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 ^{-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:

\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.

References

↑ P. S. Addison, et al., The Journal of Sound and Vibration, 2002

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