Wavelet Transforms and Their Applications pp 361-402 | Cite as

# The Wavelet Transform and Its Basic Properties

Chapter

## 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{*iϕ*(*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

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## Copyright information

© Springer Science+Business Media New York 2002