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:

:$\backslash hat\{\backslash Psi\}(\backslash omega)\; =\; \backslash begin\{cases\}$

2\sqrt{\frac{2}{3}}\pi^{-\frac{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:

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

This wavelet has $O\backslash left(|t|^\{-3\}\backslash right)$ asymptotic temporal decay in $|\backslash Psi(t)|$,

dominated by the discontinuity of the second derivative of $\backslash hat\{\backslash Psi\}(\backslash omega)$ at $\backslash omega\; =\; 0$.

This wavelet was proposed in 2002 by Addison et al.[http://sbe.napier.ac.uk/staff/paddison/wavelet.htm P. S. Addison, et al., The Journal of Sound and Vibration, 2002] {{webarchive|url=https://archive.today/20000226173323/http://sbe.napier.ac.uk/staff/paddison/wavelet.htm |date=2000-02-26 }} for applications requiring high temporal precision time-frequency analysis.