abstract: Signal compressed using classical compression methods can be acquired using brute force (i.e. searching for non-zero entries in component-wise). However, sparse solutions require combinatorial searches of high computations. In this thesis, instead, two Bayesian approaches are considered to recover a sparse vector from underdetermined noisy measurements. The first is constructed using a Bernoulli-Gaussian (BG) prior distribution and is assumed to be the true generative model. The second is constructed using a Gamma-Normal (GN) prior distribution and is, therefore, a different (i.e. misspecified) model. To estimate the posterior distribution for the correctly specified scenario, an algorithm based on generalized approximated message passing (GAMP) is constructed, while an algorithm based on sparse Bayesian learning (SBL) is used for the misspecified scenario. Recovering sparse signal using Bayesian framework is one class of algorithms to solve the sparse problem. All classes of algorithms aim to get around the high computations associated with the combinatorial searches. Compressive sensing (CS) is a widely-used terminology attributed to optimize the sparse problem and its applications. Applications such as magnetic resonance imaging (MRI), image acquisition in radar imaging, and facial recognition. In CS literature, the target vector can be recovered either by optimizing an objective function using point estimation, or recovering a distribution of the sparse vector using Bayesian estimation. Although Bayesian framework provides an extra degree of freedom to assume a distribution that is directly applicable to the problem of interest, it is hard to find a theoretical guarantee of convergence. This limitation has shifted some of researches to use a non-Bayesian framework. This thesis tries to close this gab by proposing a Bayesian framework with a suggested theoretical bound for the assumed, not necessarily correct, distribution. In the simulation study, a general lower Bayesian Cram\'er-Rao bound (BCRB) bound is extracted along with misspecified Bayesian Cram\'er-Rao bound (MBCRB) for GN model. Both bounds are validated using mean square error (MSE) performances of the aforementioned algorithms. Also, a quantification of the performance in terms of gains versus losses is introduced as one main finding of this report. / Dissertation/Thesis / Masters Thesis Computer Engineering 2019

<p>Recent advances in technology have allowed researchers to collect high-dimensional biological data simultaneously. In genomic studies, for instance, measurements from tens of thousands of genes are taken from individuals across several experimental groups. In time course microarray experiments, gene expression is measured at several time points for each individual across the whole genome resulting in massive amount of data. In such experiments, researchers are faced with two types of high-dimensionality. The first is global high-dimensionality, which is common to all genomic experiments. The global high-dimensionality arises because inference is being done on tens of thousands of genes resulting in multiplicity. This challenge is often dealt with statistical methods for multiple comparison, such as the Bonferroni correction or false discovery rate (FDR). We refer to the second type of high-dimensionality as gene specific high-dimensionality, which arises in time course microarry experiments due to the fact that, in such experiments, sample size is often smaller than the number of time points ($n</p> <p>In this thesis, we use the growth curve model (GCM), which is a generalized multivariate analysis of variance (GMANOVA) model, and propose a moderated test statistic for testing a special case of the general linear hypothesis, which is specially useful for identifying genes that are expressed. We use the trace test for the GCM and modify it so that it can be used in high-dimensional situations. We consider two types of moderation: the Moore-Penrose generalized inverse and Stein's shrinkage estimator of $ S $. We performed extensive simulations to show performance of the moderated test, and compared the results with original trace test. We calculated empirical level and power of the test under many scenarios. Although the focus is on hypothesis testing, we also provided moderated maximum likelihood estimator for the parameter matrix and assessed its performance by investigating bias and mean squared error of the estimator and compared the results with those of the maximum likelihood estimators. Since the parameters are matrices, we consider distance measures in both power and level comparisons as well as when investigating bias and mean squared error. We also illustrated our approach using time course microarray data taken from a study on Lung Cancer. We were able to filter out 1053 genes as non-noise genes from a pool of 22,277 genes which is approximately 5\% of the total number of genes. This is in sync with results from most biological experiments where around 5\% genes are found to be differentially expressed.</p> / Master of Science (MSc)

