534 lines
14 KiB
HTML
534 lines
14 KiB
HTML
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
|
|
|
<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
|
original version by: Nikos Drakos, CBLU, University of Leeds
|
|
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
|
|
* with significant contributions from:
|
|
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
|
|
<HTML>
|
|
<HEAD>
|
|
<TITLE>Fourier transform of DC</TITLE>
|
|
<META NAME="description" CONTENT="Fourier transform of DC">
|
|
<META NAME="keywords" CONTENT="book">
|
|
<META NAME="resource-type" CONTENT="document">
|
|
<META NAME="distribution" CONTENT="global">
|
|
|
|
<META NAME="Generator" CONTENT="LaTeX2HTML v2002-2-1">
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
|
|
|
<LINK REL="STYLESHEET" HREF="book.css">
|
|
|
|
<LINK REL="next" HREF="node169.html">
|
|
<LINK REL="previous" HREF="node167.html">
|
|
<LINK REL="up" HREF="node167.html">
|
|
<LINK REL="next" HREF="node169.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html3090"
|
|
HREF="node169.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/next.png"></A>
|
|
<A NAME="tex2html3084"
|
|
HREF="node167.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/up.png"></A>
|
|
<A NAME="tex2html3078"
|
|
HREF="node167.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/prev.png"></A>
|
|
<A NAME="tex2html3086"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/contents.png"></A>
|
|
<A NAME="tex2html3088"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html3091"
|
|
HREF="node169.html">Shifts and phase changes</A>
|
|
<B> Up:</B> <A NAME="tex2html3085"
|
|
HREF="node167.html">Properties of Fourier transforms</A>
|
|
<B> Previous:</B> <A NAME="tex2html3079"
|
|
HREF="node167.html">Properties of Fourier transforms</A>
|
|
<B> <A NAME="tex2html3087"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html3089"
|
|
HREF="node201.html">Index</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H2><A NAME="SECTION001321000000000000000">
|
|
Fourier transform of DC</A>
|
|
</H2>
|
|
|
|
<P>
|
|
Let <IMG
|
|
WIDTH="65" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1077.png"
|
|
ALT="$X[n]=1$"> for all <IMG
|
|
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img75.png"
|
|
ALT="$n$"> (this repeats with any desired integer period
|
|
<IMG
|
|
WIDTH="47" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1078.png"
|
|
ALT="$N>1$">). From the preceding discussion, we expect to find that
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{\cal FT} \left \{ X[n] \right \} (k) =
|
|
\left \{
|
|
\begin{array}{ll}
|
|
N & {k=0} \\
|
|
0 & {k=1, \ldots, N-1}
|
|
\end{array}
|
|
\right .
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="287" HEIGHT="45" BORDER="0"
|
|
SRC="img1079.png"
|
|
ALT="\begin{displaymath}
|
|
{\cal FT} \left \{ X[n] \right \} (k) =
|
|
\left \{
|
|
\begin{...
|
|
...}
|
|
N & {k=0} \\
|
|
0 & {k=1, \ldots, N-1}
|
|
\end{array} \right .
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
We will often need to know the answer for non-integer values of <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$"> however,
|
|
and for this there is nothing better to do than to calculate the value
|
|
directly:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{\cal FT} \left \{ X[n] \right \} (k) =
|
|
{V ^ {0}} X[0] +
|
|
{V ^ {1}} X[1] +
|
|
\cdots +
|
|
{V ^ {N-1}} X[N-1]
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="406" HEIGHT="28" BORDER="0"
|
|
SRC="img1057.png"
|
|
ALT="\begin{displaymath}
|
|
{\cal FT}\left \{ X[n] \right \} (k) =
|
|
{V ^ {0}} X[0] +
|
|
{V ^ {1}} X[1] +
|
|
\cdots +
|
|
{V ^ {N-1}} X[N-1]
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
where <IMG
|
|
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img1059.png"
|
|
ALT="$V$"> is, as before, the unit magnitude complex number with argument
|
|
<IMG
|
|
WIDTH="35" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1055.png"
|
|
ALT="$-k\omega$">. This is a geometric series; as long as <IMG
|
|
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1080.png"
|
|
ALT="$V \not= 1$"> we get:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{\cal FT} \left \{ X[n] \right \} (k) =
|
|
