Mexican hat wavelet: Difference between revisions
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In [[applied mathematics]], the '''complex Mexican hat wavelet''' is a low-oscillation, [[complex number|complex-valued]], [[wavelet]] for the [[continuous wavelet transform]]. This wavelet is formulated in terms of its [[Fourier transform]] as the [[Hilbert analytic function]] of the conventional [[Mexican hat wavelet]]: | |||
:<math>\hat{\Psi}(\omega)=\begin{cases} 2\sqrt{\frac{2}{3}}\pi^{-1/4}\omega^2 e^{-\frac{1}{2}\omega^2} & \omega\geq0 \\[10pt] | |||
0 & \omega\leq 0. \end{cases}</math> | |||
Temporally, this wavelet can be expressed in terms of the [[error function]], | |||
as: | |||
:<math>\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).</math> | |||
This wavelet has <math>O(|t|^{-3})</math> [[asymptotic]] temporal decay in <math>|\Psi(t)|</math>, | |||
dominated by the [[Discontinuity (mathematics)|discontinuity]] of the second [[derivative]] of <math>\hat{\Psi}(\omega)</math> | |||
at <math>\omega=0</math>. | |||
This wavelet was proposed in 2002 by Addison ''et al.''<ref>[http://sbe.napier.ac.uk/staff/paddison/wavelet.htm P. S. Addison, ''et al.'', ''The Journal of Sound and Vibration'', 2002]</ref> for applications requiring high temporal precision [[time-frequency analysis]]. | |||
== References == | |||
{{reflist}} | |||
[[Category:Continuous wavelets]] |
Revision as of 21:13, 18 August 2013
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 function of the conventional Mexican hat wavelet:
Temporally, this wavelet can be expressed in terms of the error function, as:
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
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