You can not select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
463 lines
13 KiB
463 lines
13 KiB
<!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>Fourier analysis of periodic signals</TITLE>
|
|
<META NAME="description" CONTENT="Fourier analysis of periodic signals">
|
|
<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="node167.html">
|
|
<LINK REL="previous" HREF="node163.html">
|
|
<LINK REL="up" HREF="node163.html">
|
|
<LINK REL="next" HREF="node165.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A ID="tex2html3031"
|
|
HREF="node165.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A ID="tex2html3025"
|
|
HREF="node163.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A ID="tex2html3019"
|
|
HREF="node163.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A ID="tex2html3027"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A ID="tex2html3029"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A ID="tex2html3032"
|
|
HREF="node165.html">Periodicity of the Fourier</A>
|
|
<B> Up:</B> <A ID="tex2html3026"
|
|
HREF="node163.html">Fourier analysis and resynthesis</A>
|
|
<B> Previous:</B> <A ID="tex2html3020"
|
|
HREF="node163.html">Fourier analysis and resynthesis</A>
|
|
<B> <A ID="tex2html3028"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A ID="tex2html3030"
|
|
HREF="node201.html">Index</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H1><A ID="SECTION001310000000000000000">
|
|
Fourier analysis of periodic signals</A>
|
|
</H1>
|
|
|
|
<P>
|
|
Suppose <IMG
|
|
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img669.png"
|
|
ALT="$X[n]$"> is a complex-valued signal that repeats every <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> samples. (We
|
|
are continuing to use complex-valued signals rather than real-valued ones
|
|
to simplify the mathematics.) Because of the period <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$">, the
|
|
values of <IMG
|
|
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img669.png"
|
|
ALT="$X[n]$"> for <!-- MATH
|
|
$n=0,\ldots,N-1$
|
|
-->
|
|
<IMG
|
|
WIDTH="119" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1042.png"
|
|
ALT="$n=0,\ldots,N-1$"> determine <IMG
|
|
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img669.png"
|
|
ALT="$X[n]$"> for all integer values
|
|
of <IMG
|
|
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img75.png"
|
|
ALT="$n$">.
|
|
|
|
<P>
|
|
Suppose further that <IMG
|
|
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img669.png"
|
|
ALT="$X[n]$"> can be written as a sum of complex sinusoids of
|
|
frequency <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img179.png"
|
|
ALT="$0$">, <IMG
|
|
WIDTH="42" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img184.png"
|
|
ALT="$2\pi/N$">, <IMG
|
|
WIDTH="42" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1043.png"
|
|
ALT="$4\pi/N$">, <IMG
|
|
WIDTH="22" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img239.png"
|
|
ALT="$\ldots$">, <IMG
|
|
WIDTH="97" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1044.png"
|
|
ALT="$2(N-1)\pi/N$">. These are the
|
|
partials, starting with the zeroth, for a signal of period <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$">. We stop at
|
|
the <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$">th term because the next one would have frequency <IMG
|
|
WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img16.png"
|
|
ALT="$2\pi $">, equivalent
|
|
to frequency <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img179.png"
|
|
ALT="$0$">, which is already on the list.
|
|
|
|
<P>
|
|
Given the values of <IMG
|
|
WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img670.png"
|
|
ALT="$X$">, we wish to find the complex amplitudes of the
|
|
partials. Suppose we want the <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$">th partial, where <IMG
|
|
WIDTH="77" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1045.png"
|
|
ALT="$0 \leq k < N$">. The
|
|
frequency of this partial is <IMG
|
|
WIDTH="51" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1046.png"
|
|
ALT="$2\pi k / N$">. We can find its complex amplitude
|
|
by modulating <IMG
|
|
WIDTH="17" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img670.png"
|
|
ALT="$X$"> downward <IMG
|
|
WIDTH="51" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1046.png"
|
|
ALT="$2\pi k / N$"> radians per sample in frequency, so
|
|
that the <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$">th partial is modulated to frequency zero. Then we pass the signal
|
|
through a low-pass filter with such a low cutoff frequency that nothing but the
|
|
zero-frequency partial remains. We can do this in effect by averaging over a
|
|
huge number of samples; but since the signal repeats every <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> samples, this
|
|
huge average is the same as the average of the first <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> samples. In short, to
|
|
measure a sinusoidal component of a periodic signal, modulate it down to DC and
|
|
then average over one period.
|
|
|
|
<P>
|
|
Let <IMG
|
|
WIDTH="74" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1047.png"
|
|
ALT="$\omega=2\pi/N$"> be the fundamental frequency for the period <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$">, and
|
|
let <IMG
|
|
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img1048.png"
|
|
ALT="$U$"> be the unit-magnitude complex number with argument <IMG
|
|
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img27.png"
|
|
ALT="$\omega $">:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
U = \cos(\omega) + i \sin(\omega)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="146" HEIGHT="28" BORDER="0"
|
|
SRC="img1049.png"
|
|
ALT="\begin{displaymath}
|
|
U = \cos(\omega) + i \sin(\omega)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
The <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$">th partial of the signal <IMG
|
|
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img669.png"
|
|
ALT="$X[n]$"> is of the form:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{P_k}[n] = {A_k}{{\left [ {U^k} \right ]} ^ {n}}
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="119" HEIGHT="30" BORDER="0"
|
|
SRC="img1050.png"
|
|
ALT="\begin{displaymath}
|
|
{P_k}[n] = {A_k}{{\left [ {U^k} \right ]} ^ {n}}
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
where <IMG
|
|
WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1051.png"
|
|
ALT="${A_k}$"> is the complex amplitude of the partial, and the frequency
|
|
of the partial is:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
\angle({U^k}) = k \angle(U) = k\omega
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="146" HEIGHT="28" BORDER="0"
|
|
SRC="img1052.png"
|
|
ALT="\begin{displaymath}
|
|
\angle({U^k}) = k \angle(U) = k\omega
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
We're assuming for the moment that the signal <IMG
|
|
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img669.png"
|
|
ALT="$X[n]$"> can actually be written
|
|
as a sum of the <IMG
|
|
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img75.png"
|
|
ALT="$n$"> partials, or in other words:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
X[n] =
|
|
{A_0}{{\left [ {U^0} \right ]} ^ {n}}
|
|
+ {A_1}{{\left [ {U^1} \right ]} ^ {n}}
|
|
+ \cdots
|
|
+ {A_{N-1}}{{\left [ {U^{N-1}} \right ]} ^ {n}}
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="355" HEIGHT="30" BORDER="0"
|
|
SRC="img1053.png"
|
|
ALT="\begin{displaymath}
|
|
X[n] =
|
|
{A_0}{{\left [ {U^0} \right ]} ^ {n}}
|
|
+ {A_1}{{\l...
