Journal articles on the topic 'Spline-Wavelets, Hilbert-Transformation, Riemann-Hilbert Probleme'

En el presente artículo se describe e implementa una solución al problema de la tomografía local, equivalente a la inversión de la transformada de Radon, utilizando la transformada wavelet. Para ello se ejecuta un algoritmo basado en wavelets B-spline cúbicos de soporte compacto con suficientes momentos de desvanecimiento para que la función de escalado filtrada, la wavelet madre y su transformada de Hilbert tengan decaimiento rápido. Lo anterior favorece la localización de la transformada wavelet de la transformada de Radon y, por tanto, la inversión, es decir; la reconstrucción de una región central de interés del fantasma Shepp-Logan.

This study proposed an enhanced time-frequency representation of audio signal using EMD-2TEMD based approach. To analyze non-stationary signal like audio, timefrequency representation is an important aspect. In case of representing or analyzing such kind of signal in time-frequency-energy distribution, hilbert spectrum is a recent approach and popular way which has several advantages over other methods like STFT, WT etc. Hilbert-Huang Transform (HHT) is a prominent method consists of Empirical Mode Decomposition (EMD) and Hilbert Spectral Analysis (HSA). An enhanced method called Turning Tangent empirical mode decomposition (2T-EMD) has recently developed to overcome some limitations of classical EMD like cubic spline problems, sifting stopping condition etc. 2T-EMD based hilbert spectrum of audio signal encountered some issues due to the generation of too many IMFs in the process where EMD produces less. To mitigate those problems, a mutual implementation of 2T-EMD & classical EMD is proposed in this paper which enhances the representation of hilbert spectrum along with significant improvements in source separation result using Independent Subspace Analysis (ISA) based clustering in case of audio signals. This refinement of hilbert spectrum not only contributes to the future work of source separation problem but also many other applications in audio signal processing.

In the introduction of this paper is presented the definition of the generalized spline functions as solutions of a variational problem and are shown some theorems regarding to the existence and uniqueness. The main result of this article consists in a remarkable equality verified by the generalized spline elements, based on the properties of the spaces, operator and interpolatory set involved, which can be used as a characterization theorem of the generalized spline functions in Hilbert spaces.

Wahba (1978) and Weinert et al. (1980), using different models, show that an optimal smoothing spline can be thought of as the conditional expectation of a stochastic process observed with noise. This observation leads to efficient computational algorithms. By going back to the Hilbert space formulation of the spline minimization problem, we provide a framework for linking the two different stochastic models. The last part of the paper reviews some new efficient algorithms for spline smoothing.

Wahba (1978) and Weinert et al. (1980), using different models, show that an optimal smoothing spline can be thought of as the conditional expectation of a stochastic process observed with noise. This observation leads to efficient computational algorithms. By going back to the Hilbert space formulation of the spline minimization problem, we provide a framework for linking the two different stochastic models. The last part of the paper reviews some new efficient algorithms for spline smoothing.

A matching pursuit method based on Dual-Tree Complex Wavelet Transform (DT-CWT) is proposed for extracting feature. Many new orthogonal wavelet bases formed Hilbert transform pairs is constructed by the method which is based on the sufficient and necessary condition on constructing wavelet, via the flat delay filter, and translated the problem into resolving algebraic equations. And taking these wavelets as choice object, a matching pursuit method based on DT-CWT is used for extracting feature. The matching pursuit method is based on series expansion of the signal by a set of elementary functions of orthogonal wavelets formed Hilbert transform pairs to match feature more effectively. Simulation testing and field experiments confirm that the proposed method is effective especially in extracting impulsive feature on high intensity noise, which matching pursuit method based on Discrete Wavelet Transform and other wavelet de-noising methods based on threshold and frequency-band, etc cannot do it completely.

The present article reviews the multiple applications of group theory to the symmetry problems in physics. In classical physics, this concerns primarily relativity: Euclidean, Galilean, and Einsteinian (special). Going over to quantum mechanics, we first note that the basic principles imply that the state space of a quantum system has an intrinsic structure of pre-Hilbert space that one completes into a genuine Hilbert space. In this framework, the description of the invariance under a group G is based on a unitary representation of G. Next, we survey the various domains of application: atomic and molecular physics, quantum optics, signal and image processing, wavelets, internal symmetries, and approximate symmetries. Next, we discuss the extension to gauge theories, in particular, to the Standard Model of fundamental interactions. We conclude with some remarks about recent developments, including the application to braid groups.