[EN] Common audio systems are designed with the intent of creating real and immersive scenarios that allow the user to experience a particular acoustic sensation that does not depend on the room he is perceiving the sound. However, acoustic devices and multichannel rendering systems working inside a room, can impair the global audio effect and thus the 3D spatial sound. In order to preserve the spatial sound characteristics of multichannel rendering techniques, adaptive filtering schemes are presented in this dissertation to compensate these electroacoustic effects and to achieve the immersive sensation of the desired acoustic system. Adaptive filtering offers a solution to the room equalization problem that is doubly interesting. First of all, it iteratively solves the room inversion problem, which can become computationally complex to obtain when direct methods are used. Secondly, the use of adaptive filters allows to follow the time-varying room conditions. In this regard, adaptive equalization (AE) filters try to cancel the echoes due to the room effects. In this work, we consider this problem and propose effective and robust linear schemes to solve this equalization problem by using adaptive filters. To do this, different adaptive filtering schemes are introduced in the AE context. These filtering schemes are based on three strategies previously introduced in the literature: the convex combination of filters, the biasing of the filter weights and the block-based filtering. More specifically, and motivated by the sparse nature of the acoustic impulse response and its corresponding optimal inverse filter, we introduce different adaptive equalization algorithms. In addition, since audio immersive systems usually require the use of multiple transducers, the multichannel adaptive equalization problem should be also taken into account when new single-channel approaches are presented, in the sense that they can be straightforwardly extended to the multichannel case. On the other hand, when dealing with audio devices, consideration must be given to the nonlinearities of the system in order to properly equalize the electroacoustic system. For that purpose, we propose a novel nonlinear filtered-x approach to compensate both room reverberation and nonlinear distortion with memory caused by the amplifier and loudspeaker devices. Finally, it is important to validate the algorithms proposed in a real-time implementation. Thus, some initial research results demonstrate that an adaptive equalizer can be used to compensate room distortions. / [ES] Los sistemas de audio actuales están diseñados con la idea de crear escenarios reales e inmersivos que permitan al usuario experimentar determinadas sensaciones acústicas que no dependan de la sala o situación donde se esté percibiendo el sonido. Sin embargo, los dispositivos acústicos y los sistemas multicanal funcionando dentro de salas, pueden perjudicar el efecto global sonoro y de esta forma, el sonido espacial 3D. Para poder preservar las características espaciales sonoras de los sistemas de reproducción multicanal, en esta tesis se presentan los esquemas de filtrado adaptativo para compensar dichos efectos electroacústicos y conseguir la sensación inmersiva del sistema sonoro deseado. El filtrado adaptativo ofrece una solución al problema de salas que es interesante por dos motivos. Por un lado, resuelve de forma iterativa el problema de inversión de salas, que puede llegar a ser computacionalmente costoso para los métodos de inversión directos existentes. Por otro lado, el uso de filtros adaptativos permite seguir las variaciones cambiantes de los efectos de la sala de escucha. A este respecto, los filtros de ecualización adaptativa (AE) intentan cancelar los ecos introducidos por la sala de escucha. En esta tesis se considera este problema y se proponen esquemas lineales efectivos y robustos para resolver el problema de ecualización mediante filtros adaptativos. Para conseguirlo, se introducen diferentes esquemas de filtrado adaptativo para AE. Estos esquemas de filtrado se basan en tres estrategias ya usadas en la literatura: la combinación convexa de filtros, el sesgado de los coeficientes del filtro y el filtrado basado en bloques. Más especificamente y motivado por la naturaleza dispersiva de las respuestas al impulso acústicas y de sus correspondientes filtros inversos óptimos, se presentan diversos algoritmos adaptativos de ecualización específicos. Además, ya que los sistemas de audio inmersivos requieren usar normalmente múltiples trasductores, se debe considerar también el problema de ecualización multicanal adaptativa cuando se diseñan nuevas estrategias de filtrado adaptativo para sistemas monocanal, ya que éstas deben ser fácilmente extrapolables al caso multicanal. Por otro lado, cuando se utilizan dispositivos acústicos, se debe considerar la existencia de no linearidades en el sistema elactroacústico, para poder ecualizarlo correctamente. Por este motivo, se propone un nuevo modelo no lineal de filtrado-x que compense a la vez la reverberación introducida por la sala y la distorsión no lineal con memoria provocada por el amplificador y el altavoz. Por último, es importante validar los algoritmos propuestos mediante implementaciones en tiempo real, para asegurarnos que pueden realizarse. Para ello, se presentan algunos resultados experimentales iniciales que muestran la idoneidad de la ecualización adaptativa en problemas de compensación de salas. / [CA] Els sistemes d'àudio actuals es dissenyen amb l'objectiu de crear ambients reals i immersius que permeten a l'usuari experimentar una sensació acústica particular