167 lines
5.6 KiB
HTML
167 lines
5.6 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>Combining sawtooth waves</TITLE>
|
|
<META NAME="description" CONTENT="Combining sawtooth waves">
|
|
<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="node199.html">
|
|
<LINK REL="previous" HREF="node197.html">
|
|
<LINK REL="up" HREF="node197.html">
|
|
<LINK REL="next" HREF="node199.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html3527"
|
|
HREF="node199.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A NAME="tex2html3521"
|
|
HREF="node197.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A NAME="tex2html3515"
|
|
HREF="node197.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A NAME="tex2html3523"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A NAME="tex2html3525"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html3528"
|
|
HREF="node199.html">Strategies for band-limiting sawtooth</A>
|
|
<B> Up:</B> <A NAME="tex2html3522"
|
|
HREF="node197.html">Examples</A>
|
|
<B> Previous:</B> <A NAME="tex2html3516"
|
|
HREF="node197.html">Examples</A>
|
|
<B> <A NAME="tex2html3524"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html3526"
|
|
HREF="node201.html">Index</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H2><A NAME="SECTION001451000000000000000">
|
|
Combining sawtooth waves</A>
|
|
</H2>
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig10.12"></A><A NAME="14498"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.12:</STRONG>
|
|
Combining sawtooth waves: (a) adding and subtracting sawtooth waves
|
|
1/2 cycle out of phase, to extract even and odd harmonics; (b) combining three
|
|
sawtooth waves with arbitrary amplitudes and phases.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="524" HEIGHT="396" BORDER="0"
|
|
SRC="img1375.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig10.12.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
Example J01.even.odd.pd (Figure <A HREF="#fig10.12">10.12</A>, part a) shows how to combine sawtooth
|
|
waves in pairs to extract the even and odd harmonics. The resulting waveforms
|
|
are as shown in Figure <A HREF="node187.html#fig10.03">10.3</A>. Example J02.trapezoids.pd (part b of the
|
|
figure) demonstrates combining three sawtooth waves at arbitrary phases and
|
|
amplitudes; the resulting classic waveform has up to three jumps and no
|
|
corners. The three line segments are horizontal as long as the three jumps add
|
|
to zero; otherwise the segments are sloped to make up for the the unbalanced
|
|
jumps so that the result repeats from one period to the next.
|
|
|
|
<P>
|
|
Example J03.pulse.width.mod.pd (not shown) combines two sawtooth waves, of opposite sign, with
|
|
slightly different frequencies so that the relative phase changes
|
|
continuously. Their sum is a rectangle wave whose width varies in time. This
|
|
is known as pulse width modulation (``PWM").
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig10.13"></A><A NAME="14505"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.13:</STRONG>
|
|
Combining parabolic waves to make a waveform with three corners.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="390" HEIGHT="464" BORDER="0"
|
|
SRC="img1376.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig10.13.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
Example J04.corners.pd (Figure <A HREF="#fig10.13">10.13</A>) shows how to add parabolic waves to make
|
|
a combined waveform with three corners. Each parabolic wave is computed from a
|
|
sawtooth wave (ranging from -0.5 to 0.5) by squaring it, multiplying by 0.5,
|
|
and subtracting the DC component of -1/12, or -0.08333. The patch combines
|
|
three such parabolic waves with controllable amplitudes and phases. As
|
|
long as the amplitudes sum to zero, the resulting waveform consists of line
|
|
segments, whose corners are located according to the three phases and have
|
|
slope changes according to the three amplitudes.
|
|
|
|
<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html3527"
|
|
HREF="node199.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A NAME="tex2html3521"
|
|
HREF="node197.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A NAME="tex2html3515"
|
|
HREF="node197.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A NAME="tex2html3523"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A NAME="tex2html3525"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html3528"
|
|
HREF="node199.html">Strategies for band-limiting sawtooth</A>
|
|
<B> Up:</B> <A NAME="tex2html3522"
|
|
HREF="node197.html">Examples</A>
|
|
<B> Previous:</B> <A NAME="tex2html3516"
|
|
HREF="node197.html">Examples</A>
|
|
<B> <A NAME="tex2html3524"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html3526"
|
|
HREF="node201.html">Index</A></B>
|
|
<!--End of Navigation Panel-->
|
|
<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|