An Introduction to Wavelets

Chui, Charles K.; Heil, Christopher

doi:10.1063/1.4823126

Cited by 1,277 publications

(786 citation statements)

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“…They first appeared in the 1980s with the work of J. Morlet [19]. The theoretical background can be found in [20,21,22], though only the main principles are stated in the following section.…”

Section: The Continuous Wavelet Transformmentioning

confidence: 99%

“…The factor k 0 is called the wavenumber. Sinceφ(0) = 0, the Morlet wavelet given here does not scrupulously respect the admissibility condition imposed on a wavelet, which is necessary to obtain the inverse of the transform [22]. A correction factor can be applied, but for a wavenumber higher than 6 the error may be considered negligible.…”

Section: The Continuous Wavelet Transformmentioning

confidence: 99%

“…A wavelet analysis starts by selecting an elementary wavelet function φ(t) whose decay is very fast and which fulfills the admissibility conditions [21,22]: the mother wavelet. A family of functions (the wavelets) are then obtained by translation and dilatation of the mother wavelet:…”

Section: The Continuous Wavelet Transformmentioning

confidence: 99%

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Nonlinear transient vibrations and coexistences of multi-instabilities induced by friction in an aircraft braking system

Chevillot¹,

Sinou²,

Hardouin³

2009

Journal of Sound and Vibration

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Hardouin. Nonlinear transient vibrations and coexistences of multi-instabilities induced by friction in an aircraft braking system. Journal of Sound and Vibration, Elsevier, 2009, 328 (4-5) AbstractFriction-induced vibration is still a cause for concern in a wide variety of mechanical systems, because it can lead to structural damage if high vibration levels are reached. Another effect is the noise produced that can be very unpleasant for end-users, thereby making it a major problem in the field of terrestrial transport. In this work the case of an aircraft braking system is examined. An analytical model with polynomial nonlinearity in the contact between rotors and stators is considered.Stability analysis is commonly used to evaluate the capacity of a nonlinear system to generate frictioninduced vibrations. With this approach, the effects of variations in the system parameters on stability can be easily estimated. However, this technique does not give the amplitude of the vibrations produced. The integration of the full set of nonlinear dynamic equations allows computing the time-history response of the system when vibration occurs. This technique, which can be time-consuming for a model with a large number of degrees of freedom (dof), is nevertheless necessary in order to calculate the transient-state behavior of the system. The use of a Continuous Wavelet Transform (CWT) is very suitable for the detailed analysis of the transient response. In this paper, the possibilities of coexistence of several instabilities at the same time will be examined. It will be shown that the behavior of the brake can be very complex and cannot be predicted by stability analysis alone.

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Section: The Continuous Wavelet Transformmentioning

confidence: 99%

Section: The Continuous Wavelet Transformmentioning

confidence: 99%

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Nonlinear transient vibrations and coexistences of multi-instabilities induced by friction in an aircraft braking system

Chevillot¹,

Sinou²,

Hardouin³

2009

Journal of Sound and Vibration

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“…For the basic terminology and fundamental concepts of wavelets, the reader is referred to the monograph of C. K. Chui [3]. A multiresolution analysis for 2π-periodic square-integrable functions consisting of finite-dimensional nested spaces of trigonometric polynomials was first studied in a paper by C. K. Chui and H. N. Mhaskar [4].…”

Section: Introductionmentioning

confidence: 99%

“…First, in §5, the inner product matrix of the scaling functions is explicitly computed as well as the entries of its inverse, which are the coefficients of the biorthogonal bases of dual functions. The usefulness of these dual functions -as described in [3] and [5] for functions on the real axis -can be seen in §6, where they are used to establish the more complicated decomposition relations. Finally, §7 provides a short numerical example illustrating practical results and offers a discussion of open questions.…”

Section: Introductionmentioning

confidence: 99%

Trigonometric wavelets for Hermite interpolation

Quak

1996

Math. Comp.

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Abstract. The aim of this paper is to investigate a multiresolution analysis of nested subspaces of trigonometric polynomials. The pair of scaling functions which span the sample spaces are fundamental functions for Hermite interpolation on a dyadic partition of nodes on the interval [0, 2π). Two wavelet functions that generate the corresponding orthogonal complementary subspaces are constructed so as to possess the same fundamental interpolatory properties as the scaling functions. Together with the corresponding dual functions, these interpolatory properties of the scaling functions and wavelets are used to formulate the specific decomposition and reconstruction sequences. Consequently, this trigonometric multiresolution analysis allows a completely explicit algorithmic treatment.

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Reproducing kernel hierarchical partition of unity, Part I?formulation and theory

Liu

1999

Int. J. Numer. Meth. Engng.

169

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SUMMARYThis work is concerned with developing the hierarchical basis for meshless methods. A reproducing kernel hierarchical partition of unity is proposed in the framework of continuous representation as well as its discretized counterpart. To form such hierarchical partition, a class of basic wavelet functions are introduced. Based upon the built-in consistency conditions, the di erential consistency conditions for the hierarchical kernel functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, which is theoretically proven and numerically validated. For a special interpolant with di erent combinations of the hierarchical kernels, a synchronized convergence e ect may be observed. Being di erent from the conventional Legendre function based p-type hierarchical basis, the new hierarchical basis is an intrinsic pseudo-spectral basis, which can remain as a partition of unity in a local region, because the discrete wavelet kernels form a 'partition of nullity'. These newly developed kernels can be used as the multiscale basis to solve partial di erential equations in numerical computation as a p-type reÿnement.

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An Introduction to Wavelets

Cited by 1,277 publications

References 0 publications

Nonlinear transient vibrations and coexistences of multi-instabilities induced by friction in an aircraft braking system

Nonlinear transient vibrations and coexistences of multi-instabilities induced by friction in an aircraft braking system

Trigonometric wavelets for Hermite interpolation

Reproducing kernel hierarchical partition of unity, Part I?formulation and theory

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