300 lines
8.7 KiB
HTML
300 lines
8.7 KiB
HTML
|
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
|
||
|
|
|
||
|
|
<!--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>
|
||
|
|
<TITLE>Fourier series of the elementary waveforms</TITLE>
|
||
|
|
<META NAME="description" CONTENT="Fourier series of the elementary waveforms">
|
||
|
|
<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="node193.html">
|
||
|
|
<LINK REL="previous" HREF="node187.html">
|
||
|
|
<LINK REL="up" HREF="node184.html">
|
||
|
|
<LINK REL="next" HREF="node189.html">
|
||
|
|
</HEAD>
|
||
|
|
|
||
|
|
<BODY >
|
||
|
|
<!--Navigation Panel-->
|
||
|
|
<A NAME="tex2html3382"
|
||
|
|
HREF="node189.html">
|
||
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/next.png"></A>
|
||
|
|
<A NAME="tex2html3376"
|
||
|
|
HREF="node184.html">
|
||
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/up.png"></A>
|
||
|
|
<A NAME="tex2html3370"
|
||
|
|
HREF="node187.html">
|
||
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/prev.png"></A>
|
||
|
|
<A NAME="tex2html3378"
|
||
|
|
HREF="node4.html">
|
||
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/contents.png"></A>
|
||
|
|
<A NAME="tex2html3380"
|
||
|
|
HREF="node201.html">
|
||
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/index.png"></A>
|
||
|
|
<BR>
|
||
|
|
<B> Next:</B> <A NAME="tex2html3383"
|
||
|
|
HREF="node189.html">Sawtooth wave</A>
|
||
|
|
<B> Up:</B> <A NAME="tex2html3377"
|
||
|
|
HREF="node184.html">Classical waveforms</A>
|
||
|
|
<B> Previous:</B> <A NAME="tex2html3371"
|
||
|
|
HREF="node187.html">Dissecting classical waveforms</A>
|
||
|
|
<B> <A NAME="tex2html3379"
|
||
|
|
HREF="node4.html">Contents</A></B>
|
||
|
|
<B> <A NAME="tex2html3381"
|
||
|
|
HREF="node201.html">Index</A></B>
|
||
|
|
<BR>
|
||
|
|
<BR>
|
||
|
|
<!--End of Navigation Panel-->
|
||
|
|
|
||
|
|
<H1><A NAME="SECTION001430000000000000000"></A>
|
||
|
|
<A NAME="sect10.spectra"></A>
|
||
|
|
<BR>
|
||
|
|
Fourier series of the elementary waveforms
|
||
|
|
</H1>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
In general, given a repeating waveform <IMG
|
||
|
|
WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img669.png"
|
||
|
|
ALT="$X[n]$">, we can evaluate its Fourier
|
||
|
|
series coefficients <IMG
|
||
|
|
WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img1251.png"
|
||
|
|
ALT="$A[k]$"> by directly evaluating the Fourier transform:
|
||
|
|
<BR><P></P>
|
||
|
|
<DIV ALIGN="CENTER">
|
||
|
|
<!-- MATH
|
||
|
|
\begin{displaymath}
|
||
|
|
A[k] = {1 \over N} {\cal FT}\{X[n]\}(k)
|
||
|
|
\end{displaymath}
|
||
|
|
-->
|
||
|
|
|
||
|
|
<IMG
|
||
|
|
WIDTH="164" HEIGHT="38" BORDER="0"
|
||
|
|
SRC="img1301.png"
|
||
|
|
ALT="\begin{displaymath}
|
||
|
|
A[k] = {1 \over N} {\cal FT}\{X[n]\}(k)
|
||
|
|
\end{displaymath}">
|
||
|
|
</DIV>
|
||
|
|
<BR CLEAR="ALL">
|
||
|
|
<P></P>
|
||
|
|
<BR><P></P>
|
||
|
|
<DIV ALIGN="CENTER">
|
||
|
|
<!-- MATH
|
||
|
|
\begin{displaymath}
|
||
|
|
= {1 \over N} \left [ X[0] +
|
||
|
|
{U^{-k}} X[1] + \cdots +
|
||
|
|
{U^{-(N-1)k}} X[N-1] \right ]
|
||
|
|
\end{displaymath}
|
||
|
|
-->
|
||
|
|
|
||
|
|
<IMG
|
||
|
|
WIDTH="351" HEIGHT="38" BORDER="0"
|
||
|
|
SRC="img1302.png"
|
||
|
|
ALT="\begin{displaymath}
|
||
|
|
= {1 \over N} \left [ X[0] +
|
||
|
|
{U^{-k}} X[1] + \cdots +
|
||
|
|
{U^{-(N-1)k}} X[N-1] \right ]
|
||
|
|
\end{displaymath}">
|
||
|
|
</DIV>
|
||
|
|
<BR CLEAR="ALL">
|
||
|
|
<P></P>
|
||
|
|
but doing this directly for sawtooth and parabolic waves will require pages of
|
||
|
|
algebra (somewhat less if we were willing resort to differential calculus).
