08-16-2017, 09:58 PM
Display of 1D continuous wavelets
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Complex Morlet wavelet
Here we take the complex Morlet wavelet as a plane wave modulated by a Gaussian envelope.
where the parameter k0 is the wavenumber associated with the Morlet wavelet, roughly
corresponding to the number of oscillations of the wavelet.
The Fourier transform of the complex Morlet wavelet is
As one can see, this is not zero at k = 0. Strictly speaking, this formula does not satisfy
the admissibility condition for a wavelet (roughly speaking it does not have a zero mean).
We can either enforce the admissibility by setting ^ (k = 0) = 0 or by making a correction
to the formula to satisfy the admissibility condition. The resulting zero mean formula for
the complex Morlet wavelet becomes
Is the barycenter of the wavelet in Fourier space.
It is important to nd the correspondence between the dilation parameter a and the scale
of the wavelet. We rst nd the relation between the central frequency of the wavelet and
the dilation parameter a.
The calculus involved in calculating k for the complex Morlet wavelet is a bit messy,
but the result is clean and should be obvious. For the complex Morlet wavelet