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The Wavelet Transform and Its Basic Properties

  • Lokenath Debnath

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)e iωτ oscillates more rapidly as the frequency ω tends to infinity. This leads to significant numerical instability in the computation of the coefficients (f,g ω,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{(t)} and then introduced the wavelet y defined by its Fourier transform
$$ \hat \psi \left( \omega \right) = \sqrt {{\rm{2}}\pi } \omega ^{\rm{2}} {\rm{exp}}\left( { - \frac{{\rm{1}}}{{\rm{2}}}\omega ^{\rm{2}} } \right), \omega > {\rm{0}}{\rm{.}} $$
(6.1.1)

Keywords

Wavelet Transform Discrete Wavelet Mother Wavelet Continuous Wavelet Transform Continuous Wavelet 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 2002

Authors and Affiliations

  • Lokenath Debnath
    • 1
  1. 1.Department of MathematicsUniversity of Texas—Pan AmericanEdinburgUSA

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