que no depèn de la sala on està percebent el so. No obstant això, els dispositius acústics i els sistemes de renderització multicanal treballant dins d'una sala poden arribar a modificar l'efecte global de l'àudio i per tant, l'efecte 3D del so a l'espai. Amb l'objectiu de conservar les característiques espacials del so obtingut amb tècniques de renderització multicanal, aquesta tesi doctoral presenta esquemes de filtrat adaptatiu per a compensar aquests efectes electroacústics i aconseguir una sensació immersiva del sistema acústic desitjat. El filtrat adaptatiu presenta una solució al problema d'equalització de sales que es interessant baix dos punts de vista. Per una banda, el filtrat adaptatiu resol de forma iterativa el problema inversió de sales, que pot arribar a ser molt complexe computacionalment quan s'utilitzen mètodes directes. Per altra banda, l'ús de filtres adaptatius permet fer un seguiment de les condicions canviants de la sala amb el temps. Més concretament, els filtres d'equalització adaptatius (EA) intenten cancel·lar els ecos produïts per la sala. A aquesta tesi, considerem aquest problema i proposem esquemes lineals efectius i robustos per a resoldre aquest problema d'equalització mitjançant filtres adaptatius. Per aconseguir-ho, diferent esquemes de filtrat adaptatiu es presenten dins del context del problema d'EA. Aquests esquemes de filtrat es basen en tres estratègies ja presentades a l'estat de l'art: la combinació convexa de filtres, el sesgat dels pesos del filtre i el filtrat basat en blocs. Més concretament, i motivat per la naturalesa dispersa de la resposta a l'impuls acústica i el corresponent filtre òptim invers, presentem diferents algorismes d'equalització adaptativa. A més a més, com que els sistemes d'àudio immersiu normalment requereixen l'ús de múltiples transductors, cal considerar també el problema d'equalització adaptativa multicanal quan es presenten noves solucions de canal simple, ja que aquestes s'han de poder estendre fàcilment al cas multicanal. Un altre aspecte a considerar quan es treballa amb dispositius d'àudio és el de les no linealitats del sistema a l'hora d'equalitzar correctament el sistema electroacústic. Amb aquest objectiu, a aquesta tesi es proposa una nova tècnica basada en filtrat-x no lineal, per a compensar tant la reverberació de la sala com la distorsió no lineal amb memòria introduïda per l'amplificador i els altaveus. Per últim, és important validar la implementació en temps real dels algorismes proposats. Amb aquest objectiu, alguns resultats inicials demostren la idoneïtat de l'equalització adaptativa en problemes de compensació de sales. / Fuster Criado, L. (2015). Linear and nonlinear room compensation of audio rendering systems [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/59459

In developing an understanding of real-world problems, researchers develop mathematical and statistical models. Various model selection methods exist which can be used to obtain a mathematical model that best describes the real-world situation in some or other sense. These methods aim to assess the merits of competing models by concentrating on a particular criterion. Each selection method is associated with its own criterion and is named accordingly. The better known ones include Akaike's Information Criterion, Mallows' Cp and cross-validation, to name a few. The value of the criterion is calculated for each model and the model corresponding to the minimum value of the criterion is then selected as the "best" model. / Mathematical Sciences / M. Sc. (Statistics)

Η παρούσα μεταπτυχιακή διατριβή εντάσσεται ερευνητικά στην περιοχή της Στατιστικής Θεωρίας Αποφάσεων και ειδικότερα στην εκτίμηση των παραμέτρων στο μοντέλο της διπαραμετρικής εκθετικής κατανομής με παράμετρο θέσης μ και παράμετρο κλίμακος σ. Θεωρούμε το πρόβλημα εκτίμησης των παραμέτρων κλίμακας μ και θέσης σ, όταν μ≤c, όπου c είναι μία γνωστή σταθερά. Αποδεικνύουμε ότι σε σχέση με το κριτήριο του Μέσου Τετραγωνικού Σφάλματος (ΜΤΣ), οι βέλτιστοι αναλλοίωτοι εκτιμητές των μ και σ, είναι μη αποδεκτοί όταν μ≤c, και προτείνουμε βελτιωμένους. Επίσης συγκρίνουμε του εκτιμητές αυτούς σε σχέση με το κριτήριο του Pitman. Επιπλέον, προτείνουμε εκτιμητές που είναι καλύτεροι από τους βέλτιστους αναλλοίωτους εκτιμητές, όταν μ≤c, ως προς την συνάρτηση ζημίας LINEX. Τέλος, η θεωρία που αναπτύσσεται εφαρμόζεται σε δύο ανεξάρτητα δείγματα προερχόμενα από εκθετική κατανομή. / The present master thesis deals with the estimation of the location parameter μ and the scale parameter σ of the two-parameter exponential distribution. We consider the problem of estimation of locasion parameter μ and the scale parameter σ, when it is known apriori that μ≤c, where c is a known constant. We establish that with respect to the mean square error (mse) criterion the best affine estimators of μ and σ in the absence of information μ≤c are inadmissible and we propose estimators which are better than these estimators. Also, we compare these estimators with respect to the Pitman Nearness criterion. We propose estimators which are better than the standard estimators in the unrestricted case with respect to the suitable choise of LINEX loss. Finally, the theory developed is applied to the problem of estimating the location and scale parameters of two exponential distributions when the location parameters are ordered.

