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271 lines
5.7 KiB
271 lines
5.7 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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<H2><A ID="SECTION001432000000000000000">
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Parabolic wave</A>
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</H2>
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The same analysis, with some differences in sign and normalization, works
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for parabolic waves. First we compute the difference:
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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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p[n] - p[n-1] = {
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{
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{{({n\over N} - {1\over 2})}^2} -
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{{({{n-1}\over N} - {1\over 2})}^2}
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} \over {
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2
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}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="285" HEIGHT="46" BORDER="0"
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SRC="img1326.png"
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ALT="\begin{displaymath}
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p[n] - p[n-1] = {
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{
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{{({n\over N} - {1\over 2})}^2} -
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{{({{n-1}\over N} - {1\over 2})}^2}
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} \over {
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2
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}}
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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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= {
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{
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{{({n\over N} - {N\over {2N}})}^2} -
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{{({{n}\over N} - {{N - 2}\over {2N}})}^2}
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} \over {
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2
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}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="199" HEIGHT="46" BORDER="0"
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SRC="img1327.png"
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ALT="\begin{displaymath}
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= {
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{
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{{({n\over N} - {N\over {2N}})}^2} -
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{{({{n}\over N} - {{N - 2}\over {2N}})}^2}
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} \over {
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2
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}}
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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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= {
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{
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{{{2n}\over {N^2}} - {1\over {N}}} + {1\over {N^2}}
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} \over {
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2
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}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="115" HEIGHT="43" BORDER="0"
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SRC="img1328.png"
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ALT="\begin{displaymath}
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= {
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{
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{{{2n}\over {N^2}} - {1\over {N}}} + {1\over {N^2}}
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} \over {
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2
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}}
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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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\approx - s[n] / N .
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\end{displaymath}
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-->
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<IMG
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WIDTH="77" HEIGHT="28" BORDER="0"
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SRC="img1329.png"
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ALT="\begin{displaymath}
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\approx - s[n] / N .
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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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So (again for <IMG
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WIDTH="41" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1317.png"
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ALT="$k \neq 0$">, small compared to <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$">) we get:
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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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{\cal FT}\{ p[n] \} (k) \approx
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{{-1} \over N} \cdot {{-iN} \over {2 \pi k}} \cdot {\cal FT}\{ s[n] \} (k)
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\end{displaymath}
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-->
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<IMG
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WIDTH="281" HEIGHT="39" BORDER="0"
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SRC="img1330.png"
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ALT="\begin{displaymath}
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{\cal FT}\{ p[n] \} (k) \approx
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{{-1} \over N} \cdot {{-iN} \over {2 \pi k}} \cdot {\cal FT}\{ s[n] \} (k)
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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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\approx {{-1} \over N} \cdot {{-iN} \over {2 \pi k}}
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\cdot {{-iN} \over {2 \pi k}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="134" HEIGHT="39" BORDER="0"
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SRC="img1331.png"
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ALT="\begin{displaymath}
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\approx {{-1} \over N} \cdot {{-iN} \over {2 \pi k}}
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\cdot {{-iN} \over {2 \pi k}}
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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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= {N \over {4 {\pi ^2} {k^2}}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="59" HEIGHT="39" BORDER="0"
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SRC="img1332.png"
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ALT="\begin{displaymath}
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= {N \over {4 {\pi ^2} {k^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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and as before we get the Fourier series:
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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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p[n] \approx {1 \over {2 {\pi^2}}} \left [
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{\cos ( \omega n )}
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+ {{\cos ( 2 \omega n)} \over 4}
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+ {{\cos ( 3 \omega n)} \over 9}
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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="354" HEIGHT="45" BORDER="0"
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SRC="img1333.png"
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ALT="\begin{displaymath}
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p[n] \approx {1 \over {2 {\pi^2}}} \left [
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{\cos ( \omega ...
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...\over 4}
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+ {{\cos ( 3 \omega n)} \over 9}
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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><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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</BODY>
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