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<B> Next:</B> <A NAME="tex2html3355"
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<H2><A NAME="SECTION001411000000000000000">
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Sawtooth waves and symmetry</A>
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</H2>
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig10.02"></A><A NAME="14265"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.2:</STRONG>
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Using a symmetry relation to extract even and odd harmonics from a
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sawtooth wave: (a) the
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original sawtooth wave; (b) shifted by 1/2 cycle; (c) their sum (another
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sawtooth wave at
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twice the frequency); (d) their difference (a square wave).</CAPTION>
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<TR><TD><IMG
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WIDTH="477" HEIGHT="467" BORDER="0"
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SRC="img1277.png"
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ALT="\begin{figure}\psfig{file=figs/fig10.02.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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As an example, we apply the shift symmetry (even and odd harmonics) to a
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sawtooth wave. Figure <A HREF="#fig10.02">10.2</A> (part a) shows the original sawtooth wave
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and part (b) shows the result of shifting by a half cycle. The sum of the two
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(part c) drops discontinuously whenever either one of the two copies does so,
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and traces a line segment whenever both component sawtooth waves do; so it in
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turn becomes a sawtooth wave, of half the original period (twice the
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fundamental frequency). Subtracting the two sawtooth waves (part d) gives a
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waveform with slope zero except at the discontinuities. The discontinuities
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coming from the original sawtooth wave jump in the same direction (negative to
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positive), but those coming from the shifted one are negated and jump from
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positive to negative. The result is a
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<A NAME="14269"></A><I>square wave</I>,
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a particular rectangle wave in which the two component segments
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have the same duration.
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<P>
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This symmetry was used to great effect in the design of Buchla analog
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synthesizers; instead of offering a single sawtooth generator,
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Buchla designed an oscillator that outputs the even and odd harmonic portions
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separately, so that cross-fading between the two allows a continuous control
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over the relative strengths of the even and odd harmonics in the analog
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waveform.
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<ADDRESS>
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Miller Puckette
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2006-12-30
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