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* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
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HREF="node201.html">Index</A></B>
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<!--End of Navigation Panel-->
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<H1><A NAME="SECTION00600000000000000000"></A>
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<A NAME="chapter-wavetable"></A>
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<BR>
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Wavetables and samplers
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</H1>
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<P>
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In Chapter 1 we treated audio signals as if they always flowed by in
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a continuous stream at some sample rate. The sample rate isn't really a
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quality of the audio signal, but rather it specifies how fast the individual
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samples should flow into or out of the computer. But audio signals are at
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bottom just sequences of numbers, and in practice there is no requirement that
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they be ``played" sequentially. Another, complementary view is that
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they can be stored in memory, and, later, they can be read back in any
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order--forward, backward, back and forth, or totally at random. An
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inexhaustible range of new possibilities opens up.
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<P>
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For many years (roughly 1950-1990), magnetic tape served as the main storage
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medium for sounds. Tapes were passed back and forth across magnetic pickups to
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play the signals back in real time. Since 1995 or so, the predominant way
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of storing sounds has been to keep them as digital audio signals, which are read
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back with much greater freedom and facility than were the magnetic tapes. Many
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modes of use dating from the tape era are still current, including cutting,
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duplication, speed change, and time reversal. Other techniques, such as
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<I>waveshaping</I>, have come into their own only in the digital era.
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<P>
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Suppose we have a stored digital audio signal, which is just a sequence of
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samples (i.e., numbers) <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="$x[n]$"> for <!-- MATH
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$n = 0, ..., N-1$
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-->
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<IMG
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WIDTH="111" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img166.png"
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ALT="$n = 0, ..., N-1$">, where <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="$N$"> is the length
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of the sequence. Then if we have an input signal <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="$y[n]$"> (which we can imagine
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to be flowing in real time), we can use its values as indices to look up values
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of the stored signal <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="$x[n]$">. This operation, called
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<A NAME="2151"></A><I>wavetable lookup</I>,
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gives us a new signal, <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img167.png"
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ALT="$z[n]$">, calculated as:
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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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z[n] = x[y[n]]
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\end{displaymath}
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-->
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<IMG
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WIDTH="91" HEIGHT="28" BORDER="0"
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SRC="img168.png"
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ALT="\begin{displaymath}
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z[n] = x[y[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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Schematically we represent this operation as shown in Figure <A HREF="#fig02.01">2.1</A>.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig02.01"></A><A NAME="2156"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2.1:</STRONG>
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Diagram for wavetable lookup. The input is in samples,
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ranging approximately from 0 to the wavetable's size <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="$N$">, depending on the
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interpolation scheme.</CAPTION>
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<TR><TD><IMG
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WIDTH="110" HEIGHT="175" BORDER="0"
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SRC="img169.png"
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ALT="\begin{figure}\psfig{file=figs/fig02.01.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Two complications arise. First, the input values, <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="$y[n]$">, might lie outside
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the range <IMG
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WIDTH="80" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img170.png"
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ALT="$0, ..., N-1$">, in which case the 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="$x[n]$"> has no value and
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the expression for the output <IMG
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WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img167.png"
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ALT="$z[n]$"> is undefined. In this situation we might
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choose to
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<A NAME="2159"></A><I>clip</I> the input, that is, to substitute 0 for
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anything negative and <IMG
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WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
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SRC="img171.png"
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ALT="$N-1$"> for anything N or greater. Alternatively, we might
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prefer to wrap the input around end to end. Here we'll adopt the convention that
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out-of-range samples are always clipped; when we need wraparound, we'll
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introduce another signal processing operation to do it for us.
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<P>
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The second complication is that the input values need not be integers; in other
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words they might fall between the points of the wavetable. In general, this
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is addressed by choosing some scheme for interpolating between the points of
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the wavetable. For the moment, though, we'll just round
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down to the nearest integer below the input. This is called
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<A NAME="2161"></A><I>non-interpolating</I> wavetable lookup, and its full definition is:
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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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z[n] = \left \{ {
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\begin{array}{ll}
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x[ \lfloor y[n] \rfloor ] & \mbox{if $0 \le y[n] < N-1$} \\
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x[0] & \mbox{if $y[n] < 0$} \\
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x[N-1] & \mbox{if $y[n] \ge N-1$} \\
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\end{array}
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} \right .
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\end{displaymath}
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-->
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<IMG
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WIDTH="279" HEIGHT="64" BORDER="0"
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SRC="img172.png"
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ALT="\begin{displaymath}
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z[n] = \left \{ {
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\begin{array}{ll}
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x[ \lfloor y[n] \rflo...
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...x[N-1] & \mbox{if $y[n] \ge N-1$} \\
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\end{array} } \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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(where <!-- MATH
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$\lfloor y[n] \rfloor$
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-->
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<IMG
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WIDTH="44" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img173.png"
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ALT="$\lfloor y[n] \rfloor$"> means, ``the greatest integer not
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exceeding <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="$y[n]$">").
