Showing posts with label Examples of Wavelets. Show all posts
Showing posts with label Examples of Wavelets. Show all posts

Monday, February 9, 2009

Examples of Wavelets

Morlet's Wavelet

The wavelet defined by Morlet is [16]:
$\displaystyle \hat{g}(\omega) = e^{ -2 \pi^2(\nu - \nu_0)^2}$ (14.7)

it is a complex wavelet which can be decomposed in two parts, one for the real part, and the other for the imaginary part.
\begin{eqnarray*}g_r(x) & = & \frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}} \cos(2\p... ... = & \frac{1}{\sqrt{2 \pi}} e^{-\frac{x^2}{2}} \sin(2\pi\nu_0 x) \end{eqnarray*}

where $\nu_0$ is a constant. The admissibility condition is verified only if 0.8$" align="middle" border="0" height="39" width="82">. Figure 14.1 shows these two functions.
Figure 14.1: Morlet's wavelet: real part at left and imaginary part at right.
\begin{figure} \centerline{ \hbox{\psfig{figure=fig_morlet.ps,bbllx=1cm,bblly=13.5cm,bburx=20.5cm,bbury=27cm,height=5cm,width=15cm,clip=} }} \end{figure}

Mexican Hat

The Mexican hat defined by Murenzi [30] is:
$\displaystyle g(x) = (1 - x^2) e^{-\frac{1}{2}x^2}$ (14.8)

it is the second derivative of a Gaussian (see figure 14.2).
Figure 14.2: Mexican Hat
\begin{figure} \centerline{ \hbox{\psfig{figure=fig_mexicain.ps,bbllx=1cm,bblly=13.5cm,bburx=10.5cm,bbury=27cm,height=5cm,width=10cm,clip=} }} \end{figure}
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