morlet wavelet

The Morlet wavelet, named after Jean Morlet, was originally formulated by Goupillaud, Grossmann and Morlet in 1984 as a constant

\kappa_{\sigma}

subtracted from a plane wave and then localised by a Gaussian:

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

where

\kappa_{\sigma}=e^{-\frac{1}{2}\sigma^{2}}

is defined by the admissibility criterion and the normalisation constant

c_{\sigma}

is:

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

The Fourier transform of the Morlet wavelet is:

\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})}

The "central frequency"

\omega_{\Psi}

is the position of the global maximum of

\hat{\Psi}_{\sigma}(\omega)

which, in this case, is given by the solution of the equation:

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

The parameter

\sigma

in the Morlet wavelet allows trade between time and frequency resolutions. Conventionally, the restriction

\sigma>5

is used to avoid problems with the Morlet wavelet at low

\sigma

(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

\sigma

. In this case,

\kappa_{\sigma}

becomes very small (e.g.

\sigma>5

\Rightarrow

\kappa_{\sigma}<10^{-5}

) and is, therefore, often neglected. Under the restriction

\sigma>5

, the frequency of the Morlet wavelet is conventionally taken to be

\omega_{\Psi}\simeq\sigma

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