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# cwtftinfo

Valid analyzing wavelets for FFT-based CWT

cwtftinfo

## Description

cwtftinfo displays expressions for the Fourier transforms of valid analyzing wavelets for use with cwtft.

## Examples

Display a list of Fourier transforms for all valid analyzing wavelets.

`cwtftinfo`

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### Wavelet Definitions

#### Morlet Wavelet

Both non-analytic and analytic Morlet wavelets are supported. The analytic Morlet wavelet, 'morl', is defined in the Fourier domain by:

$\stackrel{^}{\Psi }\left(s\omega \right)={\pi }^{-1/4}{e}^{{\left(s\omega -{\omega }_{0}\right)}^{2}/2}U\left(s\omega \right)$

where U(ω) is the Heaviside step function.

The non-analytic Morlet wavelet, 'morlex', is defined in the Fourier domain by:

$\stackrel{^}{\Psi }\left(s\omega \right)={\pi }^{-1/4}{e}^{{\left(s\omega -{\omega }_{0}\right)}^{2}/2}$

'morl0' defines a non-analytic Morlet wavelet in the Fourier domain with exact zero mean:

$\stackrel{^}{\Psi }\left(s\omega \right)={\pi }^{-1/4}\left\{{e}^{{\left(s\omega -{\omega }_{0}\right)}^{2}/2}-{e}^{{\omega }_{0}^{2}/2}\right\}$

The default value of ω0 is 6.

The scale-to-frequency Fourier factor for the Morlet wavelet is:

$\frac{4\pi s}{{\omega }_{0}+\sqrt{2+{\omega }_{0}^{2}}}$

#### m-th Order Derivative of Gaussian Wavelets

In the Fourier domain, the m-th order derivative of Gaussian wavelets, 'dog', is defined by:

$\stackrel{^}{\Psi }\left(s\omega \right)=-\frac{1}{\sqrt{\Gamma \left(m+1/2\right)}}{\left(js\omega \right)}^{m}{e}^{-{\left(s\omega \right)}^{2}/2}$

The derivative must be an even order. The default order of the derivative is 2, which is also known as the Mexican hat wavelet.

Because the unit imaginary, j, is always raised to an even power, the Fourier transform is real-valued.

The scale-to-frequency Fourier factor for the DOG wavelet is:

$\frac{2\pi s}{\sqrt{m+\frac{1}{2}}}$

#### Paul Wavelet

The Fourier transform of the Paul wavelet, 'paul', of order m is:

$\stackrel{^}{\Psi }\left(s\omega \right)={2}^{m}\sqrt{m\left(2m-1\right)!}\text{ }\text{ }{\left(s\omega \right)}^{m}{e}^{-s\omega }U\left(sw\right)$

where U(ω) is the Heaviside step function. The Paul wavelet is analytic.

The scale-to-frequency Fourier factor for the Paul wavelet is:

$\frac{4\pi s}{2m+1}$

The default order of the Paul wavelet is 4.

## References

[1] Daubechies, I. Ten Lectures on Wavelets, Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM), 1992.

[2] Farge, M. Wavelet Transforms and Their Application to Turbulence, Ann. Rev. Fluid. Mech., 1992, 24, 395–457.

[3] Mallat, S. A Wavelet Tour of Signal Processing, San Diego, CA: Academic Press, 1998.

[4] Torrence, C. and G.P. Compo A Practical Guide to Wavelet Analysis, Bull. Am. Meteorol. Soc., 79, 61–78, 1998.