421 lines
14 KiB
HTML
421 lines
14 KiB
HTML
<!DOCTYPE html>
|
|
|
|
<!--Converted with LaTeX2HTML 2002-2-1 (1.71)
|
|
original version by: Nikos Drakos, CBLU, University of Leeds
|
|
* revised and updated by: Marcus Hennecke, Ross Moore, Herb Swan
|
|
* with significant contributions from:
|
|
Jens Lippmann, Marek Rouchal, Martin Wilck and others -->
|
|
<HTML>
|
|
<HEAD>
|
|
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0">
|
|
|
|
|
|
<TITLE>Wavetables and samplers</TITLE>
|
|
<META NAME="description" CONTENT="Wavetables and samplers">
|
|
<META NAME="keywords" CONTENT="book">
|
|
<META NAME="resource-type" CONTENT="document">
|
|
<META NAME="distribution" CONTENT="global">
|
|
|
|
<META NAME="Generator" CONTENT="LaTeX2HTML v2002-2-1">
|
|
<META HTTP-EQUIV="Content-Style-Type" CONTENT="text/css">
|
|
|
|
<LINK REL="STYLESHEET" HREF="book.css">
|
|
|
|
<LINK REL="next" HREF="node40.html">
|
|
<LINK REL="previous" HREF="node7.html">
|
|
<LINK REL="up" HREF="book.html">
|
|
<LINK REL="next" HREF="node27.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html919"
|
|
HREF="node27.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A NAME="tex2html913"
|
|
HREF="book.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A NAME="tex2html907"
|
|
HREF="node25.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A NAME="tex2html915"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A NAME="tex2html917"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html920"
|
|
HREF="node27.html">The Wavetable Oscillator</A>
|
|
<B> Up:</B> <A NAME="tex2html914"
|
|
HREF="book.html">book</A>
|
|
<B> Previous:</B> <A NAME="tex2html908"
|
|
HREF="node25.html">Exercises</A>
|
|
<B> <A NAME="tex2html916"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html918"
|
|
HREF="node201.html">Index</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H1><A NAME="SECTION00600000000000000000"></A>
|
|
<A NAME="chapter-wavetable"></A>
|
|
<BR>
|
|
Wavetables and samplers
|
|
</H1>
|
|
|
|
<P>
|
|
In Chapter 1 we treated audio signals as if they always flowed by in
|
|
a continuous stream at some sample rate. The sample rate isn't really a
|
|
quality of the audio signal, but rather it specifies how fast the individual
|
|
samples should flow into or out of the computer. But audio signals are at
|
|
bottom just sequences of numbers, and in practice there is no requirement that
|
|
they be ``played" sequentially. Another, complementary view is that
|
|
they can be stored in memory, and, later, they can be read back in any
|
|
order--forward, backward, back and forth, or totally at random. An
|
|
inexhaustible range of new possibilities opens up.
|
|
|
|
<P>
|
|
For many years (roughly 1950-1990), magnetic tape served as the main storage
|
|
medium for sounds. Tapes were passed back and forth across magnetic pickups to
|
|
play the signals back in real time. Since 1995 or so, the predominant way
|
|
of storing sounds has been to keep them as digital audio signals, which are read
|
|
back with much greater freedom and facility than were the magnetic tapes. Many
|
|
modes of use dating from the tape era are still current, including cutting,
|
|
duplication, speed change, and time reversal. Other techniques, such as
|
|
<I>waveshaping</I>, have come into their own only in the digital era.
|
|
|
|
<P>
|
|
Suppose we have a stored digital audio signal, which is just a sequence of
|
|
samples (i.e., numbers) <IMG
|
|
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img80.png"
|
|
ALT="$x[n]$"> for <!-- MATH
|
|
$n = 0, ..., N-1$
|
|
-->
|
|
<IMG
|
|
WIDTH="111" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img166.png"
|
|
ALT="$n = 0, ..., N-1$">, where <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> is the length
|
|
of the sequence. Then if we have an input signal <IMG
|
|
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img2.png"
|
|
ALT="$y[n]$"> (which we can imagine
|
|
to be flowing in real time), we can use its values as indices to look up values
|
|
of the stored signal <IMG
|
|
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img80.png"
|
|
ALT="$x[n]$">. This operation, called
|
|
<A NAME="2151"></A><I>wavetable lookup</I>,
|
|
gives us a new signal, <IMG
|
|
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img167.png"
|
|
ALT="$z[n]$">, calculated as:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
z[n] = x[y[n]]
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="91" HEIGHT="28" BORDER="0"
|
|
SRC="img168.png"
|
|
ALT="\begin{displaymath}
|
|
z[n] = x[y[n]]
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
Schematically we represent this operation as shown in Figure <A HREF="#fig02.01">2.1</A>.
