Continuous Wavelet Transform¶

Complex Morlet Wavelets¶

There are several definitions of Complex Morlet Wavelets.

\[\psi(t) = \frac{1}{\sqrt[4]{\pi}} e^{j \omega_0t } e^{\frac{-t^2}{2}}\]

Its Fourier transform is:

\[\Psi(s \omega) = \frac{1}{\sqrt[4]{\pi}} H(\omega) e^{\frac{-(s\omega - \omega_0)^2}{2}}\]

where \(H(\omega)\) is the Heaviside step function.

Second definition is more general and is based on two parameters:

\[\psi(t,B, C) = \frac{1}{\sqrt{\pi B}} \ e^{\frac{-t^2}{B}} \ e^{j2 \pi C t}\]

This is Gaussian modulated by a complex sinusoid with the standard deviation:

\[\sigma = \sqrt{\frac{T_p}{2}}\]

However, this definition doesn’t have unit energy.