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 function 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 \\ 0 & \omega\leq 0. \end{cases}

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

This wavelet has asymptotic temporal decay in, dominated by the discontinuity of the second derivative of at .

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

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