{{
|
|
{V^N} - 1
|
|
} \over {
|
|
V - 1
|
|
}}
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="177" HEIGHT="43" BORDER="0"
|
|
SRC="img1081.png"
|
|
ALT="\begin{displaymath}
|
|
{\cal FT} \left \{ X[n] \right \} (k) =
|
|
{{
|
|
{V^N} - 1
|
|
} \over {
|
|
V - 1
|
|
}}
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
We now symmetrize the top and bottom in the same way as we earlier did in
|
|
Section <A HREF="node108.html#sect7.network">7.3</A>. To do this let:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
\xi = \cos(\pi k / N) - i \sin(\pi k / N)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="201" HEIGHT="28" BORDER="0"
|
|
SRC="img1082.png"
|
|
ALT="\begin{displaymath}
|
|
\xi = \cos(\pi k / N) - i \sin(\pi k / N)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
so that <IMG
|
|
WIDTH="52" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1083.png"
|
|
ALT="${\xi^2} = V$">. Then factoring appropriate powers of <IMG
|
|
WIDTH="11" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1084.png"
|
|
ALT="$\xi$"> out of the
|
|
numerator and denominator gives:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{\cal FT} \left \{ X[n] \right \} (k) =
|
|
{\xi^{N-1}}
|
|
{{
|
|
{\xi^N} - {\xi^{-N}}
|
|
} \over {
|
|
\xi - {\xi^{-1}}
|
|
}}
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="230" HEIGHT="45" BORDER="0"
|
|
SRC="img1085.png"
|
|
ALT="\begin{displaymath}
|
|
{\cal FT} \left \{ X[n] \right \} (k) =
|
|
{\xi^{N-1}}
|
|
{{
|
|
{\xi^N} - {\xi^{-N}}
|
|
} \over {
|
|
\xi - {\xi^{-1}}
|
|
}}
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
It's easy now to simplify the numerator:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{\xi^N} - {\xi^{-N}} =
|
|
\left (\cos(\pi k) - i \sin(\pi k) \right ) -
|
|
\left (\cos(\pi k) + i \sin(\pi k) \right )
|
|
= - 2 i \sin(\pi k)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="493" HEIGHT="28" BORDER="0"
|
|
SRC="img1086.png"
|
|
ALT="\begin{displaymath}
|
|
{\xi^N} - {\xi^{-N}} =
|
|
\left (\cos(\pi k) - i \sin(\pi k) ...
|
|
...eft (\cos(\pi k) + i \sin(\pi k) \right )
|
|
= - 2 i \sin(\pi k)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
and similarly for the denominator, giving:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{\cal FT} \left \{ X[n] \right \} (k) =
|
|
\left ( {
|
|
\parbox[t][0.1in]{0in}{\mbox{}}
|
|
\cos(\pi k (N-1)/N) - i \sin(\pi k (N-1)/N)
|
|
} \right )
|
|
{{
|
|
\sin(\pi k)
|
|
} \over {
|
|
\sin(\pi k / N)
|
|
}}
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="502" HEIGHT="44" BORDER="0"
|
|
SRC="img1087.png"
|
|
ALT="\begin{displaymath}
|
|
{\cal FT} \left \{ X[n] \right \} (k) =
|
|
\left ( {
|
|
\parbo...
|
|
...
|
|
} \right )
|
|
{{
|
|
\sin(\pi k)
|
|
} \over {
|
|
\sin(\pi k / N)
|
|
}}
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
Whether <IMG
|
|
WIDTH="45" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img1088.png"
|
|
ALT="$V=1$"> or not, we have
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{\cal FT} \left \{ X[n] \right \} (k) =
|
|
\left ( {
|
|
\parbox[t][0.1in]{0in}{\mbox{}}
|
|
\cos(\pi k (N-1)/N) - i \sin(\pi k (N-1)/N)
|
|
} \right )
|
|
{D_N}(k)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="473" HEIGHT="35" BORDER="0"
|
|
SRC="img1089.png"
|
|
ALT="\begin{displaymath}
|
|
{\cal FT} \left \{ X[n] \right \} (k) =
|
|
\left ( {
|
|
\parbo...
|
|
...(\pi k (N-1)/N) - i \sin(\pi k (N-1)/N)
|
|
} \right )
|
|
{D_N}(k)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
where <IMG
|
|
WIDTH="49" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1090.png"
|
|
ALT="${D_N}(k)$">, known as the
|
|
<A NAME="12394"></A><I>Dirichlet kernel</I>,
|
|
is defined as
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{D_N}(k) =
|
|
\left \{
|
|
\begin{array}{ll}
|
|
N & {k= 0} \\
|
|
{{
|
|
\sin(\pi k)
|
|
} \over {
|
|
\sin(\pi k / N)
|
|
}}
|
|
& {k\not=0,\; -N < k < N}
|
|
\end{array}
|
|
\right .
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="307" HEIGHT="54" BORDER="0"
|
|
SRC="img1091.png"
|
|
ALT="\begin{displaymath}
|
|
{D_N}(k) =
|
|
\left \{
|
|
\begin{array}{ll}
|
|
N & {k= 0} \\
|
|
{...
|
|
...pi k / N)
|
|
}}
|
|
& {k\not=0,\; -N < k < N}
|
|
\end{array} \right .
