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IEVref:103-04-11ID:
Language:enStatus: Standard
Term: wavelet
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Definition: small localized wave, represented by a function having a zero mean value and a practically finite duration

Note 1 to entry: From a mother wavelet ψ(t) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5jaacIcaca WG0bGaaiykaaaa@3972@ , daughter wavelets are obtained through shifting and scaling (expansion or compression): ψ a,b (t)= 1 a ψ( tb a ) MathType@MTEF@5@5@+=feaagCart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHXgaruavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGeaGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaqFn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpeWZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5naaBaaaleaacaWGHbGaaiilaiaadkgaaeqaaOGaaiikaiaadshacaGGPaGaeyypa0ZaaSaaaeaajugabiaaigdaaOqaamaakaaabaGaamyyaaWcbeaaaaGccqaHipqEdaqadaqaamaalaaabaGaamiDaiabgkHiTiaadkgaaeaacaWGHbaaaaGaayjkaiaawMcaaaaa@4691@ , where a is a scale parameter and b a position parameter.

Note 2 to entry: Examples (see Figures 3 and 4):

  • Haar wavelet: ψ(t)=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5jaacIcaca WG0bGaaiykaiabg2da9iabgkHiTKqzaeGaaGymaaaa@3C8F@ for −1/2 < t < 0, ψ(t)=1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5jaacIcaca WG0bGaaiykaiabg2da9KqzaeGaaGymaaaa@3BA2@ for 0 < t < 1/2, ψ(t)=0 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaiabeI8a5jaacIcaca WG0bGaaiykaiabg2da9KqzaeGaaGimaaaa@3BA1@ outside;
  • Morlet wavelet: ψ(t)= e t 2 /2 e jωt MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbbjxAHX garuavP1wzZbItLDhis9wBH5garmWu51MyVXgarqqtubsr4rNCHbGe aGqipG0dh9qqWrVepG0dbbL8F4rqqrVepeea0xe9LqFf0xc9q8qqaq Fn0lXdHiVcFbIOFHK8Feea0dXdar=Jb9hs0dXdHuk9fr=xfr=xfrpe WZqaaiqaciWacmGadaGadeaabaGaaqaaaOqaaKqzaeGaeqiYdKNaai ikaiaadshacaGGPaGaeyypa0JaciyzaOWaaWbaaSqabeaacqGHsisl caWG0bWaaWbaaWqabeaajugOaiaaikdaaaWccaGGVaqcLbmacaaIYa aaaKqzaeGaciyzaOWaaWbaaSqabeaacqGHsisljugWaiGacQgaliab eM8a3jaayIW7caWG0baaaaaa@4B8D@ (example of exponential damping; Figure 4 gives the real part).

Figure 3 – Haar wavelet

Figure 3 – Ondelette de Haar

Figure 4 – Morlet wavelet

Figure 4 – Ondelette de Morlet


Publication date:2009-12
Source:
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Internal notes:2017-02-20: Editorial revisions in accordance with the information provided in C00020 (IEV 103) - evaluation. JGO
2017-08-25: Added a <mstyle displaystyle='true'> tag. LMO
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