The Hilbert transform is a widely used transform in signal processing. In this thesis we explore its use for three different applications: electrocardiography, the Hilbert-Huang transform and modulation. For electrocardiography, we examine how and why the Hilbert transform can be used for QRS complex detection. Also, what are the advantages and limitations of this method? The Hilbert-Huang transform is a very popular method for spectral analysis for nonlinear and/or nonstationary processes. We examine its connection with the Hilbert transform and show limitations of the method. Lastly, the connection between the Hilbert transform and single-sideband modulation is investigated.

In der Arbeit werden effektive Verfahren zur Auswertung der Hilbert-Transformation entwickelt und zur Lösung nichtlinearer Randwertaufgaben der Funktionentheorie eingesetzt. Die Verwendung polynomialer Spline-Wavelets und geeignet modifizierter Wavelet-Algorithmen ermöglichen die schnelle Berechnung auf gleichmäßigen und ungleichmäßigen Gittern sowie deren automatische Anpassung an lokale Besonderheiten der Lösung. Die detaillierte Untersuchung des Zusammenhangs zwischen der Glattheit, der Größe des Trägers des Splines, der Anzahl verschwindender Momente und des asymptotischen Verhaltens der Hilbert-Transformierten erlaubt die Anpassung der Parameter des Verfahrens in Bezug auf Genauigkeit und Effektivität. Im zweiten Teil der Arbeit werden verschiedene Algorithmen zur Lösung von Riemann-Hilbert Probleme vorgeschlagen und deren Konvergenzverhalten untersucht. Die theoretischen Ergebnisse werden durch numerische Experimente bestätigt.

Η διπλωματική εργασία με τίτλο «Μέθοδοι για Ανίχνευση και Χαρακτηρισμό Βιοσημάτων σε Θορυβώδεις Χρονοσειρές βασισμένοι στο Μετασχηματισμό Hilbert-Huang» μελετάει ζητήματα που σχετίζονται με βιοϊατρικά σήματα και την ανάλυση τους. Γίνεται διερεύνηση των διαθέσιμων τεχνικών και μεθόδων ανάλυσης βιοϊατρικών σημάτων, επισημαίνονται τα ιδιαίτερα χαρακτηριστικά των χρονοσειρών που προκύπτουν από την παρατήρηση και καταγραφή των σημάτων και έμφαση δίνεται στη μη στασιμότητα, την μη γραμμικότητα των υποκείμενων φυσικών διεργασιών και την ανάγκη προσαρμοστικότητας της μεθόδου. Μια μέθοδος που ικανοποιεί αυτές τις απαιτήσεις είναι η εμπειρική μέθοδος αποσύνθεσης η οποία αναλύει ένα σήμα σε ένα σύνολο συνιστωσών (IMFs) από τις οποίες ένα υποσύνολο θεωρείται ότι έχει φυσική σημασία. Επιπλέον, με το μετασχηματισμό Hilbert ανιχνεύονται οι στιγμιαίες συχνότητες και διαμορφώνεται η χρονοσυχνοτική κατανομή του σήματος. Τα θέματα που διερευνώνται αναφορικά με την εμπειρική μέθοδο αποσύνθεσης αφορούν τη στατιστική σημαντικότητα των IMFs, την αποθορυβοποίηση βιοϊατρικών σημάτων, την εξαγωγή χαρακτηριστικών από ηλεκτροκαρδιογράφημα και την απόδοση της μεθόδου. Ειδικά η απόδοση της εμπειρικής μεθόδου αποσύνθεσης είναι κρίσιμη παράμετρος για συστήματα με περιορισμένους πόρους όπως είναι οι κόμβοι ασύρματων δικτύων αισθητήρων ή τα ενσωματωμένα συστήματα. Η μοντελοποίηση μεθόδων που υλοποιούνται στο επίπεδο κόμβων ασύρματου δικτύου αισθητήρων είναι απαραίτητη για τη βέλτιστη διαχείριση πόρων και τον προγραμματισμό διεργασιών ώστε να μην διαταραχθεί η λειτουργία και λειτουργικότητα του συστήματος / This diploma thesis entitled "Methods for Identification and Characterization of Biosignals in Noise corrupted Time Series based on Hilbert-Huang Transform " studies issues concerning biomedical signal analysis. There is a review of the available techniques and methods for biomedical signal analysis pointing at certain characteristics of biomedical time series such as non stationarity, the non linearity of the underlying physical process and the need for the adaptive nature of the analysis method. One method that meets these requirements is considered to be the Empirical Mode Decomposition (EMD) which decomposes a signal into a set of components (IMFs) that a subset of them is believed to have a physical meaning. Application of Hilbert Transform on these IMFs provides the instantaneous frequencies and forms the time-frequency distribution of the signal. Issues studied are related to the statistical significance of the IMFs, denoising of biomedical signals, characteristics extraction and feature selection out of the electrocardiogram as well as the performance of the method. Particularly, the performance of empirical mode decomposition is considered to be a critical parameter especially in the case of implementation on nodes of wireless sensor networks or generally embedded systems due to the limited amount of resources available onboard. Modeling method's performance and demand for resources is a significant task facilitating the optimum resource management and task execution schedule of these systems.

