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:

$\hat{\mathrm{\Psi }}(\omega )=\{\begin{array}{ll}2\sqrt{\frac{2}{3}}{\pi }^{-\frac{1}{4}}{\omega }^2{e}^{-\frac{1}{2}{\omega }^2}& \omega \ge 0\\ 0& \omega \le 0.\end{array}$

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

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

This wavelet has $O(|t{|}^{-3})$ asymptotic temporal decay in $|\mathrm{\Psi }(t)|$, dominated by the discontinuity of the second derivative of $\hat{\mathrm{\Psi }}(\omega )$ at $\omega =0$. This wavelet was proposed in 2002 by Addison et al. for applications requiring high temporal precision time-frequency analysis.