Cette thèse porte sur l’étude du positionnement optimale des réseaux de capteurs pour la localisation de sources. Nous avons étudié deux approches: l’approche basée sur les performances de l’estimation en termes d’erreur quadratique moyenne et l’approche basée sur le seuil statistique de résolution (SSR).Pour le première approche, nous avons considéré les bornes inférieures de l’erreur quadratique moyenne qui sont utilisés généralement pour évaluer la performance d’estimation indépendamment du type d’estimateur considéré. Nous avons étudié deux types de bornes: la borne Cramér-Rao (BCR) pour le modèle où les paramètres sont supposés déterministes et la borne Weiss-Weinstein (BWW) pour le modèle où les paramètres sont supposés aléatoires. Nous avons dérivé les expressions analytiques de ces bornes pour développer des outils statistiques afin d’optimiser la géométrie des réseaux de capteurs. Par rapport à la BCR, la borne BWW peut capturer le décrochement de l’EQM des estimateurs dans la zone non-asymptotique. De plus, les expressions analytiques de la BWW pour un modèle Gaussien général à moyenne paramétré ou à covariance matrice paramétré sont donnés explicitement. Basé sur ces expressions analytiques, nous avons étudié l’impact de la géométrie des réseaux de capteurs sur les performances d’estimation en utilisant les réseaux de capteurs 3D et 2D pour deux modèles des observations concernant les signaux sources: (i) le modèle déterministe et (ii) le modèle stochastique. Nous en avons ensuite déduit des conditions concernant les propriétés d’isotropie et de découplage.Pour la deuxième approche, nous avons considéré le seuil statistique de résolution qui caractérise la séparation minimale entre les deux sources. Dans cette thèse, nous avons étudié le SSR pour le contexte Bayésien moins étudié dans la littérature. Nous avons introduit un modèle des observations linéarisé basé sur le critère de probabilité d’erreur minimale. Ensuite, nous avons présenté deux approches Bayésiennes pour le SSR, l’une basée sur la théorie de l’information et l’autre basée sur la théorie de la détection. Ces approches pourront être utilisée pour améliorer la capacité de résolution des systèmes. / This thesis deals with the array geometry optimization problem in the context of sources localization. We have considered two approaches for the array geometry optimization: the performance estimation in terms of mean square error approach and the statistical resolution limit (SRL) approach. In the first approach, the lower bounds on the mean square error which are usually used in array processing to evaluate the estimation performance independently of the considered estimator have been considered. We have investigated two kinds of lower bounds: the well-known Cramér-Rao bound (CRB) for the deterministic model in which the parameters are assumed to be deterministic, and the Weiss-Weinstein bound (WWB) which is less studied, for the Bayesian model, in which, the parameters are assumed to be random with some prior distributions. We have proposed closed-form expressions of these bounds, which can be used as a statistical tool for array geometry design. Compared to the CRB, the WWB can predict the threshold effect of the MSE in the non-asymptotic area. Moreover, the closed-form expressions of the WWB proposed for a general Gaussian model with parameterized mean or parameterized covariance matrix can also be useful for other problems. Based on these closed-form expressions, the 3D array geometry and the classical planar array geometry have been investigated under (i) the conditional observation model in which the source signal is modeled as a deterministic sequence and under (ii) the unconditional observation model in which the source signal is modeled as a Gaussian random process. Conditions concerning the isotropic and uncoupling properties were then derived.In the second approach, we have considered the statistical resolution limit which characterizes the minimal separation between the two closed spaced sources which still allows to determine correctly the number of sources. In this thesis, we are interested in the SRL in the Bayesian context which is less studied in the literature. Based on the linearized observation model with the minimum probability of error, we have introduced the two Bayesian approaches of the SRL based on the detection and information theories which could lead to some interesting tools for the system design.

