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<H2><A NAME="SECTION001432000000000000000">
Parabolic wave</A>
</H2>
The same analysis, with some differences in sign and normalization, works
for parabolic waves.  First we compute the difference:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
p[n] - p[n-1] = {
        {
            {{({n\over N} - {1\over 2})}^2} -
            {{({{n-1}\over N} - {1\over 2})}^2}
    } \over {
        2
    }}
\end{displaymath}
 -->

<IMG
 WIDTH="285" HEIGHT="46" BORDER="0"
 SRC="img1326.png"
 ALT="\begin{displaymath}
p[n] - p[n-1] = {
{
{{({n\over N} - {1\over 2})}^2} -
{{({{n-1}\over N} - {1\over 2})}^2}
} \over {
2
}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
= {
        {
            {{({n\over N} - {N\over {2N}})}^2} -
            {{({{n}\over N} - {{N - 2}\over {2N}})}^2}
    } \over {
        2
    }}
\end{displaymath}
 -->

<IMG
 WIDTH="199" HEIGHT="46" BORDER="0"
 SRC="img1327.png"
 ALT="\begin{displaymath}
= {
{
{{({n\over N} - {N\over {2N}})}^2} -
{{({{n}\over N} - {{N - 2}\over {2N}})}^2}
} \over {
2
}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
= {
        {
            {{{2n}\over {N^2}} - {1\over {N}}} + {1\over {N^2}}
    } \over {
        2
    }}
\end{displaymath}
 -->

<IMG
 WIDTH="115" HEIGHT="43" BORDER="0"
 SRC="img1328.png"
 ALT="\begin{displaymath}
= {
{
{{{2n}\over {N^2}} - {1\over {N}}} + {1\over {N^2}}
} \over {
2
}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
\approx - s[n] / N .
\end{displaymath}
 -->

<IMG
 WIDTH="77" HEIGHT="28" BORDER="0"
 SRC="img1329.png"
 ALT="\begin{displaymath}
\approx - s[n] / N .
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
So (again for <IMG
 WIDTH="41" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
 SRC="img1317.png"
 ALT="$k \neq 0$">, small compared to <IMG
 WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img3.png"
 ALT="$N$">) we get:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
{\cal FT}\{ p[n] \} (k) \approx
    {{-1} \over N} \cdot {{-iN} \over {2 \pi k}} \cdot {\cal FT}\{ s[n] \} (k)
\end{displaymath}
 -->

<IMG
 WIDTH="281" HEIGHT="39" BORDER="0"
 SRC="img1330.png"
 ALT="\begin{displaymath}
{\cal FT}\{ p[n] \} (k) \approx
{{-1} \over N} \cdot {{-iN} \over {2 \pi k}} \cdot {\cal FT}\{ s[n] \} (k)
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
\approx {{-1} \over N} \cdot {{-iN} \over {2 \pi k}}
    \cdot {{-iN} \over {2 \pi k}}
\end{displaymath}
 -->

<IMG
 WIDTH="134" HEIGHT="39" BORDER="0"
 SRC="img1331.png"
 ALT="\begin{displaymath}
\approx {{-1} \over N} \cdot {{-iN} \over {2 \pi k}}
\cdot {{-iN} \over {2 \pi k}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
= {N \over {4 {\pi ^2} {k^2}}}
\end{displaymath}
 -->

<IMG
 WIDTH="59" HEIGHT="39" BORDER="0"
 SRC="img1332.png"
 ALT="\begin{displaymath}
= {N \over {4 {\pi ^2} {k^2}}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and as before we get the Fourier series:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
p[n] \approx {1 \over {2 {\pi^2}}} \left [
        {\cos ( \omega n )}
        + {{\cos ( 2 \omega n)} \over 4}
        + {{\cos ( 3 \omega n)} \over 9}
        + \cdots
    \right ]
\end{displaymath}
 -->

<IMG
 WIDTH="354" HEIGHT="45" BORDER="0"
 SRC="img1333.png"
 ALT="\begin{displaymath}
p[n] \approx {1 \over {2 {\pi^2}}} \left [
{\cos ( \omega ...
...\over 4}
+ {{\cos ( 3 \omega n)} \over 9}
+ \cdots
\right ]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><HR>
<ADDRESS>
Miller Puckette
2006-12-30
</ADDRESS>
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