Complex Mexican hat wavelet

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:

$\hat{\Psi}(\omega)=\begin{cases} 2\sqrt{\frac{2}{3}}\pi^{-1/4}\omega^2 e^{-\frac{1}{2}\omega^2} & \omega\geq0 \\[10pt] 0 & \omega\leq 0. \end{cases}$

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

$\Psi(t)=\frac{2}{\sqrt{3}}\pi^{-\frac{1}{4}}\left(\sqrt{\pi}(1-t^2)e^{-\frac{1}{2}t^2}-\left(\sqrt{2}it+\sqrt{\pi}\operatorname{erf}\left[\frac{i}{\sqrt{2}}t\right]\left(1-t^2\right)e^{-\frac{1}{2}t^2}\right)\right).$

This wavelet has $O(|t|^{-3})$ asymptotic temporal decay in $|\Psi(t)|$, dominated by the discontinuity of the second derivative of $\hat{\Psi}(\omega)$ at $\omega=0$.

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

References

1. P. S. Addison, et al., The Journal of Sound and Vibration, 2002