Hilbert Space has wide usefulness in signal processing research. It is pitched at a graduate student level, but relies only on undergraduate background material. The needs and concerns of the researchers In engineering differ from those of the pure science. It is difficult to put the finger on what distinguishes the engineering approach that we have taken. In the end, if a potential use emerges from any result, however abstract, then an engineer would tend to attach greater value to that result. This may serve to distinguish the emphasis given by a mathematician who may be interested in the proof of a fundamental concept that links deeply with other areas of mathematics or is a part of a long-standing human intellectual endeavor not that engineering, in comparison, concerns less intellectual pursuits. The theory of Hilbert spaces was initiated by David Hilbert (1862-1943), in the early of twentieth century in the context of the study of "Integral equations". Integral equations are a natural complement to differential equations and arise, for example, in the study of existence and uniqueness of function which are solution of partial differential equations such as wave equation. Convolution and Fourier transform equation also belongs to this class. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in Hilbert space. At a deeper level, perpendicular projection onto a subspace that is the analog of "dropping the altitude" of a triangle plays a significant role in optimization problem and other aspects of the theory. An element of Hilbert space can be uniquely specified by its co-ordinates with respect to a set of coordinate axes that is an orthonormal basis, in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought in terms of infinite sequences that are square summable. Linear operators on Hilbert space are ply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectral theory. In brief Hilbert spaces are the means by which the ordinary experience of Euclidean concepts can be extended meaningfully into idealized constructions of more complex abstract mathematics. However, in brief, the usual application demand for Hilbert spaces are integral and differential equations, generalized functions and partial differential equations, quantum mechanics, orthogonal polynomials and functions, optimization and approximation theory. In signal processing which is the main objective of the present thesis and engineering. Wavelets and optimization problem that has been dealt in the present thesis, optimal control, filtering and equalization, signal processing on 2- sphere, Shannon information theory, communication theory, linear and non-linear theory and many more is application domain of the Hilbert space.

This paper is concerned with some mathematical aspects of magnetic resonance imaging (MRI) of the beating human heart. In particular, we investigate the so-called retrospective gating technique which is a non-triggered technique for data acquisition and reconstruction of (approximately) periodically changing organs like the heart. We formulate the reconstruction problem as a moment problem in a Hilbert space and give the solution method. The stability of the solution is investigated and various error estimates are given. The reconstruction method consists of temporal interpolation followed by spatial Fourier inversion. Different choices for the Hilbert space ℋ of interpolating functions are possible. In particular, we study the case where ℋ is (i) the space of bandlimited functions, or (ii) the space of spline functions of odd degree. The theory is applied to reconstructions from synthetic data as well as real MRI data.

This paper proposes a novel approach for identification of nonlinear systems. By transforming the data space into a feature space, kernel methods can be used for modeling nonlinear systems. The spline kernel is adopted to produce a Hilbert space. However, a problem exists as the spline kernel-based identification method cannot deal with data with high dimensions well, resulting in huge computational cost and slow estimation speed. Additionally, owing to the large number of parameters to be estimated, the amount of training data required for accurate identification must be large enough to satisfy the persistence of excitation conditions. To solve the problem, a dimensionality reduction strategy is proposed. Transformation of coordinates is made with the tool of differential geometry. The purpose of the transformation is that no intersection of information with relevance to the output will exist between different new states, while the states with no impact on the output are extracted, which are then abandoned when constructing the model. Then, the dimension of the kernel-based model is reduced, and the number of parameters to be estimated is also reduced. Finally, the proposed identification approach was validated by simulations performed on experimental data from wind tunnel tests. The identification result turns out to be accurate and effective with lower dimensions.

The aim of this paper is to study the spline interpolation problem in spheroidal geometry. We follow the minimization of the norm of the iterated Beltrami-Laplace and consecutive iterated Helmholtz operators for all functions belonging to an appropriate Hilbert space defined on the spheroid. By exploiting surface Green’s functions, reproducing kernels for discrete Dirichlet and Neumann conditions are constructed in the spheroidal geometry. According to a complete system of surface spheroidal harmonics, generalized Green’s functions are also defined. Based on the minimization problem and corresponding reproducing kernel, spline interpolant which minimizes the desired norm and satisfies the given discrete conditions is defined on the spheroidal surface. The application of the results in Geodesy is explained in the gravity data interpolation over the globe.

