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<H2><A NAME="SECTION002031000000000000000">
Morlet's Wavelet</A>
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The wavelet defined by Morlet is [<A
HREF="node370.html#goupillaud">16</A>]:
<BR>
<DIV ALIGN="CENTER">
<!-- MATH: \begin{eqnarray}
\hat{g}(\omega) = e^{ -2 \pi^2(\nu - \nu_0)^2}
\end{eqnarray} -->
<TABLE ALIGN="CENTER" CELLPADDING="0" WIDTH="100%">
<TR VALIGN="MIDDLE"><TD NOWRAP ALIGN="RIGHT"><IMG
WIDTH="190" HEIGHT="56" ALIGN="MIDDLE" BORDER="0"
SRC="img594.gif"
ALT="$\displaystyle \hat{g}(\omega) = e^{ -2 \pi^2(\nu - \nu_0)^2}$"></TD>
<TD> </TD>
<TD> </TD>
<TD WIDTH=10 ALIGN="RIGHT">
(14.7)</TD></TR>
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<BR CLEAR="ALL"><P></P>
it is a complex wavelet which can be decomposed in two parts, one for
the real part, and the other for the imaginary part.
<BR><P></P>
<DIV ALIGN="CENTER">
<IMG
WIDTH="303" HEIGHT="115"
SRC="img595.gif"
ALT="\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*}">
</DIV><P></P>
<BR CLEAR="ALL">where <IMG
WIDTH="27" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img596.gif"
ALT="$\nu_0$">
is a constant.
The admissibility condition is verified only if
<!-- MATH: $\nu_0 > 0.8$ -->
<IMG
WIDTH="82" HEIGHT="39" ALIGN="MIDDLE" BORDER="0"
SRC="img597.gif"
ALT="$\nu_0 > 0.8$">.
Figure <A HREF="node312.html#morlet">14.1</A> shows these two functions.
<BR>
<DIV ALIGN="CENTER"><A NAME="morlet"> </A><A NAME="14300"> </A>
<TABLE WIDTH="50%">
<CAPTION><STRONG>Figure 14.1:</STRONG>
Morlet's wavelet: real part at left and imaginary part at right.</CAPTION>
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WIDTH="940" HEIGHT="334"
SRC="img598.gif"
ALT="\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}"></TD></TR>
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<ADDRESS>
<I>Petra Nass</I>
<BR><I>1999-06-15</I>
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