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:

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

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

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

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

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