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238 lines
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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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* with significant contributions from:
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Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
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<TITLE>Classical waveforms</TITLE>
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<META NAME="description" CONTENT="Classical waveforms">
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HREF="node185.html">Symmetries and Fourier series</A>
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<B> Up:</B> <A NAME="tex2html3306"
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HREF="book.html">book</A>
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<B> Previous:</B> <A NAME="tex2html3300"
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HREF="node183.html">Exercises</A>
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<B> <A NAME="tex2html3308"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html3310"
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HREF="node201.html">Index</A></B>
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<BR>
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<!--End of Navigation Panel-->
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<H1><A NAME="SECTION001400000000000000000"></A>
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<A NAME="chapter-waveforms"></A>
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<BR>
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Classical waveforms
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</H1>
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<P>
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Up until now we have primarily taken three approaches to synthesizing
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repetitive waveforms: additive synthesis (Chapter 1),
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wavetable synthesis (Chapter <A HREF="node26.html#chapter-wavetable">2</A>), and waveshaping
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(Chapters <A HREF="node75.html#chapter-modulation">5</A> and <A HREF="node89.html#chapter-paf">6</A>). This chapter
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introduces a fourth
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approach, in which waveforms are built up explicitly from line segments with
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controllable endpoints. This approach is historically at least as important
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as the others, and was dominant during the
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analog synthesizer period, approximately 1965-1985.
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For lack of a better name, we'll use the term
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<A NAME="14228"></A><I>classical waveforms</I>
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to denote waveforms composed of line segments.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig10.01"></A><A NAME="14232"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.1:</STRONG>
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Classical waveforms: (a) the sawtooth, (b) the triangle, and (c)
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the rectangle wave,
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shown as functions of a continuous variable (not sampled).</CAPTION>
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<TR><TD><IMG
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WIDTH="475" HEIGHT="351" BORDER="0"
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SRC="img1249.png"
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ALT="\begin{figure}\psfig{file=figs/fig10.01.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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They include the
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<A NAME="14235"></A><A NAME="14236"></A><A NAME="14237"></A><I>sawtooth</I>, <I>triangle</I>, and <I>rectangle</I> waves pictured in Figure
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<A HREF="#fig10.01">10.1</A>, among many other possibilities. The salient features of
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classical waveforms are either discontinuous jumps (changes in value) or
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corners (changes in slope). In the figure, the sawtooth and rectangle waves
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have jumps (once per cycle for the sawtooth, and twice for the rectangle), and
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constant slope elsewhere (negative for the sawtooth wave, zero for the
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rectangle wave). The triangle wave has no discontinuous jumps, but the slope
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changes discontinuously twice per cycle.
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<P>
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To use classical waveforms effectively, it is useful to understand how the
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shape of the waveform is reflected in its Fourier series. (To compute these we
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need background from Chapter <A HREF="node163.html#chapter-fft">9</A>, which is why this chapter
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appears here and not earlier.) We will also need strategies for digitally
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synthesizing classical waveforms. These waveforms prove to be much
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more susceptible to foldover problems than any we have treated before, so we
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will have to pay especially close attention to its control.
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<P>
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In general, our strategy for predicting and controlling foldover will be to
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consider first those sampled waveforms whose period is an integer <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$">. Then if
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we want to obtain a waveform of a non-integral period (call it <IMG
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WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img135.png"
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ALT="$\tau$">, say) we
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approximate <IMG
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WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img135.png"
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ALT="$\tau$"> as a quotient <IMG
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WIDTH="36" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1250.png"
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ALT="$N/R$"> of two integers. Conceptually at least,
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we can then synthesize the desired waveform with period <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$">, and then take only
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one of each <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img36.png"
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ALT="$R$"> samples of output. This last, down-sampling step is where the
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foldover is produced, and careful handling will help us control it.
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<P>
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<BR><HR>
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
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<LI><A NAME="tex2html3313"
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HREF="node185.html">Symmetries and Fourier series</A>
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<UL>
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<LI><A NAME="tex2html3314"
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HREF="node186.html">Sawtooth waves and symmetry</A>
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</UL>
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<BR>
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<LI><A NAME="tex2html3315"
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HREF="node187.html">Dissecting classical waveforms</A>
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<LI><A NAME="tex2html3316"
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HREF="node188.html">Fourier series of the elementary waveforms</A>
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<UL>
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<LI><A NAME="tex2html3317"
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HREF="node189.html">Sawtooth wave</A>
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<LI><A NAME="tex2html3318"
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HREF="node190.html">Parabolic wave</A>
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<LI><A NAME="tex2html3319"
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HREF="node191.html">Square and symmetric triangle waves</A>
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<LI><A NAME="tex2html3320"
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HREF="node192.html">General (non-symmetric) triangle wave</A>
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</UL>
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<BR>
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<LI><A NAME="tex2html3321"
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HREF="node193.html">Predicting and controlling foldover</A>
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<UL>
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<LI><A NAME="tex2html3322"
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HREF="node194.html">Over-sampling</A>
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<LI><A NAME="tex2html3323"
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HREF="node195.html">Sneaky triangle waves</A>
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<LI><A NAME="tex2html3324"
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HREF="node196.html">Transition splicing</A>
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</UL>
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<BR>
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<LI><A NAME="tex2html3325"
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HREF="node197.html">Examples</A>
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<UL>
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<LI><A NAME="tex2html3326"
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HREF="node198.html">Combining sawtooth waves</A>
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<LI><A NAME="tex2html3327"
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HREF="node199.html">Strategies for band-limiting sawtooth waves</A>
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</UL>
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<BR>
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<LI><A NAME="tex2html3328"
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HREF="node200.html">Exercises</A>
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</UL>
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<B> Next:</B> <A NAME="tex2html3312"
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HREF="node185.html">Symmetries and Fourier series</A>
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<B> Up:</B> <A NAME="tex2html3306"
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HREF="book.html">book</A>
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<B> Previous:</B> <A NAME="tex2html3300"
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HREF="node183.html">Exercises</A>
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<B> <A NAME="tex2html3308"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html3310"
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HREF="node201.html">Index</A></B>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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