Cette thèse porte sur l'analyse des performances en traitement du signal et se compose de deux parties: Premièrement, nous étudions les bornes inférieures dans la caractérisation et la prédiction des performances en termes d'erreur quadratique moyenne (EQM). Les bornes inférieures de l'EQM donne la variance minimale qu'un estimateur peut atteindre et peuvent être divisées en deux catégories: les bornes déterministes pour le modèle où les paramètres sont supposés déterministes (mais inconnus), et les bornes Bayésiennes pour le modèle où les paramètres sont supposés aléatoires. En particulier, nous dérivons les expressions analytiques de ces bornes pour deux applications différentes: (i) La première est la localisation des sources en utilisant un radar multiple-input multiple-output (MIMO). Nous considérons les bornes inférieures dans deux contextes c'est-à-dire avec ou sans erreurs de modèle. (ii) La deuxième est l'estimation de phase d'impulsion de pulsars à rayon X qui est une solution potentielle pour la navigation autonome dans l'espace. Pour cette application, nous avons calculé plusieurs bornes inférieures de l'EQM dans le contexte de données modélisées par une loi de Poisson (complétant ainsi les travaux disponibles dans la littérature où les données sont modélisées par une loi gaussienne). Deuxièmement, nous étudions le seuil statistique de résolution limite (SRL), qui est la distance minimale en termes des paramètres d'intérêts entre les deux signaux permettant de séparer / estimer correctement les paramètres d'intérêt. Plus précisément, nous dérivons le SRL dans deux contextes: le traitement d'antenne et le radar MIMO en utilisant deux approches basées sur la théorie de l'estimation et sur la théorie de l'information. Finalement, nous proposons des expressions compactes du SRL dans le cas d'erreurs de modèle. / This manuscript concerns the performance analysis in signal processing and consists into two parts : First, we study the lower bounds in characterizing and predicting the estimation performance in terms of mean square error (MSE). The lower bounds on the MSE give the minimum variance that an estimator can expect to achieve and it can be divided into two categories depending on the parameter assumption: the so-called deterministic bounds dealing with the deterministic unknown parameters, and the so-called Bayesian bounds dealing with the random unknown parameter. Particularly, we derive the closed-form expressions of the lower bounds for two applications in two different fields: (i) The first one is the target localization using the multiple-input multiple-output (MIMO) radar in which we derive the lower bounds in the contexts with and without modeling errors, respectively. (ii) The other one is the pulse phase estimation of X-ray pulsars which is a potential solution for autonomous deep space navigation. In this application, we show the potential universality of lower bounds to tackle problems with parameterized probability density function (pdf) different from classical Gaussian pdf since in X-ray pulse phase estimation, observations are modeled with a Poisson distribution. Second, we study the statistical resolution limit (SRL) which is the minimal distance in terms of the parameter of interest between two signals allowing to correctly separate/estimate the parameters of interest. More precisely, we derive the SRL in two contexts: array processing and MIMO radar by using two approaches based on the estimation theory and information theory. We also present in this thesis the usefulness of SRL in optimizing the array system.

