141 lines
3.7 KiB
HTML
141 lines
3.7 KiB
HTML
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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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<H2><A ID="SECTION001311000000000000000">
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Periodicity of the Fourier transform</A>
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</H2>
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<P>
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If X[n] is, as above, a signal that repeats every <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="$N$"> samples, the Fourier
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transform of X[n] also repeats itself every <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="$N$"> units of frequency, that is,
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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+N) =
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{\cal FT}\left \{ X[n] \right \} (k)
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\end{displaymath}
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-->
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<IMG
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WIDTH="256" HEIGHT="28" BORDER="0"
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SRC="img1061.png"
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ALT="\begin{displaymath}
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{\cal FT}\left \{ X[n] \right \} (k+N) =
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{\cal FT}\left \{ X[n] \right \} (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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for all real 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="$k$">. This follows immediately from the definition
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of the Fourier transform, since the factor
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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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V = \cos(-k\omega) + i\sin(-k\omega)
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\end{displaymath}
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-->
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<IMG
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WIDTH="188" HEIGHT="28" BORDER="0"
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SRC="img1060.png"
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ALT="\begin{displaymath}
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V = \cos(-k\omega) + i\sin(-k\omega)
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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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is unchanged when we add <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="$N$"> (or any multiple of <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="$N$">) to <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="$k$">.
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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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