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<P>
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Pictorally, we use <IMG
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WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img174.png"
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ALT="$y[0]$"> (a number) as a
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location on the horizontal axis of the wavetable shown in Figure <A HREF="#fig02.01">2.1</A>,
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and the output, <IMG
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WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img175.png"
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ALT="$z[0]$">, is whatever we get on the vertical axis; and the
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same for <IMG
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WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img176.png"
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ALT="$y[1]$"> and <IMG
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WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img177.png"
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ALT="$z[1]$"> and so on. The ``natural" range for the input <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="$y[n]$">
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is <!-- MATH
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$0 \le y[n] < N$
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-->
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<IMG
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WIDTH="95" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img178.png"
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ALT="$0 \le y[n] < N$">. This is different from the usual range of an audio
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signal suitable for output from the computer, which ranges from -1 to 1 in
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our units. We'll see later that the usable range of input values,
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from 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="$N$"> for non-interpolating lookup, shrinks slightly if interpolating lookup is used.
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<P>
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Figure <A HREF="#fig02.02">2.2</A> (part a) shows a wavetable and the result of using two
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different input signals as lookup indices into it. The wavetable
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contains 40 points, which are numbered from 0 to 39. In part (b), a
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<A NAME="2171"></A><I>sawtooth wave</I> is used as the input signal <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="$y[n]$">. A sawtooth wave
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is nothing but a ramp function repeated end to end. In this example the
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sawtooth's range is 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
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<IMG
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WIDTH="19" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img180.png"
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ALT="$40$"> (this is shown in the vertical axis). The sawtooth wave thus scans
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the wavetable from left to right--from the beginning point 0 to the endpoint
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39--and does so every time it repeats. Over the fifty points shown
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in Figure <A HREF="#fig02.02">2.2</A> (part b) the sawtooth wave makes
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two and a half cycles. Its period is twenty samples, or in other
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words the frequency (in cycles per second) is <IMG
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WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img181.png"
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ALT="$R/20$">.
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig02.02"></A><A NAME="2176"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2.2:</STRONG>
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Wavetable lookup: (a) a wavetable; (b) and (d) signal inputs for
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lookup; (c) and (e) the corresponding outputs.</CAPTION>
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<TR><TD><IMG
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WIDTH="471" HEIGHT="664" BORDER="0"
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SRC="img182.png"
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ALT="\begin{figure}\psfig{file=figs/fig02.02.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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Part (c) of Figure <A HREF="#fig02.02">2.2</A> shows the result of applying wavetable
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lookup, using the table <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="$x[n]$">, to the signal <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="$y[n]$">. Since the sawtooth input
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simply reads out the contents of the wavetable from left to right, repeatedly,
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at a constant rate of precession, the result will be a new periodic signal,
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whose waveform (shape) is derived from <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="$x[n]$"> and whose frequency is determined
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by the sawtooth wave <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="$y[n]$">.
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<P>
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Parts (d) and (e) show an example where the wavetable is read in a nonuniform
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way; since the input signal rises 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="$N$"> and then later recedes to
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<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="$0$">, we see the wavetable appear first forward, then frozen at its endpoint,
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then backward. The table is scanned from left to right and then, more quickly,
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from right to left. As in the previous example the incoming signal controls
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the speed of precession while the output's amplitudes are those of the wavetable.
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<P>
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<BR><HR>
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<!--Table of Child-Links-->
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<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
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<UL>
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<LI><A NAME="tex2html921"
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HREF="node27.html">The Wavetable Oscillator</A>
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<LI><A NAME="tex2html922"
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HREF="node28.html">Sampling</A>
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<LI><A NAME="tex2html923"
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HREF="node29.html">Enveloping samplers</A>
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<LI><A NAME="tex2html924"
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HREF="node30.html">Timbre stretching</A>
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<LI><A NAME="tex2html925"
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HREF="node31.html">Interpolation</A>
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<LI><A NAME="tex2html926"
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HREF="node32.html">Examples</A>
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<UL>
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<LI><A NAME="tex2html927"
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HREF="node33.html">Wavetable oscillator</A>
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<LI><A NAME="tex2html928"
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HREF="node34.html">Wavetable lookup in general</A>
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<LI><A NAME="tex2html929"
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HREF="node35.html">Using a wavetable as a sampler</A>
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<LI><A NAME="tex2html930"
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HREF="node36.html">Looping samplers</A>
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<LI><A NAME="tex2html931"
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HREF="node37.html">Overlapping sample looper</A>
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<LI><A NAME="tex2html932"
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HREF="node38.html">Automatic read point precession</A>
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</UL>
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<BR>
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<LI><A NAME="tex2html933"
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HREF="node39.html">Exercises</A>
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</UL>
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<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
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<BR>
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<B> <A NAME="tex2html916"
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Miller Puckette
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2006-12-30
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