141 lines
3.7 KiB
HTML
141 lines
3.7 KiB
HTML
<!DOCTYPE html>
|
|
|
|
<!--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>
|
|
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0">
|
|
|
|
|
|
<TITLE>Periodicity of the Fourier transform</TITLE>
|
|
<META NAME="description" CONTENT="Periodicity of the Fourier transform">
|
|
<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="node166.html">
|
|
<LINK REL="previous" HREF="node164.html">
|
|
<LINK REL="up" HREF="node164.html">
|
|
<LINK REL="next" HREF="node166.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html3047"
|
|
HREF="node166.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A NAME="tex2html3041"
|
|
HREF="node164.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A NAME="tex2html3035"
|
|
HREF="node164.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A NAME="tex2html3043"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A NAME="tex2html3045"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html3048"
|
|
HREF="node166.html">Fourier transform as additive</A>
|
|
<B> Up:</B> <A NAME="tex2html3042"
|
|
HREF="node164.html">Fourier analysis of periodic</A>
|
|
<B> Previous:</B> <A NAME="tex2html3036"
|
|
HREF="node164.html">Fourier analysis of periodic</A>
|
|
<B> <A NAME="tex2html3044"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html3046"
|
|
HREF="node201.html">Index</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H2><A NAME="SECTION001311000000000000000">
|
|
Periodicity of the Fourier transform</A>
|
|
</H2>
|
|
|
|
<P>
|
|
If X[n] is, as above, a signal that repeats every <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> samples, the Fourier
|
|
transform of X[n] also repeats itself every <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> units of frequency, that is,
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{\cal FT}\left \{ X[n] \right \} (k+N) =
|
|
{\cal FT}\left \{ X[n] \right \} (k)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="256" HEIGHT="28" BORDER="0"
|
|
SRC="img1061.png"
|
|
ALT="\begin{displaymath}
|
|
{\cal FT}\left \{ X[n] \right \} (k+N) =
|
|
{\cal FT}\left \{ X[n] \right \} (k)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
for all real values of <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$">. This follows immediately from the definition
|
|
of the Fourier transform, since the factor
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
V = \cos(-k\omega) + i\sin(-k\omega)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="188" HEIGHT="28" BORDER="0"
|
|
SRC="img1060.png"
|
|
ALT="\begin{displaymath}
|
|
V = \cos(-k\omega) + i\sin(-k\omega)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
is unchanged when we add <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> (or any multiple of <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$">) to <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$">.
|
|
|
|
<P>
|
|
<BR><HR>
|
|
<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|