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<H2><A ID="SECTION001451000000000000000">
Combining sawtooth waves</A>
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<DIV ALIGN="CENTER"><A ID="fig10.12"></A><A ID="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>
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<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
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is known as pulse width modulation ("PWM").
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<DIV ALIGN="CENTER"><A ID="fig10.13"></A><A ID="14505"></A>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.13:</STRONG>
Combining parabolic waves to make a waveform with three corners.</CAPTION>
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WIDTH="390" HEIGHT="464" BORDER="0"
SRC="img1376.png"
ALT="\begin{figure}\psfig{file=figs/fig10.13.ps}\end{figure}"></TD></TR>
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<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.
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<ADDRESS>
Miller Puckette
2006-12-30
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