Ten Lectures on Wavelets

Abstract

The wavelet transform is a tool that cuts up data or functions or operators into different frequency components, and then studies each component with a resolution matched to its scale. Forerunners of this technique were invented independently in pure mathematics (Calderón's resolution of the identity in harmonic analysis-see e.g., Calderón (1964)), physics (coherent states for the (ax + b)group in quantum mechanics, first constructed by Aslaksen and Klauder (1968), and linked to the hydrogen atom Hamiltonian by Paul (1985)) and engineering (QMF filters by Esteban and Galland (1977), and later QMF filters with exact reconstruction property by Smith and Barnwell (1986), Vetterli (1986) in electrical engineering; wavelets were proposed for the analysis of seismic data by J. Morlet (1983)). The last five years have seen a synthesis between all these different approaches, which has been very fertile for all the fields concerned.Let us stay for a moment within the signal analysis framework. (The discussion can easily be translated to other fields.) The wavelet transform of a signal evolving in time (e.g., the amplitude of the pressure on an eardrum, for acoustical applications) depends on two variables: scale (or frequency) and time; wavelets provide a tool for time-frequency localization. The first section tells us what time-frequency localization means and why it is of interest. The remaining sections describe different types of wavelets. Time-frequency localization.In many applications, given a signal f (t) (for the moment, we assume that t is a continuous variable), one is interested in its frequency content locally in time. This is similar to music notation, for example, which tells the player which notes (= frequency information) to play at any given moment. The standard Fourier transform,also gives a representation of the frequency content of f, but information concerning time-localization of, e.g., high frequency bursts cannot be read off easily from .F f . Time-localization can be achieved by first windowing the signal f, so Downloaded 10/05/12 to 132.206.27.25. Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php CHAPTER 1 as to cut off only a well-localized slice of f, and then taking its Fourier transform:This is the windowed Fourier transform, which is a standard technique for timefrequency localization. 1 It is even more familiar to signal analysts in its discrete version, where t and w are assigned regularly spaced values: t = nto , w = mwo , where m, n range over Z, and wo, to > 0 are fixed. Then (1.1.1) becomes Twin (f) = f ds ƒ(s) g(s -nto) e-imw08

Quantum Wavepacket Ab Initio Molecular Dynamics: An Approach for Computing Dynamically Averaged Vibrational Spectra Including Critical Nuclear Quantum Effects

“…167 On the basis of eqs 5 and 6, a variety of hierarchical wavelet bases have been developed. 111,115,117,[142][143][144][145] Here, we expand the multidimensional, positive semidefinite TDDS function as a multiconfigurational (sum-of-products) expansion of Haar scaling functions where the Haar scaling function, H(x), is a square function equal to 1, for 0 e x e 1, and zero otherwise. The quantity N GEN is the number of wavelet generations, and the underline below the summations is meant to indicate that there are N Dim summations, [j 1 ,j 2 , ..., j NDim ], and c i,{j} implies that the coefficients depend on i and the entire set of j-indices.…”

Estimation of reproduction number and non stationary spectral analysis of dengue epidemic

“…Wavelets consists of a family of basis functions, which can be localized in time and varied in scale. These wavelets ψ a,τ (called daughter wavelets) are derived from time shifts τ and scaling a of a single function called the mother wavelet ψ(t) [27,13] as below :…”

A wavelet based algorithm for the identification of oscillatory event-related potential components

“…Wavelets behave exactly like filters and a wavelet function ψ when convolved with an input signal f (t) will project the signal onto an orthogonal subspace ξ asf (ξ). In terms of symmetry, a wavelet filter with coefficients a n is linear if the phase of the function a(ξ) = n a n e inξ is a linear function of ξ for some l ∈ Z (Daubechies, 1992). This essentially means that the filter delays each frequency in the input signal in equal amounts at the output.…”