Orientador: Ricardo Tokio Higuti / Banca: Alexandre César Rodrigues da Silva / Banca: Flávio Buiochi / Resumo: A estimação do tempo de atraso entre dois sinais de ultra-som é uma tarefa muito comum e importante em diversas aplicações, como em sistemas de posicionamento para medição de distâncias, medidores de espessura em ensaios não-destrutivos, células de medição de propriedades de materiais, entre outros. Em algumas aplicações há necessidade de elevada acurácia e precisão na determinação do tempo de atraso, que dependem de diversos parâmetros do transdutor, de sua excitação e do meio em que a onda se propaga, além do método de estimação e representação numérica. Neste trabalho são avaliados três estimadores de tempo de atraso, com implementações em ponto-fixo e ponto-flutuante: correlação cruzada com interpolação parabólica, transformada de Hilbert da correlação e envoltória do sinal analítico. Os estimadores são avaliados em MATLAB, em ponto-flutuante, com sinais sintetizados e com sinais reais obtidos em laboratório, e em ponto-fixo, usando um processador digital de sinais TMS320VC5416, da Texas Instruments. São explorados parâmetros como freqüência central do transdutor, freqüência de amostragem, largura de banda, relação sinal-ruído e atenuação do meio. O desempenho dos métodos é comparado por meio dos erros médios e desvios-padrão das medidas / Abstract: Time-delay estimation between two ultrasonic signals is a very common and important task in several applications, such as distance measurement in positioning systems, thickness measurement in nondestructive testing, measurement cells of materials properties, among others. Some applications require high accuracy and precision on the determination of the time-delay, which depend on several transducer parameters, excitation and medium of propagation, as well as the estimation method and numerical representation. In this work, three time-delay estimators are evaluated, with fixed- and floating-point implementations: cross-correlation with parabolic interpolation, Hilbert transform of correlation and analytic signal envelope. The estimators are evaluated in MATLAB with floating-point representation, using synthesized signals and real signals acquired in laboratory, and in fixed-point using a Texas Instruments TMS320VC5416 digital signal processor. Parameters as transducer central frequency, sampling frequency, bandwidth, signal-to-noise ratio and medium attenuation are considered. The performances of the methods are compared by means of errors (or bias) and standard deviations / Mestre

Orientador: Janito Vaqueiro Ferreira / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Eletrica / Made available in DSpace on 2018-08-04T09:11:42Z (GMT). No. of bitstreams: 1 Mariano_SilvioLuiz_M.pdf: 10538477 bytes, checksum: b5e594a3783a5571c8face1a74dae202 (MD5) Previous issue date: 2005 / Resumo: Este trabalho apresenta a análise numérica e experimental do índice de vibro-impacto em alavancas de transmissão mecânica. Foram desenvolvidos modelos de alavanca que consideram não-linearidades na estrutura. Métodos para quantificação do vibro-impacto foram obtidos na literatura e implementados para o levantamento de curvas, as quais indicam a variação do índice de Vibro-impacto com relação à variação dos parâmetros dos modelos desenvolvidos. A técnica da Transformada de Hilbert também foi implementada para levantamento das forças não-lineares atuantes nos modelos. Foram utilizadas três bancadas para identificação das forças não-lineares atuantes e validação da técnica salientada, que utiliza sinais de força, deslocamento, velocidade e aceleração. As curvas de rigidez não-Hnear foram identificadas coerentemente, as quais tiveram validação pelo levantamento da rigidez das mesmas bancadas por medições estáticas. Dois modelos numéricos de alavanca, utilizando massas concentradas, foram estudados. Nestes identificaram-se curvas de força não-Hnear e levantaram-se curvas de sensibilidade do índice de Vibro-impacto à variação dos parâmetros dos modelos. Mostrou-se que a técnica da Transformada de Hilbert pode ser utilizada para a identificação de sistemas não-lineares e que o estudo de sensibilidade em modelos de vibro-impacto pode ser obtido sem muita dificuldade / Abstract: This work presents a study about a numerical and experimental vibro-impact level analisys in mechanical gearshift levers. It was developed gearshift lever models which consider nonlinearities in the structure. Methods to vibro-impact quantification were obtained in the available literature and were implemented to obtain curves which show the vibro-impact level change with the modification of the parameters of the developed models. Hubert Transform techniques were also implemented to obtain nonlinear forces implemented on the models. To verify the efficiency of these techniques, three test rigs with nonlinearities were developed and the nonlinear forces were identified. The techniques need the displacement, velocity and acceleration signals to calculate the forces. The nonlinear forces curves were identified and validated with the static forces measured. Two models of gearshift levers were developed, using lumped masses. In these models the nonlinear forces curves and the sensibility to the vibro-impact level with modifications in the model parameters were obtained. It was showed that the Hubert Transform techniques are reliable to identify nonlinear systems and that the sensibility study in vibro-impact can be obtained easily / Mestrado / Mecanica dos Sólidos e Projeto Mecanico / Mestre em Engenharia Mecânica