|
||
|
|
Instead, we rely on properties of the Fourier transform to relate the transform
|
||
|
|
of a signal <IMG
|
||
|
|
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img80.png"
|
||
|
|
ALT="$x[n]$"> with its
|
||
|
|
<A NAME="14332"></A><I>first difference</I>,
|
||
|
|
defined as <IMG
|
||
|
|
WIDTH="105" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img1303.png"
|
||
|
|
ALT="$x[n] - x[n-1]$">. The first difference of the parabolic wave will
|
||
|
|
turn out to be a sawtooth, and that of a sawtooth will be simple enough to
|
||
|
|
evaluate directly, and thus we'll get the desired Fourier series.
|
||
|
|
|
||
|
|
<P>
|
||
|
|
In general, to evaluate the strength of the <IMG
|
||
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img58.png"
|
||
|
|
ALT="$k$">th harmonic, we'll make the
|
||
|
|
assumption that <IMG
|
||
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img3.png"
|
||
|
|
ALT="$N$"> is much larger than <IMG
|
||
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img58.png"
|
||
|
|
ALT="$k$">, or equivalently, that <IMG
|
||
|
|
WIDTH="33" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img1304.png"
|
||
|
|
ALT="$k/N$"> is
|
||
|
|
negligible.
|
||
|
|
|
||
|
|
<P>
|
||
|
|
We start from the Time Shift Formula for Fourier Transforms
|
||
|
|
(Page <A HREF="node169.html#sect9.shift"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>) setting the time shift to one sample:
|
||
|
|
<BR><P></P>
|
||
|
|
<DIV ALIGN="CENTER">
|
||
|
|
<!-- MATH
|
||
|
|
\begin{displaymath}
|
||
|
|
{\cal FT}\{ x[n-1] \} =
|
||
|
|
\left [ \cos(k \omega) - i \sin (k \omega) \right ]
|
||
|
|
{\cal FT}\{ x[n] \}
|
||
|
|
\end{displaymath}
|
||
|
|
-->
|
||
|
|
|
||
|
|
<IMG
|
||
|
|
WIDTH="329" HEIGHT="28" BORDER="0"
|
||
|
|
SRC="img1305.png"
|
||
|
|
ALT="\begin{displaymath}
|
||
|
|
{\cal FT}\{ x[n-1] \} =
|
||
|
|
\left [ \cos(k \omega) - i \sin (k \omega) \right ]
|
||
|
|
{\cal FT}\{ x[n] \}
|
||
|
|
\end{displaymath}">
|
||
|
|
</DIV>
|
||
|
|
<BR CLEAR="ALL">
|
||
|
|
<P></P>
|
||
|
|
<BR><P></P>
|
||
|
|
<DIV ALIGN="CENTER">
|
||
|
|
<!-- MATH
|
||
|
|
\begin{displaymath}
|
||
|
|
\approx (1 - i \omega k) {\cal FT}\{ x[n] \}
|
||
|
|
\end{displaymath}
|
||
|
|
-->
|
||
|
|
|
||
|
|
<IMG
|
||
|
|
WIDTH="148" HEIGHT="28" BORDER="0"
|
||
|
|
SRC="img1306.png"
|
||
|
|
ALT="\begin{displaymath}
|
||
|
|
\approx (1 - i \omega k) {\cal FT}\{ x[n] \}
|
||
|
|
\end{displaymath}">
|
||
|
|
</DIV>
|
||
|
|
<BR CLEAR="ALL">
|
||
|
|
<P></P>
|
||
|
|
Here we're using the assumption that, because <IMG
|
||
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img3.png"
|
||
|
|
ALT="$N$"> is much larger than <IMG
|
||
|
|
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img58.png"
|
||
|
|
ALT="$k$">,
|
||
|
|
<!-- MATH
|
||
|
|
$k \omega = 2\pi k / N$
|
||
|
|
-->
|
||
|
|
<IMG
|
||
|
|
WIDTH="91" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img1307.png"
|
||
|
|
ALT="$k \omega = 2\pi k / N$"> is much smaller than unity and we can make
|
||
|
|
approximations:
|
||
|
|
<BR><P></P>
|
||
|
|
<DIV ALIGN="CENTER">
|
||
|
|
<!-- MATH
|
||
|
|
\begin{displaymath}
|
||
|
|
\cos(k \omega) \approx 1 \; , \; \sin(k \omega) \approx k \omega
|
||
|
|
\end{displaymath}
|
||
|
|
-->
|
||
|
|
|
||
|
|
<IMG
|
||
|
|
WIDTH="187" HEIGHT="28" BORDER="0"
|
||
|
|
SRC="img1308.png"
|
||
|
|
ALT="\begin{displaymath}
|
||
|
|
\cos(k \omega) \approx 1 \; , \; \sin(k \omega) \approx k \omega
|
||
|
|
\end{displaymath}">
|
||
|
|
</DIV>
|
||
|
|
<BR CLEAR="ALL">
|
||
|
|
<P></P>
|
||
|
|
which are good to within a small error, on the order of <IMG
|
||
|
|
WIDTH="53" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img1309.png"
|
||
|
|
ALT="$(k/N)^2$">.
