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268 lines
6.1 KiB
268 lines
6.1 KiB
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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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* with significant contributions from:
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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HREF="node192.html">General (non-symmetric) triangle wave</A>
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HREF="node188.html">Fourier series of the</A>
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HREF="node190.html">Parabolic wave</A>
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<BR>
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<H2><A ID="SECTION001433000000000000000">
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Square and symmetric triangle waves</A>
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</H2>
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<P>
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<DIV ALIGN="CENTER"><A ID="fig10.06"></A><A ID="14409"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.6:</STRONG>
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Symmetric triangle wave, obtained by superposing parabolic waves with
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<IMG
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WIDTH="46" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img66.png"
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ALT="$(M, c)$"> pairs equal to <IMG
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WIDTH="38" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img67.png"
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ALT="$(0, 8)$"> and <IMG
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WIDTH="72" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img68.png"
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ALT="$(N/2, -8)$">.</CAPTION>
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<TR><TD><IMG
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WIDTH="473" HEIGHT="97" BORDER="0"
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SRC="img1334.png"
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ALT="\begin{figure}\psfig{file=figs/fig10.06.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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To see how to obtain Fourier series for classical waveforms in general,
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consider first the square wave,
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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x[n] = s[n] - s[n-{N \over 2}]
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\end{displaymath}
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-->
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<IMG
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WIDTH="155" HEIGHT="39" BORDER="0"
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SRC="img1335.png"
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ALT="\begin{displaymath}
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x[n] = s[n] - s[n-{N \over 2}]
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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equal to <IMG
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WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img98.png"
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ALT="$1/2$"> for the first half cycle (<IMG
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WIDTH="104" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1336.png"
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ALT="$0 <= n < N/2$">) and <IMG
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WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img212.png"
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ALT="$-1/2$"> for the rest.
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We get the Fourier series by plugging in the Fourier series for <IMG
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WIDTH="29" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1311.png"
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ALT="$s[n]$"> twice:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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x[n] \approx {1 \over \pi} \left [
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{\sin ( \omega n )}
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+ {{\sin ( 2 \omega n)} \over 2}
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+ {{\sin ( 3 \omega n)} \over 3}
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+ \cdots
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\right .
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\end{displaymath}
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-->
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<IMG
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WIDTH="327" HEIGHT="45" BORDER="0"
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SRC="img1337.png"
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ALT="\begin{displaymath}
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x[n] \approx {1 \over \pi} \left [
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{\sin ( \omega n )}
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+ ...
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...\over 2}
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+ {{\sin ( 3 \omega n)} \over 3}
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+ \cdots
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\right .
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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\left .
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-{\sin ( \omega n )}
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+ {{\sin ( 2 \omega n)} \over 2}
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- {{\sin ( 3 \omega n)} \over 3}
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\pm \cdots
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\right ]
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\end{displaymath}
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-->
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<IMG
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WIDTH="272" HEIGHT="45" BORDER="0"
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SRC="img1338.png"
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ALT="\begin{displaymath}
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\left .
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-{\sin ( \omega n )}
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+ {{\sin ( 2 \omega n)} \over 2}
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- {{\sin ( 3 \omega n)} \over 3}
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\pm \cdots
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\right ]
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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= {2 \over \pi} \left [
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{\sin ( \omega n )}
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+ {{\sin ( 3 \omega n)} \over 3}
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+ {{\sin ( 5 \omega n)} \over 5}
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+ \cdots
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\right ]
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\end{displaymath}
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-->
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<IMG
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WIDTH="300" HEIGHT="45" BORDER="0"
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SRC="img1339.png"
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ALT="\begin{displaymath}
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= {2 \over \pi} \left [
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{\sin ( \omega n )}
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+ {{\sin ( 3 ...
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...\over 3}
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+ {{\sin ( 5 \omega n)} \over 5}
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+ \cdots
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\right ]
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<P>
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The symmetric triangle wave (Figure <A HREF="#fig10.06">10.6</A>) given by
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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x[n] = 8 p[n] - 8 p[n-{N \over 2}]
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\end{displaymath}
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-->
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<IMG
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WIDTH="173" HEIGHT="39" BORDER="0"
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SRC="img1340.png"
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ALT="\begin{displaymath}
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x[n] = 8 p[n] - 8 p[n-{N \over 2}]
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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similarly comes to
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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x[n] \approx {8 \over {{\pi^2}}} \left [
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{\cos ( \omega n )}
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+ {{\cos ( 3 \omega n)} \over 9}
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+ {{\cos ( 5 \omega n)} \over 25}
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+ \cdots
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\right ]
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\end{displaymath}
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-->
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<IMG
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WIDTH="345" HEIGHT="45" BORDER="0"
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SRC="img1341.png"
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ALT="\begin{displaymath}
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x[n] \approx {8 \over {{\pi^2}}} \left [
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{\cos ( \omega n ...
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...over 9}
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+ {{\cos ( 5 \omega n)} \over 25}
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+ \cdots
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\right ]
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<P>
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<BR><HR>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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</ADDRESS>
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