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig02.01"></A><A NAME="2156"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2.1:</STRONG>
|
|
Diagram for wavetable lookup. The input is in samples,
|
|
ranging approximately from 0 to the wavetable's size <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$">, depending on the
|
|
interpolation scheme.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="110" HEIGHT="175" BORDER="0"
|
|
SRC="img169.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig02.01.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
Two complications arise. First, the input values, <IMG
|
|
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img2.png"
|
|
ALT="$y[n]$">, might lie outside
|
|
the range <IMG
|
|
WIDTH="80" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img170.png"
|
|
ALT="$0, ..., N-1$">, in which case the wavetable <IMG
|
|
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img80.png"
|
|
ALT="$x[n]$"> has no value and
|
|
the expression for the output <IMG
|
|
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img167.png"
|
|
ALT="$z[n]$"> is undefined. In this situation we might
|
|
choose to
|
|
<A NAME="2159"></A><I>clip</I> the input, that is, to substitute 0 for
|
|
anything negative and <IMG
|
|
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img171.png"
|
|
ALT="$N-1$"> for anything N or greater. Alternatively, we might
|
|
prefer to wrap the input around end to end. Here we'll adopt the convention that
|
|
out-of-range samples are always clipped; when we need wraparound, we'll
|
|
introduce another signal processing operation to do it for us.
|
|
|
|
<P>
|
|
The second complication is that the input values need not be integers; in other
|
|
words they might fall between the points of the wavetable. In general, this
|
|
is addressed by choosing some scheme for interpolating between the points of
|
|
the wavetable. For the moment, though, we'll just round
|
|
down to the nearest integer below the input. This is called
|
|
<A NAME="2161"></A><I>non-interpolating</I> wavetable lookup, and its full definition is:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
z[n] = \left \{ {
|
|
\begin{array}{ll}
|
|
x[ \lfloor y[n] \rfloor ] & \mbox{if $0 \le y[n] < N-1$} \\
|
|
x[0] & \mbox{if $y[n] < 0$} \\
|
|
x[N-1] & \mbox{if $y[n] \ge N-1$} \\
|
|
\end{array}
|
|
} \right .
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="279" HEIGHT="64" BORDER="0"
|
|
SRC="img172.png"
|
|
ALT="\begin{displaymath}
|
|
z[n] = \left \{ {
|
|
\begin{array}{ll}
|
|
x[ \lfloor y[n] \rflo...
|
|
...x[N-1] & \mbox{if $y[n] \ge N-1$} \\
|
|
\end{array} } \right .
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
(where <!-- MATH
|
|
$\lfloor y[n] \rfloor$
|
|
-->
|
|
<IMG
|
|
WIDTH="44" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img173.png"
|
|
ALT="$\lfloor y[n] \rfloor$"> means, ``the greatest integer not
|
|
exceeding <IMG
|
|
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img2.png"
|
|
ALT="$y[n]$">").
|
|
|
|
<P>
|
|
Pictorally, we use <IMG
|
|
WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img174.png"
|
|
ALT="$y[0]$"> (a number) as a
|
|
location on the horizontal axis of the wavetable shown in Figure <A HREF="#fig02.01">2.1</A>,
|
|
and the output, <IMG
|
|
WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img175.png"
|
|
ALT="$z[0]$">, is whatever we get on the vertical axis; and the
|
|
same for <IMG
|
|
WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img176.png"
|
|
ALT="$y[1]$"> and <IMG
|
|
WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img177.png"
|
|
ALT="$z[1]$"> and so on. The ``natural" range for the input <IMG
|
|
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img2.png"
|
|
ALT="$y[n]$">
|
|
is <!-- MATH
|
|
$0 \le y[n] < N$
|
|
-->
|
|
<IMG
|
|
WIDTH="95" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img178.png"
|
|
ALT="$0 \le y[n] < N$">. This is different from the usual range of an audio
|
|
signal suitable for output from the computer, which ranges from -1 to 1 in
|
|
our units. We'll see later that the usable range of input values,
|
|
from 0 to <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> for non-interpolating lookup, shrinks slightly if interpolating lookup is used.
|
|
|
|
<P>
|
|
Figure <A HREF="#fig02.02">2.2</A> (part a) shows a wavetable and the result of using two
|
|
different input signals as lookup indices into it. The wavetable
|
|
contains 40 points, which are numbered from 0 to 39. In part (b), a
|
|
<A NAME="2171"></A><I>sawtooth wave</I> is used as the input signal <IMG
|
|
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img2.png"
|
|
ALT="$y[n]$">. A sawtooth wave
|
|
is nothing but a ramp function repeated end to end. In this example the
|
|
sawtooth's range is from <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img179.png"
|
|
ALT="$0$"> to
|
|
<IMG
|
|
WIDTH="19" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img180.png"
|
|
ALT="$40$"> (this is shown in the vertical axis). The sawtooth wave thus scans
|
|
the wavetable from left to right--from the beginning point 0 to the endpoint
|
|
39--and does so every time it repeats. Over the fifty points shown
|
|
in Figure <A HREF="#fig02.02">2.2</A> (part b) the sawtooth wave makes
|
|
two and a half cycles. Its period is twenty samples, or in other
|
|
words the frequency (in cycles per second) is <IMG
|
|
WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img181.png"
|
|
ALT="$R/20$">.
