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587 lines
18 KiB
587 lines
18 KiB
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<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
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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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HREF="node29.html">Enveloping samplers</A>
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HREF="node26.html">Wavetables and samplers</A>
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<H1><A ID="SECTION00620000000000000000"></A>
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<A ID="sect2.sampling"></A>
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<BR>
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Sampling
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</H1>
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<P>
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"Sampling"
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<A ID="2203"></A>
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is nothing more than recording a live signal into a wavetable, and then later
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playing it out again. (In commercial samplers the entire wavetable is
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usually called a "sample" but to avoid confusion we'll only use the word
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"sample" here to mean a single number in an audio signal.)
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<P>
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At its simplest, a sampler is simply a wavetable oscillator, as was shown in
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Figure <A HREF="node27.html#fig02.03">2.3</A>. However, in the earlier discussion we imagined playing
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the oscillator back at a frequency high enough to be perceived as a pitch, at
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least 30 Hertz or so. In the case of sampling, the frequency is usually lower
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than 30 Hertz, and so the period, at least 1/30 second and perhaps much more, is
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long enough that you can hear the individual cycles as separate events.
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<P>
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Going back to Figure <A HREF="node26.html#fig02.02">2.2</A>, suppose that instead of 40 points the
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wavetable <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$"> is a one-second recording, at an original
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sample rate of 44100, so that it has 44100 points; and let
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<IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$"> in part (b) of the figure have a period of 22050 samples. This
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corresponds to a frequency of 2 Hertz. But what we hear is not a pitched sound
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at 2 cycles per second (that's too slow to hear as a pitch) but rather, we
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hear the original recording <IMG
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WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img80.png"
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ALT="$x[n]$"> played back repeatedly at double speed. We've
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just reinvented the sampler.
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<P>
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In general, if we assume the sample rate <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img36.png"
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ALT="$R$"> of the recording is the same as
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the output sample rate, if the wavetable has <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$"> samples, and if we index
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it with a sawtooth wave of period <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="$M$">,
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the sample is sped up or slowed down by
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a factor of <IMG
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WIDTH="40" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img191.png"
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ALT="$N/M$">, equal to <IMG
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WIDTH="47" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img192.png"
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ALT="$N f / R$"> if <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="$f$"> is the frequency in Hertz of the
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sawtooth. If we denote the transposition factor by <IMG
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WIDTH="9" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img82.png"
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ALT="$t$"> (so that, for instance,
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<IMG
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WIDTH="54" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img193.png"
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ALT="$t=3/2$"> means transposing upward a perfect fifth), and if we denote the
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transposition in half-steps by <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img194.png"
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ALT="$h$">, then we get the
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<A ID="2206"></A>Transposition Formulas for Looping Wavetables:
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<A ID="sect2.transpositionformula"></A>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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t = {N / M} = {{N f} / R}
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\end{displaymath}
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-->
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<IMG
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WIDTH="130" HEIGHT="28" BORDER="0"
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SRC="img195.png"
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ALT="\begin{displaymath}
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t = {N / M} = {{N f} / R}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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h = 12 \, {\log _ 2} \left ( {N \over M} \right ) =
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12 \, {\log _ 2} \left ( {N f \over R} \right )
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\end{displaymath}
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-->
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<IMG
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WIDTH="241" HEIGHT="45" BORDER="0"
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SRC="img196.png"
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ALT="\begin{displaymath}
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h = 12 \, {\log _ 2} \left ( {N \over M} \right ) =
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12 \, {\log _ 2} \left ( {N f \over R} \right )
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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Frequently the desired transposition in half-steps (<IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img194.png"
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ALT="$h$">) is known and the
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formula must be solved for either <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="$f$"> or <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$">:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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f = {{2^{h/12} R} \over N}
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\end{displaymath}
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-->
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<IMG
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WIDTH="79" HEIGHT="42" BORDER="0"
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SRC="img197.png"
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ALT="\begin{displaymath}
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f = {{2^{h/12} R} \over N}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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N = {{2^{h/12} R} \over f}
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\end{displaymath}
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-->
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<IMG
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WIDTH="86" HEIGHT="45" BORDER="0"
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SRC="img198.png"
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ALT="\begin{displaymath}
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N = {{2^{h/12} R} \over f}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<P>
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So far we have used a sawtooth as the input wave <IMG
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WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img199.png"
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ALT="$y[t]$">, but, as suggested in
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parts (d) and (e) of Figure <A HREF="node26.html#fig02.02">2.2</A>, we could use anything we like as an
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input signal. In general, the transposition may be time dependent and
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is controlled by the rate of change of the input signal.
