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.
152 lines
5.1 KiB
152 lines
5.1 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>Sawtooth waves and symmetry</TITLE>
|
|
<META NAME="description" CONTENT="Sawtooth waves and symmetry">
|
|
<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="previous" HREF="node185.html">
|
|
<LINK REL="up" HREF="node185.html">
|
|
<LINK REL="next" HREF="node187.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A ID="tex2html3354"
|
|
HREF="node187.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A ID="tex2html3348"
|
|
HREF="node185.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A ID="tex2html3344"
|
|
HREF="node185.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A ID="tex2html3350"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A ID="tex2html3352"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A ID="tex2html3355"
|
|
HREF="node187.html">Dissecting classical waveforms</A>
|
|
<B> Up:</B> <A ID="tex2html3349"
|
|
HREF="node185.html">Symmetries and Fourier series</A>
|
|
<B> Previous:</B> <A ID="tex2html3345"
|
|
HREF="node185.html">Symmetries and Fourier series</A>
|
|
<B> <A ID="tex2html3351"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A ID="tex2html3353"
|
|
HREF="node201.html">Index</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H2><A ID="SECTION001411000000000000000">
|
|
Sawtooth waves and symmetry</A>
|
|
</H2>
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A ID="fig10.02"></A><A ID="14265"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.2:</STRONG>
|
|
Using a symmetry relation to extract even and odd harmonics from a
|
|
sawtooth wave: (a) the
|
|
original sawtooth wave; (b) shifted by 1/2 cycle; (c) their sum (another
|
|
sawtooth wave at
|
|
twice the frequency); (d) their difference (a square wave).</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="477" HEIGHT="467" BORDER="0"
|
|
SRC="img1277.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig10.02.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
As an example, we apply the shift symmetry (even and odd harmonics) to a
|
|
sawtooth wave. Figure <A HREF="#fig10.02">10.2</A> (part a) shows the original sawtooth wave
|
|
and part (b) shows the result of shifting by a half cycle. The sum of the two
|
|
(part c) drops discontinuously whenever either one of the two copies does so,
|
|
and traces a line segment whenever both component sawtooth waves do; so it in
|
|
turn becomes a sawtooth wave, of half the original period (twice the
|
|
fundamental frequency). Subtracting the two sawtooth waves (part d) gives a
|
|
waveform with slope zero except at the discontinuities. The discontinuities
|
|
coming from the original sawtooth wave jump in the same direction (negative to
|
|
positive), but those coming from the shifted one are negated and jump from
|
|
positive to negative. The result is a
|
|
<A ID="14269"></A><I>square wave</I>,
|
|
a particular rectangle wave in which the two component segments
|
|
have the same duration.
|
|
|
|
<P>
|
|
This symmetry was used to great effect in the design of Buchla analog
|
|
synthesizers; instead of offering a single sawtooth generator,
|
|
Buchla designed an oscillator that outputs the even and odd harmonic portions
|
|
separately, so that cross-fading between the two allows a continuous control
|
|
over the relative strengths of the even and odd harmonics in the analog
|
|
waveform.
|
|
|
|
<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A ID="tex2html3354"
|
|
HREF="node187.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A ID="tex2html3348"
|
|
HREF="node185.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A ID="tex2html3344"
|
|
HREF="node185.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A ID="tex2html3350"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A ID="tex2html3352"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A ID="tex2html3355"
|
|
HREF="node187.html">Dissecting classical waveforms</A>
|
|
<B> Up:</B> <A ID="tex2html3349"
|
|
HREF="node185.html">Symmetries and Fourier series</A>
|
|
<B> Previous:</B> <A ID="tex2html3345"
|
|
HREF="node185.html">Symmetries and Fourier series</A>
|
|
<B> <A ID="tex2html3351"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A ID="tex2html3353"
|
|
HREF="node201.html">Index</A></B>
|
|
<!--End of Navigation Panel-->
|
|
<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|
|
|