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537 lines
13 KiB
537 lines
13 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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<!--End of Navigation Panel-->
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<H2><A ID="SECTION001321000000000000000">
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Fourier transform of DC</A>
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</H2>
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<P>
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Let <IMG
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WIDTH="65" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1077.png"
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ALT="$X[n]=1$"> for all <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$"> (this repeats with any desired integer period
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<IMG
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WIDTH="47" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1078.png"
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ALT="$N>1$">). From the preceding discussion, we expect to find that
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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} \left \{ X[n] \right \} (k) =
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\left \{
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\begin{array}{ll}
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N & {k=0} \\
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0 & {k=1, \ldots, N-1}
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\end{array}
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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="287" HEIGHT="45" BORDER="0"
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SRC="img1079.png"
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ALT="\begin{displaymath}
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{\cal FT} \left \{ X[n] \right \} (k) =
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\left \{
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\begin{...
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...}
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N & {k=0} \\
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0 & {k=1, \ldots, N-1}
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\end{array} \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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We will often need to know the answer for non-integer values of <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img58.png"
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ALT="$k$"> however,
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and for this there is nothing better to do than to calculate the value
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directly:
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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} \left \{ X[n] \right \} (k) =
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{V ^ {0}} X[0] +
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{V ^ {1}} X[1] +
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\cdots +
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{V ^ {N-1}} X[N-1]
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\end{displaymath}
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-->
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<IMG
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WIDTH="406" HEIGHT="28" BORDER="0"
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SRC="img1057.png"
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ALT="\begin{displaymath}
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{\cal FT}\left \{ X[n] \right \} (k) =
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{V ^ {0}} X[0] +
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{V ^ {1}} X[1] +
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\cdots +
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{V ^ {N-1}} X[N-1]
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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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where <IMG
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WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1059.png"
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ALT="$V$"> is, as before, the unit magnitude complex number with argument
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<IMG
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WIDTH="35" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1055.png"
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ALT="$-k\omega$">. This is a geometric series; as long as <IMG
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WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1080.png"
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ALT="$V \not= 1$"> 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} \left \{ X[n] \right \} (k) =
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{{
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{V^N} - 1
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} \over {
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V - 1
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}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="177" HEIGHT="43" BORDER="0"
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SRC="img1081.png"
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ALT="\begin{displaymath}
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{\cal FT} \left \{ X[n] \right \} (k) =
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{{
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{V^N} - 1
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} \over {
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V - 1
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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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We now symmetrize the top and bottom in the same way as we earlier did in
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Section <A HREF="node108.html#sect7.network">7.3</A>. To do this let:
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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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\xi = \cos(\pi k / N) - i \sin(\pi k / N)
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\end{displaymath}
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-->
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<IMG
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WIDTH="201" HEIGHT="28" BORDER="0"
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SRC="img1082.png"
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ALT="\begin{displaymath}
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\xi = \cos(\pi k / N) - i \sin(\pi k / 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 that <IMG
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WIDTH="52" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img1083.png"
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ALT="${\xi^2} = V$">. Then factoring appropriate powers of <IMG
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WIDTH="11" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1084.png"
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ALT="$\xi$"> out of the
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numerator and denominator gives:
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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} \left \{ X[n] \right \} (k) =
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{\xi^{N-1}}
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{{
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{\xi^N} - {\xi^{-N}}
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} \over {
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\xi - {\xi^{-1}}
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}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="230" HEIGHT="45" BORDER="0"
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SRC="img1085.png"
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ALT="\begin{displaymath}
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{\cal FT} \left \{ X[n] \right \} (k) =
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{\xi^{N-1}}
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{{
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{\xi^N} - {\xi^{-N}}
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} \over {
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\xi - {\xi^{-1}}
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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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It's easy now to simplify the numerator:
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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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{\xi^N} - {\xi^{-N}} =
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\left (\cos(\pi k) - i \sin(\pi k) \right ) -
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\left (\cos(\pi k) + i \sin(\pi k) \right )
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= - 2 i \sin(\pi k)
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\end{displaymath}
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-->
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<IMG
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WIDTH="493" HEIGHT="28" BORDER="0"
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SRC="img1086.png"
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ALT="\begin{displaymath}
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{\xi^N} - {\xi^{-N}} =
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\left (\cos(\pi k) - i \sin(\pi k) ...
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...eft (\cos(\pi k) + i \sin(\pi k) \right )
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= - 2 i \sin(\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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and similarly for the denominator, giving:
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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} \left \{ X[n] \right \} (k) =
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\left ( {
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\parbox[t][0.1in]{0in}{\mbox{}}
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\cos(\pi k (N-1)/N) - i \sin(\pi k (N-1)/N)
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} \right )
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{{
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\sin(\pi k)
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} \over {
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\sin(\pi k / N)
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}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="502" HEIGHT="44" BORDER="0"
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SRC="img1087.png"
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ALT="\begin{displaymath}
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{\cal FT} \left \{ X[n] \right \} (k) =
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\left ( {
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\parbo...
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...