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<P>
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The
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transposition multiple <IMG
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WIDTH="9" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img82.png"
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ALT="$t$"> and the transposition in half-steps <IMG
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WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img194.png"
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ALT="$h$"> are then
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given by the
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<A ID="2217"></A>Momentary Transposition Formulas for Wavetables:
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<A ID="sect2.momentaryformula"></A>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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t[n] = \left | y[n] - y[n-1] \right |
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\end{displaymath}
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-->
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<IMG
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WIDTH="153" HEIGHT="28" BORDER="0"
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SRC="img200.png"
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ALT="\begin{displaymath}
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t[n] = \left \vert y[n] - y[n-1] \right \vert
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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h[n] = 12 {{\log_2} \left | y[n] - y[n-1] \right |}
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\end{displaymath}
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-->
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<IMG
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WIDTH="202" HEIGHT="28" BORDER="0"
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SRC="img201.png"
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ALT="\begin{displaymath}
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h[n] = 12 {{\log_2} \left \vert y[n] - y[n-1] \right \vert}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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(Here the enclosing bars "<IMG
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WIDTH="7" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img202.png"
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ALT="$\vert$">" mean absolute value.)
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For example, if
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<IMG
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WIDTH="61" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img203.png"
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ALT="$y[n] = n$">, then <IMG
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WIDTH="78" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img204.png"
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ALT="$z[n] = x[n]$"> so we hear the wavetable at its original pitch,
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and this is what the formula predicts since, in that case,
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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y[n]-y[n-1] = 1
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\end{displaymath}
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-->
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<IMG
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WIDTH="129" HEIGHT="28" BORDER="0"
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SRC="img205.png"
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ALT="\begin{displaymath}
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y[n]-y[n-1] = 1
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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On the other hand, if <IMG
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WIDTH="69" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img206.png"
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ALT="$y[n] = 2n$">, then the wavetable is transposed up an
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octave, consistent with
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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y[n]-y[n-1] = 2
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\end{displaymath}
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-->
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<IMG
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WIDTH="129" HEIGHT="28" BORDER="0"
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SRC="img207.png"
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ALT="\begin{displaymath}
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y[n]-y[n-1] = 2
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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If values of <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img2.png"
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ALT="$y[n]$"> are
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decreasing with <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$">, you hear the sample backward, but the transposition
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formula still gives a positive multiplier. This all agrees with the
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earlier Transposition Formula for Looping Wavetables; if a sawtooth ranges
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from <IMG
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WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img179.png"
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ALT="$0$"> to <IMG
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WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img3.png"
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ALT="$N$">, <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="$f$"> times per second,
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the difference of successive samples is just <IMG
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WIDTH="47" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img192.png"
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ALT="$N f / R$">--except at the sample
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at the beginning of each new cycle.
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<P>
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It's well known that transposing a recording also transposes its timbre--this
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is the "chipmunk" effect. Not only are any periodicities (such as might
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give rise to pitch) transposed, but so are the frequencies of
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the overtones. Some timbres, notably those of vocal sounds, have characteristic
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frequency ranges in which overtones are stronger than other nearby ones.
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Such frequency ranges are also transposed, and this is is heard as
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a timbre change. In language that will be made more precise in Section
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<A HREF="node76.html#sect5-spectra">5.1</A>,
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we say that the
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<A ID="2221"></A><I>spectral envelope</I> is transposed along with the pitch or
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pitches.
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<P>
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In both this and the preceding section, we have considered playing
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wavetables periodically. In Section <A HREF="node27.html#sect2.oscillator">2.1</A> the playback
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repeated quickly enough that the repetition gives rise to a pitch, say between
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30 and 4000 times per second, roughly the range of a piano. In the current
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section we assumed a wavetable one second long, and in this case
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"reasonable" transposition factors (less than four octaves up) would give rise
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to a rate of repetition below 30, usually much lower, and going down as low as
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we wish.
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<P>
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The number 30 is significant for another reason: it is roughly the maximum
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number of separate events the ear can discern per second; for instance, 30
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vocal phonemes, or melodic notes, or attacks of a snare drum are about the most
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we can hope to crowd into a second before our ability to distinguish them
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breaks down.
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<P>
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A continuum exists between samplers and wavetable oscillators, in that the
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patch of Figure <A HREF="node27.html#fig02.03">2.3</A> can either be regarded as a sampler (if the
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frequency of repetition is less than about 20 Hertz) or as a wavetable oscillator
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(if the frequency is greater than about 40 Hertz). It is possible to move
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continuously between the two regimes.
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Furthermore, it is not necessary to play an entire wavetable in a loop; with a bit
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more arithmetic we can choose sub-segments of the wavetable, and these can change
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in length and location continuously as the wavetable is played.
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<P>
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The practice of playing many small segments of a wavetable in rapid succession
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is often called
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<A ID="2225"></A><I>granular synthesis</I>.
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For much more discussion of the possibilities, see [<A
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HREF="node202.html#r-roads01">Roa01</A>].
