155 lines
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155 lines
5.9 KiB
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HREF="node193.html">Predicting and controlling foldover</A>
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<B> Previous:</B> <A NAME="tex2html3474"
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HREF="node194.html">Over-sampling</A>
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<H2><A NAME="SECTION001442000000000000000">
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Sneaky triangle waves</A>
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</H2>
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<P>
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For low fundamental frequencies, over-sampling is an easy way to get adequate
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foldover protection. If we wish to allow higher frequencies, we will need a
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more sophisticated approach. One possibility is to replace discontinuities by
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ramps, or in other words, to replace component sawtooth waves by triangle
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waves, as treated in Section <A HREF="node192.html#sect10.gentriangle">10.3.4</A>,
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with values of <IMG
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WIDTH="41" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1356.png"
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ALT="$M/N$"> small enough
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that the result sounds like a sawtooth wave, but large enough to control
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foldover.
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<P>
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Returning to Figure <A HREF="node194.html#fig10.08">10.8</A>, suppose for example we imitate a sawtooth
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wave with a triangle wave with <IMG
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WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img86.png"
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ALT="$M$"> equal to two samples, so that the first
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notch falls on the Nyquist frequency. Partials above the first notch (the 17th
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partial in the figure) will fold over; the worst of them is about 40 dB below
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the fundamental. On the other hand, the partial strengths start dropping faster
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than those of a true sawtooth wave at about half the Nyquist frequency. This
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is acceptable in some, but not all, situations.
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<P>
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The triangle wave strategy can be combined with over-sampling to improve the
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situation further. Again in the context of Figure <A HREF="node194.html#fig10.08">10.8</A>, suppose we
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over-sample by a factor of 4, and set the first notch at the original sample
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rate. The partials up to the Nyquist frequency (partial 8, at the fundamental
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frequency shown in the figure) follow those of the true sawtooth wave fairly
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well. Foldover sets in only at partial number 48, and is 52 dB below the
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fundamental. This overall behavior holds for any fundamental frequency up to
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about one quarter the sample rate (after which <IMG
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WIDTH="20" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img86.png"
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ALT="$M$"> exceeds <IMG
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WIDTH="32" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1265.png"
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ALT="$N/2$">). Setting the
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notch frequency to the original sample rate is equivalent to setting the
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segment of length <IMG
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ALT="$2M$"> to one sample (at the original sample rate).
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<BR>
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<B> Next:</B> <A NAME="tex2html3486"
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HREF="node196.html">Transition splicing</A>
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<B> Up:</B> <A NAME="tex2html3480"
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HREF="node193.html">Predicting and controlling foldover</A>
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<B> Previous:</B> <A NAME="tex2html3474"
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HREF="node194.html">Over-sampling</A>
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<B> <A NAME="tex2html3482"
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HREF="node4.html">Contents</A></B>
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<B> <A NAME="tex2html3484"
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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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