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
|
|
<P>
|
|
Figure <A HREF="#fig09.01">9.1</A> shows the Fourier transform of <IMG
|
|
WIDTH="65" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1077.png"
|
|
ALT="$X[n]=1$">, with <IMG
|
|
WIDTH="63" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img63.png"
|
|
ALT="$N=100$">. The
|
|
transform repeats every 100 samples, with a peak at <IMG
|
|
WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img1092.png"
|
|
ALT="$k=0$">, another at
|
|
<IMG
|
|
WIDTH="57" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img1093.png"
|
|
ALT="$k=100$">, and so on. The figure endeavors to show both the magnitude and phase
|
|
behavior using a 3-dimensional graph projected onto the page. The phase
|
|
term
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
\cos(\pi k (N-1)/N) - i \sin(\pi k (N-1)/N)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="280" HEIGHT="28" BORDER="0"
|
|
SRC="img1094.png"
|
|
ALT="\begin{displaymath}
|
|
\cos(\pi k (N-1)/N) - i \sin(\pi k (N-1)/N)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
acts to twist the values of <!-- MATH
|
|
${\cal FT} \left \{ X[n] \right \} (k)$
|
|
-->
|
|
<IMG
|
|
WIDTH="104" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1095.png"
|
|
ALT="${\cal FT} \left \{ X[n] \right \} (k)$"> around
|
|
the <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$"> axis with a period of approximately two. The Dirichlet kernel
|
|
<IMG
|
|
WIDTH="49" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1090.png"
|
|
ALT="${D_N}(k)$">, shown in Figure <A HREF="#fig09.02">9.2</A>, controls the magnitude of
|
|
<!-- MATH
|
|
${\cal FT} \left \{ X[n] \right \} (k)$
|
|
-->
|
|
<IMG
|
|
WIDTH="104" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1095.png"
|
|
ALT="${\cal FT} \left \{ X[n] \right \} (k)$">. It has a peak, two units wide, around
|
|
<IMG
|
|
WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img1092.png"
|
|
ALT="$k=0$">. This is surrounded by one-unit-wide
|
|
<A NAME="12409"></A><I>sidelobes</I>,
|
|
alternating in sign and gradually decreasing in magnitude as <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$"> increases or
|
|
decreases away from zero. The phase term rotates by almost <IMG
|
|
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img41.png"
|
|
ALT="$\pi $"> radians
|
|
each time the Dirichlet kernel changes sign, so that the product of the
|
|
two stays roughly in the same complex half-plane for <IMG
|
|
WIDTH="41" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1096.png"
|
|
ALT="$k>1$"> (and in the
|
|
opposite half-plane for <IMG
|
|
WIDTH="53" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1097.png"
|
|
ALT="$k < -1$">). The phase rotates by almost <IMG
|
|
WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img16.png"
|
|
ALT="$2\pi $">
|
|
radians over the peak from <IMG
|
|
WIDTH="53" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1098.png"
|
|
ALT="$k=-1$"> to <IMG
|
|
WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img259.png"
|
|
ALT="$k=1$">.
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig09.01"></A><A NAME="12413"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.1:</STRONG>
|
|
The Fourier transform of a signal consisting of all ones. Here
|
|
N=100, and values are shown for <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$"> ranging from -5 to 10. The result
|
|
is complex-valued and shown as a projection, with the real axis pointing up the
|
|
page and the imaginary axis pointing away from it.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="470" HEIGHT="265" BORDER="0"
|
|
SRC="img1099.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig09.01.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig09.02"></A><A NAME="12418"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.2:</STRONG>
|
|
The Dirichlet kernel, for <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> = 100.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="448" HEIGHT="175" BORDER="0"
|
|
SRC="img1100.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig09.02.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html3090"
|
|
HREF="node169.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/next.png"></A>
|
|
<A NAME="tex2html3084"
|
|
HREF="node167.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/up.png"></A>
|
|
<A NAME="tex2html3078"
|
|
HREF="node167.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/prev.png"></A>
|
|
<A NAME="tex2html3086"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/contents.png"></A>
|
|
<A NAME="tex2html3088"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html3091"
|
|
HREF="node169.html">Shifts and phase changes</A>
|
|
<B> Up:</B> <A NAME="tex2html3085"
|
|
HREF="node167.html">Properties of Fourier transforms</A>
|
|
<B> Previous:</B> <A NAME="tex2html3079"
|
|
HREF="node167.html">Properties of Fourier transforms</A>
|
|
<B> <A NAME="tex2html3087"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html3089"
|
|
HREF="node201.html">Index</A></B>
|
|
<!--End of Navigation Panel-->
|
|
<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|