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<P>
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Figure <A HREF="#fig02.05">2.5</A> shows how to build a very simple looping sampler. In the
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figure, if the frequency is <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="$f$"> and the segment size in samples is <IMG
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WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img208.png"
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ALT="$s$">,
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the output transposition factor is given by <IMG
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WIDTH="67" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img209.png"
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ALT="$t = fs/R$">, where R is the sample
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rate at which the wavetable was recorded (which need not equal the sample rate
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the block diagram is working at.) In practice, this equation must usually
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be solved for either <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="$f$"> or <IMG
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WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img208.png"
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ALT="$s$"> to attain a desired transposition.
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<P>
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In the figure, a sawtooth oscillator controls the location of wavetable lookup,
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but the lower and upper values of the sawtooth aren't statically
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specified as they were in Figure <A HREF="node27.html#fig02.03">2.3</A>; rather, the sawtooth
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oscillator simply ranges from 0 to 1 in value and the range is adjusted
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to select a desired segment of samples in the wavetable.
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<P>
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It might be desirable to specify the segment's location <IMG
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WIDTH="8" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img210.png"
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ALT="$l$"> either as its
|
|
left-hand edge (its lower bound) or else as the segment's midpoint; in either
|
|
case we specify the length <IMG
|
|
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img208.png"
|
|
ALT="$s$"> as a separate parameter. In the first
|
|
case, we start by
|
|
multiplying the sawtooth by <IMG
|
|
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img208.png"
|
|
ALT="$s$">, so that it then ranges from <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img179.png"
|
|
ALT="$0$"> to <IMG
|
|
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img208.png"
|
|
ALT="$s$">; then
|
|
we add <IMG
|
|
WIDTH="8" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img210.png"
|
|
ALT="$l$"> so that it now ranges from <IMG
|
|
WIDTH="8" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img210.png"
|
|
ALT="$l$"> to <IMG
|
|
WIDTH="35" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img211.png"
|
|
ALT="$l+s$">. In order to specify the
|
|
location as the segment's midpoint, we first subtract <IMG
|
|
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img98.png"
|
|
ALT="$1/2$"> from the sawtooth
|
|
(so that it ranges from <IMG
|
|
WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img212.png"
|
|
ALT="$-1/2$"> to <IMG
|
|
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img98.png"
|
|
ALT="$1/2$">), and then as before multiply by <IMG
|
|
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img208.png"
|
|
ALT="$s$">
|
|
(so that it now ranges from <IMG
|
|
WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img213.png"
|
|
ALT="$-s/2$"> to <IMG
|
|
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img214.png"
|
|
ALT="$s/2$">) and add <IMG
|
|
WIDTH="8" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img210.png"
|
|
ALT="$l$"> to give a range from
|
|
<IMG
|
|
WIDTH="51" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img215.png"
|
|
ALT="$l-s/2$"> to <IMG
|
|
WIDTH="51" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img216.png"
|
|
ALT="$l+s/2$">.
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A ID="fig02.05"></A><A ID="2232"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2.5:</STRONG>
|
|
A simple looping sampler, as yet with no amplitude control.
|
|
There are inputs to control the frequency and the segment size and location.
|
|
The "-" operation is included if we wish the segment location to be specified
|
|
as the segment's midpoint; otherwise we specify the location of the left
|
|
end of the segment.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="320" HEIGHT="398" BORDER="0"
|
|
SRC="img217.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig02.05.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
In the looping sampler,
|
|
we will need to worry about maintaining continuity between the beginning and the
|
|
end of segments of the wavetable; we'll take this up in the next section.
|
|
|
|
<P>
|
|
A further detail is that, if the segment size and location are changing
|
|
with time (they might be digital audio signals themselves, for instance),
|
|
they will affect the transposition factor, and the pitch or timbre of the
|
|
output signal might waver up and down as a result. The simplest way to
|
|
avoid this problem is to synchronize changes in the values of <IMG
|
|
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img208.png"
|
|
ALT="$s$"> and <IMG
|
|
WIDTH="8" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img210.png"
|
|
ALT="$l$">
|
|
with the regular discontinuities of the sawtooth; since the signal jumps
|
|
discontinuously there, the transposition is not really defined there anyway,
|
|
and, if you are enveloping to hide the discontinuity, the effects of changes
|
|
in <IMG
|
|
WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img208.png"
|
|
ALT="$s$"> and <IMG
|
|
WIDTH="8" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img210.png"
|
|
ALT="$l$"> are hidden as well.
|
|
|
|
<P>
|
|
<HR>
|
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<B> Next:</B> <A ID="tex2html961"
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HREF="node29.html">Enveloping samplers</A>
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<B> Up:</B> <A ID="tex2html955"
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HREF="node26.html">Wavetables and samplers</A>
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<B> Previous:</B> <A ID="tex2html949"
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HREF="node27.html">The Wavetable Oscillator</A>
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<B> <A ID="tex2html957"
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Miller Puckette
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2006-12-30
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