Abstract
Morlet et al. (1982a,b) modified the Gabor wavelets to study the layering of sediments in a geophysical problem of oil exploration. He recognized certain difficulties of the Gabor wavelets in the sense that the Gabor analyzing function g t,ω (τ) = g(τ-t)eiωτ oscillates more rapidly as the frequency ω tends to infinity. This leads to significant numerical instability in the computation of the coefficients (f,gω,t). On the other hand, g t,ω oscillates very slowly at low frequencies. These difficulties led to a problem of finding a suitable reconstruction formula. In order to resolve these difficulties, Morlet first made an attempt to use analytic signals f(t) = a(t) exp{iϕ(t)} and then introduced the wavelet y defined by its Fourier transform
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© 2002 Springer Science+Business Media New York
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Debnath, L. (2002). The Wavelet Transform and Its Basic Properties. In: Wavelet Transforms and Their Applications. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0097-0_6
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DOI: https://doi.org/10.1007/978-1-4612-0097-0_6
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6610-5
Online ISBN: 978-1-4612-0097-0
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