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig02.02"></A><A NAME="2176"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2.2:</STRONG>
|
|
Wavetable lookup: (a) a wavetable; (b) and (d) signal inputs for
|
|
lookup; (c) and (e) the corresponding outputs.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="471" HEIGHT="664" BORDER="0"
|
|
SRC="img182.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig02.02.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
Part (c) of Figure <A HREF="#fig02.02">2.2</A> shows the result of applying wavetable
|
|
lookup, using the table <IMG
|
|
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img80.png"
|
|
ALT="$x[n]$">, to the signal <IMG
|
|
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img2.png"
|
|
ALT="$y[n]$">. Since the sawtooth input
|
|
simply reads out the contents of the wavetable from left to right, repeatedly,
|
|
at a constant rate of precession, the result will be a new periodic signal,
|
|
whose waveform (shape) is derived from <IMG
|
|
WIDTH="31" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img80.png"
|
|
ALT="$x[n]$"> and whose frequency is determined
|
|
by the sawtooth wave <IMG
|
|
WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img2.png"
|
|
ALT="$y[n]$">.
|
|
|
|
<P>
|
|
Parts (d) and (e) show an example where the wavetable is read in a nonuniform
|
|
way; since the input signal rises from <IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img179.png"
|
|
ALT="$0$"> to <IMG
|
|
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img3.png"
|
|
ALT="$N$"> and then later recedes to
|
|
<IMG
|
|
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img179.png"
|
|
ALT="$0$">, we see the wavetable appear first forward, then frozen at its endpoint,
|
|
then backward. The table is scanned from left to right and then, more quickly,
|
|
from right to left. As in the previous example the incoming signal controls
|
|
the speed of precession while the output's amplitudes are those of the wavetable.
|
|
|
|
<P>
|
|
<BR><HR>
|
|
<!--Table of Child-Links-->
|
|
<A NAME="CHILD_LINKS"><STRONG>Subsections</STRONG></A>
|
|
|
|
<UL>
|
|
<LI><A NAME="tex2html921"
|
|
HREF="node27.html">The Wavetable Oscillator</A>
|
|
<LI><A NAME="tex2html922"
|
|
HREF="node28.html">Sampling</A>
|
|
<LI><A NAME="tex2html923"
|
|
HREF="node29.html">Enveloping samplers</A>
|
|
<LI><A NAME="tex2html924"
|
|
HREF="node30.html">Timbre stretching</A>
|
|
<LI><A NAME="tex2html925"
|
|
HREF="node31.html">Interpolation</A>
|
|
<LI><A NAME="tex2html926"
|
|
HREF="node32.html">Examples</A>
|
|
<UL>
|
|
<LI><A NAME="tex2html927"
|
|
HREF="node33.html">Wavetable oscillator</A>
|
|
<LI><A NAME="tex2html928"
|
|
HREF="node34.html">Wavetable lookup in general</A>
|
|
<LI><A NAME="tex2html929"
|
|
HREF="node35.html">Using a wavetable as a sampler</A>
|
|
<LI><A NAME="tex2html930"
|
|
HREF="node36.html">Looping samplers</A>
|
|
<LI><A NAME="tex2html931"
|
|
HREF="node37.html">Overlapping sample looper</A>
|
|
<LI><A NAME="tex2html932"
|
|
HREF="node38.html">Automatic read point precession</A>
|
|
</UL>
|
|
<BR>
|
|
<LI><A NAME="tex2html933"
|
|
HREF="node39.html">Exercises</A>
|
|
</UL>
|
|
<!--End of Table of Child-Links-->
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html919"
|
|
HREF="node27.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A NAME="tex2html913"
|
|
HREF="book.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A NAME="tex2html907"
|
|
HREF="node25.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A NAME="tex2html915"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A NAME="tex2html917"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html920"
|
|
HREF="node27.html">The Wavetable Oscillator</A>
|
|
<B> Up:</B> <A NAME="tex2html914"
|
|
HREF="book.html">book</A>
|
|
<B> Previous:</B> <A NAME="tex2html908"
|
|
HREF="node25.html">Exercises</A>
|
|
<B> <A NAME="tex2html916"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html918"
|
|
HREF="node201.html">Index</A></B>
|
|
<!--End of Navigation Panel-->
|
|
<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|