|
|
...n}}
|
|
+ \cdots
|
|
+ {A_{N-1}}{{\left [ {U^{N-1}} \right ]} ^ {n}}
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
By the heterodyne-filtering argument above, we expect to be able to measure
|
|
each <IMG
|
|
WIDTH="23" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1054.png"
|
|
ALT="$A_k$"> by multiplying by the sinusoid of frequency <IMG
|
|
WIDTH="35" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1055.png"
|
|
ALT="$-k\omega$"> and
|
|
averaging over a period:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{A_k} = {1\over N} \left (
|
|
{{\left [ {U^{-k}} \right ]} ^ {0}} X[0] +
|
|
{{\left [ {U^{-k}} \right ]} ^ {1}} X[1] +
|
|
\cdots +
|
|
{{\left [ {U^{-k}} \right ]} ^ {N-1}} X[N-1]
|
|
\right )
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="457" HEIGHT="38" BORDER="0"
|
|
SRC="img1056.png"
|
|
ALT="\begin{displaymath}
|
|
{A_k} = {1\over N} \left (
|
|
{{\left [ {U^{-k}} \right ]} ^ ...
|
|
...dots +
|
|
{{\left [ {U^{-k}} \right ]} ^ {N-1}} X[N-1]
|
|
\right )
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
This is such a useful formula that it gets its own notation. The
|
|
<A ID="12325"></A><I>Fourier transform</I>
|
|
of a signal <IMG
|
|
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img669.png"
|
|
ALT="$X[n]$">, over <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> samples, is defined as:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
{\cal FT}\left \{ X[n] \right \} (k) =
|
|
{V ^ {0}} X[0] +
|
|
{V ^ {1}} X[1] +
|
|
\cdots +
|
|
{V ^ {N-1}} X[N-1]
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="406" HEIGHT="28" BORDER="0"
|
|
SRC="img1057.png"
|
|
ALT="\begin{displaymath}
|
|
{\cal FT}\left \{ X[n] \right \} (k) =
|
|
{V ^ {0}} X[0] +
|
|
{V ^ {1}} X[1] +
|
|
\cdots +
|
|
{V ^ {N-1}} X[N-1]
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
where <IMG
|
|
WIDTH="67" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img1058.png"
|
|
ALT="$V = {U^{-k}}$">. The Fourier transform is a function of the variable <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$">,
|
|
equal to <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> times the amplitude of the input's <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$">th partial. So far <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$">
|
|
has taken integer values but the formula makes sense for any value of <IMG
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img58.png"
|
|
ALT="$k$"> if we
|
|
define <IMG
|
|
WIDTH="16" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img1059.png"
|
|
ALT="$V$"> more generally as:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
V = \cos(-k\omega) + i\sin(-k\omega)
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="188" HEIGHT="28" BORDER="0"
|
|
SRC="img1060.png"
|
|
ALT="\begin{displaymath}
|
|
V = \cos(-k\omega) + i\sin(-k\omega)
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
where, as before, <IMG
|
|
WIDTH="74" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1047.png"
|
|
ALT="$\omega=2\pi/N$"> is the (angular) fundamental
|
|
frequency associated with the period <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$">.
|
|
|
|
<P>
|
|
<BR><HR>
|
|
<!--Table of Child-Links-->
|
|
<A ID="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
|
|
|
<UL>
|
|
<LI><A ID="tex2html3033"
|
|
HREF="node165.html">Periodicity of the Fourier transform</A>
|
|
<LI><A ID="tex2html3034"
|
|
HREF="node166.html">Fourier transform as additive synthesis</A>
|
|
</UL>
|
|
<!--End of Table of Child-Links-->
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A ID="tex2html3031"
|
|
HREF="node165.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A ID="tex2html3025"
|
|
HREF="node163.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A ID="tex2html3019"
|
|
HREF="node163.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A ID="tex2html3027"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A ID="tex2html3029"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A ID="tex2html3032"
|
|
HREF="node165.html">Periodicity of the Fourier</A>
|
|
<B> Up:</B> <A ID="tex2html3026"
|
|
HREF="node163.html">Fourier analysis and resynthesis</A>
|
|
<B> Previous:</B> <A ID="tex2html3020"
|
|
HREF="node163.html">Fourier analysis and resynthesis</A>
|
|
<B> <A ID="tex2html3028"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A ID="tex2html3030"
|
|
HREF="node201.html">Index</A></B>
|
|
<!--End of Navigation Panel-->
|
|
<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|
|
|