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166 lines
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166 lines
5.6 KiB
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<H2><A NAME="SECTION001451000000000000000">
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Combining sawtooth waves</A>
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</H2>
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig10.12"></A><A NAME="14498"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.12:</STRONG>
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Combining sawtooth waves: (a) adding and subtracting sawtooth waves
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1/2 cycle out of phase, to extract even and odd harmonics; (b) combining three
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sawtooth waves with arbitrary amplitudes and phases.</CAPTION>
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<TR><TD><IMG
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WIDTH="524" HEIGHT="396" BORDER="0"
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SRC="img1375.png"
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ALT="\begin{figure}\psfig{file=figs/fig10.12.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Example J01.even.odd.pd (Figure <A HREF="#fig10.12">10.12</A>, part a) shows how to combine sawtooth
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waves in pairs to extract the even and odd harmonics. The resulting waveforms
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are as shown in Figure <A HREF="node187.html#fig10.03">10.3</A>. Example J02.trapezoids.pd (part b of the
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figure) demonstrates combining three sawtooth waves at arbitrary phases and
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amplitudes; the resulting classic waveform has up to three jumps and no
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corners. The three line segments are horizontal as long as the three jumps add
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to zero; otherwise the segments are sloped to make up for the the unbalanced
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jumps so that the result repeats from one period to the next.
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<P>
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Example J03.pulse.width.mod.pd (not shown) combines two sawtooth waves, of opposite sign, with
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slightly different frequencies so that the relative phase changes
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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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<P>
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<DIV ALIGN="CENTER"><A NAME="fig10.13"></A><A NAME="14505"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.13:</STRONG>
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Combining parabolic waves to make a waveform with three corners.</CAPTION>
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<TR><TD><IMG
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WIDTH="390" HEIGHT="464" BORDER="0"
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SRC="img1376.png"
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ALT="\begin{figure}\psfig{file=figs/fig10.13.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Example J04.corners.pd (Figure <A HREF="#fig10.13">10.13</A>) shows how to add parabolic waves to make
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a combined waveform with three corners. Each parabolic wave is computed from a
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sawtooth wave (ranging from -0.5 to 0.5) by squaring it, multiplying by 0.5,
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and subtracting the DC component of -1/12, or -0.08333. The patch combines
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three such parabolic waves with controllable amplitudes and phases. As
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long as the amplitudes sum to zero, the resulting waveform consists of line
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segments, whose corners are located according to the three phases and have
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slope changes according to the three amplitudes.
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Miller Puckette
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2006-12-30
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