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8.6 KiB
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8.6 KiB
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<H2><A ID="SECTION001443000000000000000">
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Transition splicing</A>
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</H2>
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<P>
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In the point of view developed in this chapter, the energy of the spectral
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components of classical waves can be attributed entirely to their jumps and
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corners. This is artificial, of course: the energy really emanates from the
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entire waveform. Our derivation of the spectrum of the classical waveforms
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uses the jumps and corners as a bookkeeping device, and this is possible
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because the entire waveform is determined by their positions and magnitudes.
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<P>
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Taking this ruse even further, the problem of making band-limited versions of
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classical waveforms can be attacked by making band-limited versions of the
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jumps and corners. Since the jumps are the more serious foldover threat, we
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will focus on them here, although the approach described here works perfectly
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well for corners as well.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig10.09"></A><A ID="14478"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.9:</STRONG>
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A square wave, band-limited to partials 1, 3, 5, 7, 9, and 11. This
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can be regarded approximately as a series of band-limited step functions arranged
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end to end.</CAPTION>
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<TR><TD><IMG
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WIDTH="474" HEIGHT="109" BORDER="0"
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SRC="img1370.png"
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ALT="\begin{figure}\psfig{file=figs/fig10.09.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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To construct a band-limited step function, all we have to do is add the Fourier
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components of a square wave, as many as we like, and then harvest the step
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function at any one of the jumps. Figure <A HREF="#fig10.09">10.9</A> shows the partial
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Fourier sum corresponding to a square wave, using partials 1, 3, 5, 7, 9, and
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11. The cutoff frequency can be taken as <IMG
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WIDTH="29" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img1371.png"
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ALT="$12\omega$"> (if <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $"> is the
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fundamental frequency).
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<P>
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If we double the period of the square wave, to arrive at the same cutoff
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frequency, we would add twice as many Fourier partials, up to number 23, for
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instance. Extending this process forever, we would eventually see the ideal
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band-limited step function, twice per (arbitrarily long) period.
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<P>
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In practice we can do quite well using only the first two partials (one and
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three times the fundamental). Figure <A HREF="#fig10.10">10.10</A> (part a) shows
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a two-partial approximation of a square wave. The cutoff frequency is
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four times the fundamental; so if the period of the waveform is eight
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samples, the cutoff is at the Nyquist frequency. Part (b) of the
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figure shows how we could use this step function to synthesize, approximately,
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a square wave of twice the period. If the cutoff frequency is the Nyquist
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frequency, the period of the waveform of part (b) is 16 samples. Each
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transition lasts 4 samples, because the band-limited square wave has a period
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of eight samples.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig10.10"></A><A ID="14485"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.10:</STRONG>
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Stretching a band-limited square wave: (a) the original waveform;
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(b) after splicing in horizontal segments; (c) using the same step transition
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for a sawtooth wave.</CAPTION>
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<TR><TD><IMG
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WIDTH="474" HEIGHT="376" BORDER="0"
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SRC="img1372.png"
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ALT="\begin{figure}\psfig{file=figs/fig10.10.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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We can make a band-limited sawtooth wave by adding the four-sample-long
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transition to a ramp function so that the end of the resulting function meets
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smoothly with itself end to end, as shown in part (c) of the
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figure.
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There is one transition per period, so the period must be at least four
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samples; the highest fundamental frequency we can synthesize this way is half
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the Nyquist frequency. For this or lower fundamental frequency, the foldover
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products all turn out to be at least 60 dB quieter than the fundamental.
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<P>
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<DIV ALIGN="CENTER"><A ID="fig10.11"></A><A ID="14490"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.11:</STRONG>
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Block diagram for making a sawtooth wave with a spliced transition.</CAPTION>
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<TR><TD><IMG
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WIDTH="199" HEIGHT="389" BORDER="0"
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SRC="img1373.png"
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ALT="\begin{figure}\psfig{file=figs/fig10.11.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Figure <A HREF="#fig10.11">10.11</A> shows how to generate a sawtooth wave with a spliced
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transition. The two parameters are <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img112.png"
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ALT="$f$">, the fundamental frequency, and <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$">,
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the band limit, assumed to be at least as large as <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img112.png"
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ALT="$f$">. We start with a
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digital sawtooth wave (a phasor) ranging from -0.5 to 0.5 in value. The
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transition will take place at the middle of the cycle, when the phasor crosses
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0. The wavetable is traversed in a constant amount of time, <IMG
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WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1374.png"
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ALT="$1/b$">, regardless
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of <IMG
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WIDTH="13" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img112.png"
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ALT="$f$">. The table lookup is taken to be non-wraparound, so that inputs out of
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range output either -0.5 or 0.5.
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<P>
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At the end of the cycle the phasor discontinuously jumps from -0.5 to 0.5, but
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the output of the transition table jumps an equal and opposite amount, so the
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result is continuous. During the portion of the waveform in which the
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transition table is read at one or the other end-point, the output describes
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a straight line segment.
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<P>
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HREF="node193.html">Predicting and controlling foldover</A>
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<B> Previous:</B> <A ID="tex2html3488"
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HREF="node195.html">Sneaky triangle waves</A>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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