Complex Mexican hat wavelet

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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.

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