The Wavelet Transform and Its Basic Properties

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 6.1.1$$ \hat \psi \left( \omega \right) = \sqrt {{\rm{2}}\pi } \omega ^{\rm{2}} {\rm{exp}}\left( { - \frac{{\rm{1}}}{{\rm{2}}}\omega ^{\rm{2}} } \right), \omega > {\rm{0}}{\rm{.}} $$