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 ^{-{\frac {1}{4}}}\omega ^{2}e^{-{\frac {1}{2}}\omega ^{2}}&\omega \geq 0\\0&\omega \leq 0.\end{cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fb22acdf0b8444bd6baf553dbbdecad4df276652)
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 }}\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).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9d659dbeb69b395a6b79b7445b01f138e17a8e4)
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.[1] for applications requiring high temporal precision time-frequency analysis.
References
- ^ P. S. Addison, et al., The Journal of Sound and Vibration, 2002 Archived 2000-02-26 at archive.today