O objetivo deste trabalho e o desenvolvimento de uma ferramenta estatistica que sirva de base para o estudo do preco spot da energia eletrica do subsistema Sudeste/Centro-Oeste do Sistema Interligado Nacional, utilizando a estimacao por regressao linear e teste de razao de verossimilhanca como instrumentos para desenvolvimento e avaliacao dos modelos. Na analise dos resultados estatsticos descritivos dos modelos, diferentemente do que e observado na literatura, a primeira conclusao e a verificacao de que as variaveis sazonais, quando analisadas isoladamente, apresentam resultados pouco aderentes ao preco spot PLD. Apos a analise da componente sazonal e verificada a influencia da energia fornecida e a energia demandada como variaveis de entrada, com o qual conclui-se que especificamente a energia armazenada e producao de energia termeletrica sao as variaveis que mais influenciam os precos spot no subsistema estudado. Entre os modelos testados, o que particularmente ofereceu os melhores resultados foi um modelo misto criado a partir da escolha das melhores variaveis de entrada dos modelos testados preliminarmente, alcancando um coeficiente de determinacao R2 de 0.825, resultado esse que pode ser considerado aderente ao preco spot. No ultimo capitulo e apresentada uma introducao ao modelo de predicao do preco spot, possibilitando dessa forma a analise do comportamento do preco a partir da alteracao das variaveis de entrada. / The objective of this work is the development of a statistical method to study the spot prices of the electrical energy of the Southeast/Middle-West (SE-CO) subsystem of the The Brazilian National Connected System, using the Least Squares Estimation and Likelihood Ratio Test as tools to perform and evaluate the models. Verifying the descriptive statistical results of the models, differently from what is observed in the literature, the first observation is that the seasonal component, when analyzed alone, presented results loosely adherent to the spot price PLD. It is then evaluated the influence of the energy supply and the energy demand as input variables, verifying that specifically the stored water and the thermoelectric power production are the variables that the most influence the spot prices in the studied subsystem. Among the models, the one that offered the best result was a mixed model created from the selection of the best input variables of the preliminarily tested models, achieving a coeficient of determination R2 of 0.825, a result that can be considered adherent to the spot price. At the last part of the work It is presented an introduction to the spot price prediction model, allowing the analysis of the price behavior by the changing of the input variables.

In general, PN code synchronization consists of two steps: PN code acquisition (coarse alignment) and PN code tracking (fine alignment), to estimate the delay offset between received and locally generated codes. Recently, the schemes with a joint adaptive process of PN code acquisition and the weight coefficients of smart antenna have been proposed for improving the received signal-to-interference-plus-noise ratio (SINR) and simultaneously achieving better mean-acquisition-time (MAT) performance in direct-sequence code-division multiple access (DS-CDMA) systems. In which, the setting of the threshold plays an important role on the MAT performance. Often, the received SINR is varying, using the fixed threshold acquisition algorithms may result in undesirable performance. To improve the above problem, in this thesis, a new adaptive threshold scheme is devised in a joint adaptive code acquisition and beam-forming DS-CDMA receiver for code acquisition under a fading multipath and additive white Gaussian-noise (AWGN) channels. The basic idea of this new adaptive threshold scheme is to estimate the averaged output power of smart antenna to scale a reference threshold for each observation interval, such that it can approximately achieve a constant false alarm rate (CFAR) criteria. The system probabilities of the proposed scheme are derived for evaluating MAT under a slowly fading two-paths channels. Numerical analyses and simulation results demonstrate that the proposed adaptive threshold scheme does achieve better performance, in terms of the output SINR, the detection probability and the MAT, compared to a fixed threshold method.

Power system generators are interconnected in a distributed network to allow sharing of power. If one of the generators cannot meet the power demand, spare power is diverted from neighbouring generators. However, this approach also allows for propagation of electric disturbances. An oscillation arising from a disturbance at a given generator site will affect the normal operation of neighbouring generators and might cause them to fail. Hours of production time will be lost in the time it takes to restart the power plant. If the disturbance is detected early, appropriate control measures can be applied to ensure system stability. The aim of this study is to improve existing algorithms that estimate the oscillation parameters from acquired generator data to detect potentially dangerous power system disturbances. When disturbances occur in power systems (due to load changes or faults), damped oscillations (or &quotmodes") are created. Modes which are heavily damped die out quickly and pose no threat to system stability. Lightly damped modes, by contrast, die out slowly and are more problematic. Of more concern still are &quotnegatively damped" modes which grow exponentially with time and can ultimately cause the power system to fail. Widespread blackouts are then possible. To avert power system failures it is necessary to monitor the damping of the oscillating modes. This thesis proposes a number of damping estimation algorithms for this task. If the damping is found to be very small or even negative, then additional damping needs to be introduced via appropriate control strategies. This thesis presents a number of new algorithms for estimating the damping of modal oscillations in power systems. The first of these algorithms uses multiple orthogonal sliding windows along with least-squares techniques to estimate the modal damping. This algorithm produces results which are superior to those of earlier sliding window algorithms (that use only one pair of sliding windows to estimate the damping). The second algorithm uses a different modification of the standard sliding window damping estimation algorithm - the algorithm exploits the fact that the Signal to Noise Ratio (SNR) within the Fourier transform of practical power system signals is typically constant across a wide frequency range. Accordingly, damping estimates are obtained at a range of frequencies and then averaged. The third algorithm applied to power system analysis is based on optimal estimation theory. It is computationally efficient and gives optimal accuracy, at least for modes which are well separated in frequency.