For the output of wind power system has the characteristics of randomness, volatility and intermittence, the voltage of wind power system low frequency oscillation is one of the most common fluctuations in the system. For the problem of low frequency oscillation, the limitations of the detection methods such as the Lyapunov linearization method, the Prony method, wavelet transform method are summed up, and a new detecting method named Hilbert-huang Transform (HHT) is put forward in this paper, which can detect the oscillation accurately and timely. To solve the problem of end effect in the process of empirical mode decomposition (EMD), B-spline empirical mode decomposition based on support vector machine is applied in dealing with the end issue. an extension of the original signal is applied. Then, calculating the average curve of the signal by B-spline interpolation method. Finally getting the intrinsic mode function (IMF) by empirical mode decomposition (EMD). The practicality of the method is verified by Matlab simulation.

Casazza and Kutyniok [Frames of subspaces, in Wavelets, Frames and Operator Theory, Contemporary Mathematics, Vol. 345 (American Mathematical Society, Providence, RI, 2004), pp. 87–113] defined fusion frames in Hilbert spaces to split a large frame system into a set of (overlapping) much smaller systems and being able to process the data effectively locally within each sub-system. In this paper, we handle this problem using block sequences and generalized block sequences with respect to g-frames. Examples have been given to show their existence. A necessary and sufficient condition for a block sequence with respect to a g-frame to be a g-frame has been given. Finally, a sufficient condition for a generalized block sequence with respect to a g-frame to be a g-frame has been given.

The local mean decomposition method is effective in analyzing non-linear and non-stationary data, and it is suitable for the detection of power quality disturbance signals. The endpoint effect caused by the method is studied, and the original method is improved for the problem that the disturbance signal cannot be accurately located. An improved Local Mean Decomposition (ILMD) method is proposed. ILMD uses cubic spline interpolation instead of smoothing to obtain local mean function and envelope estimation function. Radial Basis Function (RBF) neural network is used to extend the information at both ends of the data, which improves the endpoint effect. Combined with Hilbert transform, the instantaneous frequency of power quality disturbance signal can be more accurately calculated. The improved method is also applicable to disturbance signals with weak periodic law, and has less requirement for disturbance signal conditions and universal applicability. The effectiveness of ILMD is validated by simulation examples and the measured data of voltage signal at low voltage side of 35kV bus transformer in a wind farm.