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} \right )
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{{
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\sin(\pi k)
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} \over {
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\sin(\pi k / N)
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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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Whether <IMG
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WIDTH="45" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1088.png"
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ALT="$V=1$"> or not, we have
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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} \left \{ X[n] \right \} (k) =
|
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\left ( {
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\parbox[t][0.1in]{0in}{\mbox{}}
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\cos(\pi k (N-1)/N) - i \sin(\pi k (N-1)/N)
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} \right )
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{D_N}(k)
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\end{displaymath}
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-->
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<IMG
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WIDTH="473" HEIGHT="35" BORDER="0"
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SRC="img1089.png"
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ALT="\begin{displaymath}
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{\cal FT} \left \{ X[n] \right \} (k) =
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\left ( {
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\parbo...
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...(\pi k (N-1)/N) - i \sin(\pi k (N-1)/N)
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} \right )
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{D_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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where <IMG
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WIDTH="49" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1090.png"
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ALT="${D_N}(k)$">, known as the
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<A ID="12394"></A><I>Dirichlet kernel</I>,
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is defined as
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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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{D_N}(k) =
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\left \{
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\begin{array}{ll}
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N & {k= 0} \\
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{{
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\sin(\pi k)
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} \over {
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\sin(\pi k / N)
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}}
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& {k\not=0,\; -N < k < N}
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\end{array}
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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="307" HEIGHT="54" BORDER="0"
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SRC="img1091.png"
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ALT="\begin{displaymath}
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{D_N}(k) =
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\left \{
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\begin{array}{ll}
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N & {k= 0} \\
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{...
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...pi k / N)
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}}
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& {k\not=0,\; -N < k < N}
|
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\end{array} \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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Figure <A HREF="#fig09.01">9.1</A> shows the Fourier transform of <IMG
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WIDTH="65" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1077.png"
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ALT="$X[n]=1$">, with <IMG
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WIDTH="63" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img63.png"
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ALT="$N=100$">. The
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transform repeats every 100 samples, with a peak at <IMG
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WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1092.png"
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ALT="$k=0$">, another at
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<IMG
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WIDTH="57" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1093.png"
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ALT="$k=100$">, and so on. The figure endeavors to show both the magnitude and phase
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behavior using a 3-dimensional graph projected onto the page. The phase
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term
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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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\cos(\pi k (N-1)/N) - i \sin(\pi k (N-1)/N)
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\end{displaymath}
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-->
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<IMG
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WIDTH="280" HEIGHT="28" BORDER="0"
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SRC="img1094.png"
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ALT="\begin{displaymath}
|
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\cos(\pi k (N-1)/N) - i \sin(\pi k (N-1)/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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acts to twist the values of <!-- MATH
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${\cal FT} \left \{ X[n] \right \} (k)$
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-->
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<IMG
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WIDTH="104" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img1095.png"
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ALT="${\cal FT} \left \{ X[n] \right \} (k)$"> around
|
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the <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
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SRC="img58.png"
|
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ALT="$k$"> axis with a period of approximately two. The Dirichlet kernel
|
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<IMG
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WIDTH="49" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
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SRC="img1090.png"
|
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ALT="${D_N}(k)$">, shown in Figure <A HREF="#fig09.02">9.2</A>, controls the magnitude of
|
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<!-- MATH
|
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${\cal FT} \left \{ X[n] \right \} (k)$
|
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-->
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<IMG
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WIDTH="104" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1095.png"
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ALT="${\cal FT} \left \{ X[n] \right \} (k)$">. It has a peak, two units wide, around
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<IMG
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WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img1092.png"
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ALT="$k=0$">. This is surrounded by one-unit-wide
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<A ID="12409"></A><I>sidelobes</I>,
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alternating in sign and gradually decreasing in magnitude as <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
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SRC="img58.png"
|
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ALT="$k$"> increases or
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decreases away from zero. The phase term rotates by almost <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img41.png"
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ALT="$\pi $"> radians
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each time the Dirichlet kernel changes sign, so that the product of the
|
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two stays roughly in the same complex half-plane for <IMG
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WIDTH="41" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1096.png"
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ALT="$k>1$"> (and in the
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opposite half-plane for <IMG
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WIDTH="53" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1097.png"
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ALT="$k < -1$">). The phase rotates by almost <IMG
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WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img16.png"
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ALT="$2\pi $">
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radians over the peak from <IMG
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WIDTH="53" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img1098.png"
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ALT="$k=-1$"> to <IMG
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WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img259.png"
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ALT="$k=1$">.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig09.01"></A><A ID="12413"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.1:</STRONG>
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The Fourier transform of a signal consisting of all ones. Here
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N=100, and values are shown for <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
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SRC="img58.png"
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ALT="$k$"> ranging from -5 to 10. The result
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is complex-valued and shown as a projection, with the real axis pointing up the
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page and the imaginary axis pointing away from it.</CAPTION>
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<TR><TD><IMG
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WIDTH="470" HEIGHT="265" BORDER="0"
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SRC="img1099.png"
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ALT="\begin{figure}\psfig{file=figs/fig09.01.ps}\end{figure}"></TD></TR>
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<DIV ALIGN="CENTER"><A ID="fig09.02"></A><A ID="12418"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 9.2:</STRONG>
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The Dirichlet kernel, for <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> = 100.</CAPTION>
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<TR><TD><IMG
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WIDTH="448" HEIGHT="175" BORDER="0"
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SRC="img1100.png"
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ALT="\begin{figure}\psfig{file=figs/fig09.02.ps}\end{figure}"></TD></TR>
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