using generic signals and then a set of measurements is taken in the open terrain. In this thesis, the limitations and problems encountered during the establishment of the system are explained in detail. The spectral and statistical analyses performed on the recorded data are examined. The temporal and spatial behavior of the measured clutter data are explored. The hypothetical models proposed so far in the literature are tested on the experimental data and the fitting of models to the experimental data is confirmed using various goodness-of-fit tests. Finally, the results of the analyses are interpreted in the light of the radar system parameters and the characteristics of the illuminated terrain.

La connaissance de la climatologie des états de mer est primordiale pour le dimensionnement de structures marines, la gestion des zones côtières ou encore la récupération de l’énergie des vagues. L'estimation de la climatologie nécessite de disposer de données d'observation sur une longue durée, ce qui n'est pas le cas de l'Afrique de l'Ouest. Pour dépasser les limites en durée imposées par les observations, nous proposons dans ces travaux une approche stochastique pour estimer une climatologie de vagues en Afrique de l’Ouest, en s’appuyant sur une représentation “par événements” des données d’états de mer. Un “événement” désigne un système de vagues (houle ou mer du vent) en évolution au cours du temps, observable pendant une durée significative et que l’on peut relier à un unique phénomène météorologique source (e.g. dépressions, tempêtes, etc.). La représentation par événements permet de reproduire la cohérence temporelle des systèmes de vagues et de structurer les données d'états de mer avec une base physique. La démarche adoptée peut se décomposer suivant trois étapes. Nous avons d'abord extrait les événements à partir d’une série temporelle de spectres directionnels d’états de mer. Nous avons ensuite développé un modèle pour représenter chacun des événements par un nombre réduit de paramètres. Enfin, nous avons construit un générateur stochastique permettant la simulation d’événements individuels et la reconstitution de climatologies sur des durées de longueurs arbitraires. Les résultats ont montré un bon accord entre la climatologie reconstituée et celle de référence, permettant de conclure que le générateur peut valablement servir à la simulation de données d’états de mer en Afrique de l’Ouest pour les applications en génie océanique. / Accurate estimation of long-term sea conditions is a major issue in design of coastal and offshore structures, coastal zone management or wave energy harvesting. An estimation of long-term sea conditions requires long duration observational data while in West Africa, only a few (3 years) years of observational data are available. To overcome the limits in duration that observations impose, a stochastic approach, event-based representation of sea state data, is proposed to model the wave climate in West Africa. An “event” refers to a wave system (swell or wind sea) evolving over time, that can be observed for a finite, yet significant duration and that can be linked to a single meteorological source phenomenon (e.g. low pressure systems, storms, etc.). Event-based approach provides structures with physical meaning and temporal consistence for the representation of sea states data. The procedure we have used is decomposed into three following steps. First, we have extracted events from a time series of directional spectra. We have then developed a model to represent each event by a reduced number of parameters. In the last step, we have constructed the stochastic events generator which allows for simulation of individual events and for reconstruction of wave climate over durations of arbitrary lengths. Results showed good agreement between reconstructed climate and that of reference and allow to conclude that the stochastic events generator can reliably be used to simulate sea state data in West Africa for a ocean engineering applications.

