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original version by: Nikos Drakos, CBLU, University of Leeds
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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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<TITLE>Over-sampling</TITLE>
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HREF="node193.html">Predicting and controlling foldover</A>
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HREF="node193.html">Predicting and controlling foldover</A>
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<H2><A NAME="SECTION001441000000000000000">
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Over-sampling</A>
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</H2>
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<P>
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As a first line of defense against foldover, we can synthesize the waveform at
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a much higher sample rate, apply a low-pass filter whose cutoff frequency is
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set to the Nyquist frequency (for the original sample rate), then
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down-sample. For example, in the above scenario (44100 sample rate, 440 Hertz
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tone) we could generate the sawtooth at a sample rate of <!-- MATH
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$16\cdot 44100 =
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705600$
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-->
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<IMG
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WIDTH="139" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img1366.png"
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ALT="$16\cdot 44100 =
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705600$"> Hertz. We need only worry about frequencies in excess of
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<!-- MATH
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$705600-20000=685600$
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-->
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<IMG
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WIDTH="179" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img1367.png"
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ALT="$705600-20000=685600$"> Hertz (so that they fold over into audible frequencies;
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foldover to ultrasonic frequencies normally won't concern us) so the first
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problematic partial is <!-- MATH
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$685600/440=1558$
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-->
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<IMG
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WIDTH="136" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img1368.png"
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ALT="$685600/440=1558$">, whose amplitude is -64dB relative
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to that of the fundamental.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig10.08"></A><A NAME="14592"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.8:</STRONG>
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Magnitude spectrum of a triangle wave with <IMG
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WIDTH="90" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img71.png"
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ALT="$M/N=0.03$">. The two
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line segments show <IMG
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WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img72.png"
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ALT="$1/k$"> and <IMG
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WIDTH="35" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
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SRC="img73.png"
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ALT="$1/{k^2}$"> behavior at low and high frequencies.</CAPTION>
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<TR><TD><IMG
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WIDTH="480" HEIGHT="344" BORDER="0"
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SRC="img1369.png"
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ALT="\begin{figure}\psfig{file=figs/fig10.08.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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This attenuation degrades by 6 dB for every octave the fundamental
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is raised, so that a 10 kHz. sawtooth only enjoys a 37 dB drop from the
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fundamental to the loudest foldover partial. On the other hand, raising the
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sample rate by an additional factor of two reduces foldover by the same
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amount. If we really wish to get 60 decibels of foldover rejection--all the
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way up to a 10 kHz. fundamental--we will have to over-sample by a factor of 256,
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to a sample rate of about 11 million Hertz.
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