Complex Mexican hat wavelet explained

In applied mathematics, the complex Mexican hat wavelet is a low-oscillation, complex-valued, wavelet for the continuous wavelet transform. This wavelet is formulated in terms of its Fourier transform as the Hilbert analytic signal of the conventional Mexican hat wavelet:

\hat{\Psi}(\omega)=\begin{cases} 2\sqrt{

	2
	3

}\pi^\omega^2 e^ & \omega\geq0 \\ 0 & \omega\leq 0.\end

Temporally, this wavelet can be expressed in terms of the error function,as:

\Psi(t)=

	2
	\sqrt{3

}\pi^\left(\sqrt\left(1 - t^2\right)e^ - \left(\sqrtit + \sqrt\operatorname\left[\frac{i}{\sqrt{2}}t\right]\left(1 - t^2\right)e^\right)\right).

This wavelet has

O\left(|t|^-3\right)

asymptotic temporal decay in

|\Psi(t)|

,dominated by the discontinuity of the second derivative of

\hat{\Psi}(\omega)

\omega=0

This wavelet was proposed in 2002 by Addison et al.^[1] for applications requiring high temporal precision time-frequency analysis.

Notes and References

http://sbe.napier.ac.uk/staff/paddison/wavelet.htm P. S. Addison, et al., The Journal of Sound and Vibration, 2002