# Morlet wavelet

The Morlet wavelet, named after Jean Morlet, was originally formulated by Goupillaud, Grossmann and Morlet in 1984 as a constant [itex]\kappa_{\sigma}[itex] subtracted from a plane wave and then localised by a Gaussian:

[itex]\Psi_{\sigma}(t)=c_{\sigma}\pi^{-\frac{1}{4}}e^{-\frac{1}{2}t^{2}}(e^{i\sigma t}-\kappa_{\sigma})[itex]

where [itex]\kappa_{\sigma}=e^{-\frac{1}{2}\sigma^{2}}[itex] is defined by the admissibility criterion and the normalisation constant [itex]c_{\sigma}[itex] is:

[itex]c_{\sigma}=\left(1+e^{-\sigma^{2}}-2e^{-\frac{3}{4}\sigma^{2}}\right)^{-\frac{1}{2}}[itex]

The Fourier transform of the Morlet wavelet is:

[itex]\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})}[itex]

The "central frequency" [itex]\omega_{\Psi}[itex] is the position of the global maximum of [itex]\hat{\Psi}_{\sigma}(\omega)[itex] which, in this case, is given by the solution of the equation:

[itex](\omega_{\Psi}-\sigma)^{2}-1=(\omega_{\Psi}^{2}-1)e^{-\sigma\omega_{\Psi}}[itex]

The parameter [itex]\sigma[itex] in the Morlet wavelet allows trade between time and frequency resolutions. Conventionally, the restriction [itex]\sigma>5[itex] is used to avoid problems with the Morlet wavelet at low [itex]\sigma[itex] (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 [itex]\sigma[itex]. In this case, [itex]\kappa_{\sigma}[itex] becomes very small (e.g. [itex]\sigma>5[itex] [itex]\Rightarrow[itex] [itex]\kappa_{\sigma}<10^{-5}[itex]) and is, therefore, often neglected. Under the restriction [itex]\sigma>5[itex], the frequency of the Morlet wavelet is conventionally taken to be [itex]\omega_{\Psi}\simeq\sigma[itex].

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