|
||
|
|
Now we plug this result in to evaluate:
|
||
|
|
<BR><P></P>
|
||
|
|
<DIV ALIGN="CENTER">
|
||
|
|
<!-- MATH
|
||
|
|
\begin{displaymath}
|
||
|
|
{\cal FT}\{ x[n] - x[n-1] \} \approx i \omega k {\cal FT}\{ x[n] \}
|
||
|
|
\end{displaymath}
|
||
|
|
-->
|
||
|
|
|
||
|
|
<IMG
|
||
|
|
WIDTH="257" HEIGHT="28" BORDER="0"
|
||
|
|
SRC="img1310.png"
|
||
|
|
ALT="\begin{displaymath}
|
||
|
|
{\cal FT}\{ x[n] - x[n-1] \} \approx i \omega k {\cal FT}\{ x[n] \}
|
||
|
|
\end{displaymath}">
|
||
|
|
</DIV>
|
||
|
|
<BR CLEAR="ALL">
|
||
|
|
<P></P>
|
||
|
|
<BR><HR>
|
||
|
|
<!--Table of Child-Links-->
|
||
|
|
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
||
|
|
|
||
|
|
<UL>
|
||
|
|
<LI><A NAME="tex2html3384"
|
||
|
|
HREF="node189.html">Sawtooth wave</A>
|
||
|
|
<LI><A NAME="tex2html3385"
|
||
|
|
HREF="node190.html">Parabolic wave</A>
|
||
|
|
<LI><A NAME="tex2html3386"
|
||
|
|
HREF="node191.html">Square and symmetric triangle waves</A>
|
||
|
|
<LI><A NAME="tex2html3387"
|
||
|
|
HREF="node192.html">General (non-symmetric) triangle wave</A>
|
||
|
|
</UL>
|
||
|
|
<!--End of Table of Child-Links-->
|
||
|
|
<HR>
|
||
|
|
<!--Navigation Panel-->
|
||
|
|
<A NAME="tex2html3382"
|
||
|
|
HREF="node189.html">
|
||
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/next.png"></A>
|
||
|
|
<A NAME="tex2html3376"
|
||
|
|
HREF="node184.html">
|
||
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/up.png"></A>
|
||
|
|
<A NAME="tex2html3370"
|
||
|
|
HREF="node187.html">
|
||
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/prev.png"></A>
|
||
|
|
<A NAME="tex2html3378"
|
||
|
|
HREF="node4.html">
|
||
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/contents.png"></A>
|
||
|
|
<A NAME="tex2html3380"
|
||
|
|
HREF="node201.html">
|
||
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
||
|
|
SRC="file:/usr/local/share/lib/latex2html/icons/index.png"></A>
|
||
|
|
<BR>
|
||
|
|
<B> Next:</B> <A NAME="tex2html3383"
|
||
|
|
HREF="node189.html">Sawtooth wave</A>
|
||
|
|
<B> Up:</B> <A NAME="tex2html3377"
|
||
|
|
HREF="node184.html">Classical waveforms</A>
|
||
|
|
<B> Previous:</B> <A NAME="tex2html3371"
|
||
|
|
HREF="node187.html">Dissecting classical waveforms</A>
|
||
|
|
<B> <A NAME="tex2html3379"
|
||
|
|
HREF="node4.html">Contents</A></B>
|
||
|
|
<B> <A NAME="tex2html3381"
|
||
|
|
HREF="node201.html">Index</A></B>
|
||
|
|
<!--End of Navigation Panel-->
|
||
|
|
<ADDRESS>
|
||
|
|
Miller Puckette
|
||
|
|
2006-12-30
|
||
|
|
</ADDRESS>
|
||
|
|
</BODY>
|
||
|
|
</HTML>
|