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[pt] No cenário atual das telecomunicações móveis, os arranjos de antenas voltaram a receber grande atenção dos pesquisadores, especialmente quando esquemas adaptativos de modificação de seus diagramas de radiação são utilizados. Uma das aplicações que exploram o potencial dos arranjos de antenas é o seu uso como forma de aumentar consideravelmente a eficiência espectral dos sistemas móveis atuais e da próxima geração. A outra aplicação em evidência está voltada para sistemas de localização de posição, pois algumas das técnicas conhecidas envolvem a estimação de ângulos-de-chegada usando arranjos de antenas. Diante destas possibilidades, cresce em importância o estudo das variações do canal de propagação rádio móvel no domínio em que o uso dos arranjos de antenas atua: o espacial. O presente trabalho procura contribuir para o contexto em questão, com uma investigação experimental do canal real rádio-móvel nos domínios temporal (retardos) e espacial (ângulos-de-chegada). No que se refere ao contexto nacional, contribuições similares baseadas em simulações já são encontradas; baseadas em medidas não. Em particular, sondagens na faixa de 1,8 GHz em ambientes internos típicos foram realizadas. Duas técnicas distintas de sondagem temporalespacial foram implementadas, tomando por base uma sonda de canal faixa-larga montada e testada com sucesso, como contribuição principal de uma dissertação de mestrado recentemente apresentada por um integrante do mesmo grupo de pesquisa ao qual esta tese está vinculada. Uma das técnicas sintetiza o arranjo realizando as sondagens com uma única antena que é sucessivamente deslocada para ocupar as posições correspondentes às dos elementos do arranjo. A outra técnica emprega um arranjo real. Em ambas, a configuração mais simples para um arranjo foi utilizada: a linear uniforme. As sondagens não forneciam diretamente os espectros espaciais-temporais. As estimativas dos espectros foram processadas posteriormente, aplicando técnicas como o correlograma para o domínio do retardo, e quatro técnicas distintas para o domínio espacial, que foi o foco principal deste trabalho: duas convencionais; e duas paramétricas, com potencial de aumentar a resolução das estimativas, assumindo hipóteses razoáveis sobre as respostas esperadas. De posse das respostas espectrais estimadas, comparações com estimativas teóricas permitiram uma análise de desempenho das técnicas utilizadas. Adicionalmente à investigação experimental do canal espacial, procurou-se verificar o potencial da aplicação da teoria de wavelets ao estudo do canal rádiomóvel. Em especial, uma das principais aplicações daquela teoria foi testada como técnica de pós-processamento das respostas espectrais no domínio do retardo. A supressão de ruído por decomposição wavelet foi aplicada a um vasto conjunto de medidas de canal disponíveis, fruto de trabalhos anteriores do grupo de pesquisa ao qual esta tese está vinculada, com resultados expressivos. / [en] In the present mobile communications scenario, researchers have turned once again special attention to antennae arrays, particularly when adaptive schemes are employed to modify its radiation patterns. One of its main applications results in considerable increases to the spectral efficiency of present and next generation mobile systems. The other major application is headed towards position location systems, since some of the known techniques comprise angle-of-arrival estimation using antennae arrays. Under such possibilities, mobile radio propagation channel variations studies grow in relevance, specially regarding the antennae arrays main domain of action: the spatial domain. The present work tries to contribute to the overstated context, experimentally investigating the actual mobile radio channel over the temporal (delays) and spatial (angles of arrival) domains. Regionally speaking, similar contributions based on simulations are already found, but none based on measurements. In special, 1.8 GHz indoor soundings have been carried out. Two different temporal spatial sounding techniques have been deployed, based on na available wideband channel sounder successfully assembled and tested as the major contribution of a MSc. dissertation recently presented by a member of the same research team to which this thesis belongs. One of such techniques sinthesyzes the array carrying the sounding out with a single antenna, which is successively moved to occupy the spots corresponding to the array elements. The other method employs an actual array. For both cases, the simplest array configuration has been used: the uniform linear one. Space-time spectra were not directly available in real time during the soundings. Its estimates have been processed later, applying techniques such as the correlogram over the delay domain, and four distinct methods over the spatial domain, the main focus of the present work. Two conventional methods have been used, as well as two parametric ones, potentially capable to increase the estimates resolution, assuming reasonable hypotheses regarding the expected responses. With the estimated spectral responses in hands, comparisons with theoretical estimates allowed a performance assessment of the employed methods. In addition to the spatial channel experimental investigation, the wavelets theory potential of application to the mobile-radio channel study has been checked out. Notably, one of the wavelets theory major applications has been tested as a post-processing technique to improve delay-domain spectral responses. Wavelet decomposition based de-noising has been applied to a huge measurements ensemble, available as the product of previous works of the research group to which this thesis is attached, leading to remarkable results.