Topic
Multiresolution analysis
About: Multiresolution analysis is a research topic. Over the lifetime, 4032 publications have been published within this topic receiving 140743 citations. The topic is also known as: Multiresolution analysis, MRA.
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TL;DR: In this paper, a family of multi-wavelets is constructed from the classical finite element basis functions using the technique of intertwining, and the resulting multiwavelets are piecewise-polynomial, orthonormal, compactly-supported and can be constructed with arbitrary approximation order.
Abstract: The Volterra series is commonly used for the modeling of nonlinear dynamical systems. In general, however, a large number of terms are needed to represent Volterra kernels, with the number of required terms increasing exponentially with the order of the kernel. Therefore, reduced-order kernel representations are needed in order to employ the Volterra series in engineering practice. This paper presents an approach whereby multiwavelets are used to obtain low-order estimates of first-, second-, and third-order Volterra kernels. A family of multiwavelets is constructed from the classical finite element basis functions using the technique of intertwining. The resulting multiwavelets are piecewise-polynomial, orthonormal, compactly-supported, and can be constructed with arbitrary approximation order. Furthermore, these multiwavelets are easily adapted to the domains of support of the Volterra kernels. In contrast, most wavelet families do not possess this characteristic. Higher-dimensional multiwavelets can easily be constructed by taking tensor products of the original one-dimensional functions. Therefore, it is straightforward to extend this approach to the representation of higher-order Volterra kernels. This kernel identification algorithm is demonstrated on a prototypical oscillator with a quadratic stiffness nonlinearity. For this system, it is shown that accurate kernel estimates can be obtained in terms of a relatively small number of wavelet coefficients. These results indicate the potential of the multiwavelet-based algorithm for obtaining reduced-order models for a large class of weakly nonlinear systems.
33 citations
01 Oct 2017
TL;DR: The results show clearly various forms of changes in amplitude and frequency of the signals, which shows that this method is fast, sensitive, and practical for detection and identification of power quality disturbance.
Abstract: With the growing use of sensitive and susceptive electronic and computing equipment, power quality is foreseen to cause a great concern to electric utilities. The best analysis on power quality is vital to provide better service to customers. Disturbances in power system usually produce continuity changes in the power signal. Wavelet transform is particularly useful in detecting discontinuities in signals, and this makes it appropriate for detection of disturbances in power quality. Wavelet transform is proposed to detect and identify the power quality disturbance at its instance of occurrence. Power quality disturbances are sag, swell, interruption, transient and harmonic. This study reviews various kinds of power quality disturbances with the goal of detecting them using wavelet transform. The results show clearly various forms of changes in amplitude and frequency of the signals. The application shows that this method is fast, sensitive, and practical for detection and identification of power quality disturbance.
33 citations
TL;DR: This contribution extends previous results of Blu and Unser (2006) to the multivariate case, showing that polyharmonic splines and fBm processes can be seen as the (deterministic vs stochastic) solutions to an identical fractional partial differential equation that involves a fractional Laplacian operator.
Abstract: In this contribution, we study the notion of affine invariance (specifically, invariance to the shifting, scaling, and rotation of the coordinate system) as a starting point for the development of mathematical tools and approaches useful in the characterization and analysis of multivariate fractional Brownian motion (fBm) fields. In particular, using a rigorous and powerful distribution theoretic formulation, we extend previous results of Blu and Unser (2006) to the multivariate case, showing that polyharmonic splines and fBm processes can be seen as the (deterministic vs stochastic) solutions to an identical fractional partial differential equation that involves a fractional Laplacian operator. We then show that wavelets derived from polyharmonic splines have a behavior similar to the fractional Laplacian, which also turns out to be the whitening operator for fBm fields. This fact allows us to study the probabilistic properties of the wavelet transform coefficients of fBm-like processes, leading for instance to ways of estimating the Hurst exponent of a multiparameter process from its wavelet transform coefficients. We provide theoretical and experimental verification of these results. To complement the toolbox available for multiresolution processing of stochastic fractals, we also introduce an extended family of multidimensional multiresolution spaces for a large class of (separable and nonseparable) lattices of arbitrary dimensionality.
33 citations
30 Oct 1997
TL;DR: In this article, the general idea of wavelet representation, in its continuous and discrete versions, as well as in terms of a multiresolution approximation, is discussed, and a general expression for the affine class, and the relationship between affine and Cohen's classes are presented.
Abstract: In this paper, we will discuss the general idea of the wavelet representation, in its continuous and discrete versions, as well as in terms of a multiresolution approximation. In addition, the general expression for the affine class, and the relationship between the affine and Cohen's classes are presented. Also, the shift-scale invariant class is defined. This class basically combines the properties of both classes. Finally a recent development, namely, the use of unitary transformations in both Cohen's and the affine classes, with the consequent generation of even more specific tools for signal analysis will be discussed.
32 citations
TL;DR: Jiang et al. as discussed by the authors established the orthonormal wavelet construction from multiresolution analysis on local fields, which is a necessary condition and four sufficient conditions for wavelet frame on local field.
32 citations