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 function of the conventional Mexican hat wavelet:

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

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.

References

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.