Morlet wavelet
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The Morlet wavelet, named after Jean Morlet, was originally formulated by Goupillaud, Grossmann and Morlet in 1984 as a constant <math>\kappa_{\sigma}<math> subtracted from a plane wave and then localised by a Gaussian:
<math>\Psi_{\sigma}(t)=c_{\sigma}\pi^{-\frac{1}{4}}e^{-\frac{1}{2}t^{2}}(e^{i\sigma t}-\kappa_{\sigma})<math>
where <math>\kappa_{\sigma}=e^{-\frac{1}{2}\sigma^{2}}<math> is defined by the admissibility criterion and the normalisation constant <math>c_{\sigma}<math> is:
<math>c_{\sigma}=\left(1+e^{-\sigma^{2}}-2e^{-\frac{3}{4}\sigma^{2}}\right)^{-\frac{1}{2}}<math>
The Fourier transform of the Morlet wavelet is:
<math>\hat{\Psi}_{\sigma}(\omega)=c_{\sigma}\pi^{-\frac{1}{4}}\left(\left(\sigma-\omega\right)e^{\sigma\omega}+\omega\right)e^{-\frac{1}{2}(\sigma^{2}+\omega^{2})}<math>
The "central frequency" <math>\omega_{\Psi}<math> is the position of the global maximum of <math>\hat{\Psi}_{\sigma}(\omega)<math> which, in this case, is given by the solution of the equation:
<math>(\omega_{\Psi}-\sigma)^{2}-1=(\omega_{\Psi}^{2}-1)e^{-\sigma\omega_{\Psi}}<math>
The parameter <math>\sigma<math> in the Morlet wavelet allows trade between time and frequency resolutions. Conventionally, the restriction <math>\sigma>5<math> is used to avoid problems with the Morlet wavelet at low <math>\sigma<math> (high temporal resolution).
For signals containing only slowly varying frequency and amplitude modulations (audio, for example) it is not necessary to use small values of <math>\sigma<math>. In this case, <math>\kappa_{\sigma}<math> becomes very small (e.g. <math>\sigma>5<math> <math>\Rightarrow<math> <math>\kappa_{\sigma}<10^{-5}<math>) and is, therefore, often neglected. Under the restriction <math>\sigma>5<math>, the frequency of the Morlet wavelet is conventionally taken to be <math>\omega_{\Psi}\simeq\sigma<math>.