We investigate optimal partnership of rearrangement-invariant Banach func- tion spaces for the Hilbert transform and the Riesz potential. We establish sharp theorems which characterize optimal action of these operators on such spaces. These results enable us to construct optimal domain (i.e. the largest) and op- timal range (i.e. the smallest) partner spaces when the other space is given. We illustrate the obtained results by non-trivial examples involving Generalized Lorentz-Zygmund spaces with broken logarithmic functions. The method is pre- sented in such a way that it should be easily adaptable to other appropriate operators. 1

Parametric acoustic arrays are ultrasonic-based loudspeakers that produce highly directive audio. The audio must first be preprocessed and modulated into an ultrasonic carrier before being emitted into the air, where it will self-demodulate in the far field. The resulting audio wave is proportional to the double time-derivative of the square of the modulation envelope. This thesis presents a fully analog processor which encodes the audio into two Pulse Width Modulated (PWM) signals in quadrature phase and sums them together to produce a Single Sideband (SSB) spectrum around the fundamental frequency of the PWM signals. The two signals are modulated between 8% and 24% duty cycle to maintain a quasi-linear relationship between the duty cycle and the output signal level. This also allows the signals to sum without overlapping each other, maintaining a two-level output. The system drives a network of narrowband transducers with a center frequency equal to the PWM fundamental. Because the transducers are voltage driven, they have a bandpass frequency response which behaves as a first-order integrator on the SSB signal, eliminating the need for two integrators in the processor. Results show that the “SSB-PWM” output wave has a consistent 20-30dB difference in magnitude between the upper sideband and lower sideband. In simulation, a single tone test shows higher total harmonic distortion for lower frequencies and higher modulation depth. A two-tone test creates a 2nd order intermodulation term that increases with the frequencies of the input signals.

The diploma work deals with techniques of optical contactless distance and velocity measurement. A basic summary of the methods are involved. The problematic of interferometric methods for vibration measurements is analysed in detail. It contains division of interferometers, description of their function principles and also chapters dealing with elements used in interferometers such as lasers, photodetectors and elements in the ray optical way - polarizers, retarders, optical isolators. The vibration and length measurement methods are described, as well as the conception of homodyne and heterodyne detection. Part of this work focuses on the quadrature signal processing and on the proposal of algorithm for demodulation of velocity/displacement and undergoing simple motioning object deviation. This algorithm is implemented in Labview and the whole software instrument served also for visualisation of measured data of the interferometer model constructed in the laboratory. The way how to build up a model, its setting and two possible configurations suitable for homodyne detection are described. Model of interferometer is built-up on the optical breadboard. Particular components are fixed by the help of mounts. The model and software enable to measure the velocity and the vibration deviation with the light wavelength exactness. Functionality and the exactness of the laboratory model are verified by vibrometer. Effects on the measurement uncertainty are discussed here and ways how to restrain them are proposed.

In der Arbeit werden effektive Verfahren zur Auswertung der Hilbert-Transformation entwickelt und zur Lösung nichtlinearer Randwertaufgaben der Funktionentheorie eingesetzt. Die Verwendung polynomialer Spline-Wavelets und geeignet modifizierter Wavelet-Algorithmen ermöglichen die schnelle Berechnung auf gleichmäßigen und ungleichmäßigen Gittern sowie deren automatische Anpassung an lokale Besonderheiten der Lösung. Die detaillierte Untersuchung des Zusammenhangs zwischen der Glattheit, der Größe des Trägers des Splines, der Anzahl verschwindender Momente und des asymptotischen Verhaltens der Hilbert-Transformierten erlaubt die Anpassung der Parameter des Verfahrens in Bezug auf Genauigkeit und Effektivität. Im zweiten Teil der Arbeit werden verschiedene Algorithmen zur Lösung von Riemann-Hilbert Probleme vorgeschlagen und deren Konvergenzverhalten untersucht. Die theoretischen Ergebnisse werden durch numerische Experimente bestätigt.

We present Lax pairs and functions called Lax functions corresponding to Calogero- Moser-Sutherland (CMS) systems. We present the Benjamin-Ono (BO) equation and a pole ansatz to the BO equation, constructed from a specific type of Lax function called a special Lax function corresponding to Rational and Trigonometric CMS systems. We present a generalization of the BO equation called the non-chiral Intermediate wave (ncILW) equation and show that a family of solutions to the ncILW equation can be constructed from the special Lax function corresponding to the hyperbolic CMS system. We present the Szegö equation on the circle and the real line. We obtain a family of solutions to the Szegö equation on the real line using a pole ansatz. Using numerical methods, we display solution plots to the BO equation and Szegö equation.