Complex Mexican hat wavelet
# 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:

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

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

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

This wavelet has $O\left(|t|^{-3}\right)$ 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. for applications requiring high temporal precision time-frequency analysis.