Una teoría generalizada de señales y sistemas A generalized signals and systems theory Emilio Gago Ribas, Juan Heredia Juesas y José Luis Ganoza Quintana Área de Teoría de la Señal y Comunicaciones, Universidad de Oviedo, Edificio Polivalente de Viesques, 33203, Gijón - España DOI: https://doi.org/10.33017/RevECIPeru2014.0015/ Resumen La Teoría de Señales y Sistemas Teoría (SST) desempeña un papel fundamental en la formación académica y profesional en diferentes áreas de la ingeniería eléctrica (procesado de la señal, electromagnetismo, acústica, mecánica cuántica, etc.), así como en muchas otras áreas científicas. Muchos autores presentan esta teoría siguiendo un esquema que es válido para el análisis práctico de muchos sistemas siguiendo el esquema habitual de la SST. Esta forma de presentar dicha teoría suele evitar tener que tratar con conceptos más generales que son fundamentales en las explicaciones asociadas con la resolución de un gran número de problemas físicos. Estas limitaciones suelen estar relacionadas con la interpretación matemática y física de muchos conceptos importantes inherentes a la SST, por ejemplo, (i) la definición de funciones generalizadas, como la delta de Dirac o sus derivadas, sin considerar el rigor matemático de la teoría de distribuciones, (ii) el análisis de sistemas lineales invariantes en los dominios tiempo-frecuencia mediante la realización del análisis espectral bajo la transformada de Fourier solamente, (iii) el análisis de problemas en variable continua y discreta por separado, (iv) el hecho de no considerar el análisis de sistemas lineales no invariantes de una manera rigurosa, etc. Estas simplificaciones dejan de lado muchos problemas importantes que deberían ser analizados bajo la SST. Esto es particularmente importante si el análisis se centra en los problemas físicos (generalmente definidos por ecuaciones diferenciales más ciertas condiciones de contorno) bajo la SST, por ejemplo: (i) los problemas en el dominio espacial, que a menudo son lineales no invariantes, (ii) el análisis en el dominio del tiempo de los sistemas lineales no invariantes (modulador de amplitud, por ejemplo), (iii) el análisis espectral bajo otras transformadas, en relación con las representaciones habituales utilizando diferentes funciones de onda como funciones de base (ondas cilíndricas, ondas esféricas, haces gaussianos, haces complejos, wavelets, etc.), (iv) el análisis de la teoría de funciones de Green como un caso particular de la SST, (v) la consideración de la teoría de las distribuciones junto con las funciones ordinarias a través de la teoría de los espacios de Hilbert equipados (RHS), (vi) la extensión de la SST a las funciones de variable compleja con el fin de entender la continuación de coordenadas reales a coordenadas complejas, o (vii) la generalización del análisis de los operadores no lineales, así como muchos otros tipos de problemas. El objetivo final del trabajo presentado en este artículo es desarrollar una teoría general que puede incluir todos estos casos de una manera rigurosa. Esto conduce a una Teoría Generalizada de Señales y Sistemas (GSST) que se ha construido teniendo en cuenta que cualquier problema físico pueda analizarse particularizando los conceptos generales de esta teoría con los parámetros concretos del problema en cuestión. Este esquema se revisa continuamente y se actualiza con nuevos resultados. En este artículo se presentará una versión actualizada de este esquema, versión que es usada hoy en día para la presentación de la SST tanto para estudiantes de grado (en una versión simplificada) como para estudiantes de postgrado. El esquema de la GSST está construido considerando inicialmente espacios vectoriales de señales de dimensión finita e infinita junto con la teoría de los operadores y de la teoría de distribuciones, considerando variables generales que pueden representar cualquier magnitud física (tiempo, espacio, etc.) Con estas consideraciones iniciales en mente, se introducirán varios conceptos generalizados importantes, tales como la Combinación Lineal Generalizada (LC), la Transformada Generalizada (GT), los Cambios Generalizados de Transformadas (GTC) y el Análisis Espectral Generalizado (GSA) de sistemas lineales (invariantes y no invariantes). Descriptores: señales, sistemas, distribuciones, transformadas generalizadas, análisis espectral generalizado Abstract The Signals and Systems Theory (SST) plays a fundamental role in the academic and professional background in different areas of electrical engineering (signal processing, electromagnetics, acoustics, quantum mechanics, etc.) as well as in many other scientific areas. Many authors present this theory following a scheme which is valid for the practical analysis of many systems following the usual scheme of the SST, for instance. This way of presenting this theory usually avoids dealing with more general concepts that are fundamental in the explanations associated with the resolution of a large number of physical problems. These limitations use to be related to the mathematical and physical interpretation of many important concepts underlying the SST, for instance (i) the definition of generalized functions such as the Dirac delta or its derivatives without considering the mathematical rigor of the theory of distributions, (ii) the analysis of linear invariant system in the timefrequency domain by performing spectral analysis under the Fourier transform only, (iii) the analysis of continuous and discrete variable problems separately, (iv) the lack of considering the analysis of linear non invariant systems in a rigorous way, etc. These simplifications leave out many important problems that should be analyzed under the SST. This is particularly important if the analysis focuses on physical problems (usually defined by differential equations plus some boundary conditions) under the SST, for instance: (i) problems in the spatial domain, which are often linear non invariant, (ii) the analysis in the time domain of linear non-invariant systems (amplitude modulator, for instance), (iii) the spectral analysis under other transforms, in connection with the usual representations using different wave functions as base functions (cylindrical waves, spherical waves, Gaussian beams, complex beams, wavelets, etc.), (iv) the analysis of the Green’s functions theory as a particular case of the SST, (v) the consideration of the theory of distributions together with the ordinary functions through the theory of rigged Hilbert spaces (RHS), (vi) the extension of the usual SST to complex variable functions in order to understand the continuation of real coordinates to complex ones, or (vii) the generalization of the analysis of nonlinear operators, as well as many other types of problems. The final aim of the work presented in this paper is to develop a general theory which can include all of these cases in a rigorous way. This leads to a Generalized Signals and Systems Theory (GSST) that has been built keeping in mind that any physical problem may be analyzed particularizing the general concepts of this theory to the concrete parameters of the problem at hand. This scheme is continuously revised and updated with new results. The up-to-date version of this scheme will be presented in this paper and it is used nowadays for presenting the SST to both undergraduates (in a simplified version) and postgraduate students. The GSST scheme is built under finite or infinite dimension signal vector spaces together with the theory of operators and the theory of distributions, considering general variables that may represent any physical magnitude (time, space, etc.). With these initial considerations in mind, several generalized important concepts will be introduced, such as the Generalized Linear Combination (LC), the Generalized Transform (GT), the Generalized Transform Changes (GTC) and the Generalized Spectral Analysis (GSA) of linear (invariant and non-invariant) systems. Keywords: signals, systems, distributions, generalized transforms, generalized spectral analysis.