<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 3.2 Final//EN">
<!--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>
<TITLE>Interpolation</TITLE>
<META NAME="description" CONTENT="Interpolation">
<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="node32.html">
<LINK REL="previous" HREF="node30.html">
<LINK REL="up" HREF="node26.html">
<LINK REL="next" HREF="node32.html">
</HEAD>
<BODY >
<!--Navigation Panel-->
<A NAME="tex2html1002"
HREF="node32.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="file:/usr/local/share/lib/latex2html/icons/next.png"></A>
<A NAME="tex2html996"
HREF="node26.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="file:/usr/local/share/lib/latex2html/icons/up.png"></A>
<A NAME="tex2html990"
HREF="node30.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="file:/usr/local/share/lib/latex2html/icons/prev.png"></A>
<A NAME="tex2html998"
HREF="node4.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="file:/usr/local/share/lib/latex2html/icons/contents.png"></A>
<A NAME="tex2html1000"
HREF="node201.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
SRC="file:/usr/local/share/lib/latex2html/icons/index.png"></A>
<BR>
<B> Next:</B> <A NAME="tex2html1003"
HREF="node32.html">Examples</A>
<B> Up:</B> <A NAME="tex2html997"
HREF="node26.html">Wavetables and samplers</A>
<B> Previous:</B> <A NAME="tex2html991"
HREF="node30.html">Timbre stretching</A>
<B> <A NAME="tex2html999"
HREF="node4.html">Contents</A></B>
<B> <A NAME="tex2html1001"
HREF="node201.html">Index</A></B>
<BR>
<BR>
<!--End of Navigation Panel-->
<H1><A NAME="SECTION00650000000000000000"></A>
<A NAME="sect2.interpolation"></A>
<BR>
Interpolation
</H1>
<P>
As mentioned before, interpolation schemes are often used to increase the
accuracy of table lookup. Here we will give a somewhat simplified account of
the effects of table sizes and interpolation schemes on the result of table
lookup.
<P>
To speak of error in table lookup, we must view the wavetable as a sampled
version of an underlying function. When we ask for a value of the
underlying function which lies between the points of the wavetable, the error
is the difference between the result of the wavetable lookup and the ``ideal"
value of the function at that point. The most revealing study of wavetable
lookup error assumes that the underlying function is a sinusoid (Page
<A HREF="node7.html#eq-realsinusoid"><IMG ALIGN="BOTTOM" BORDER="1" ALT="[*]"
SRC="file:/usr/local/share/lib/latex2html/icons/crossref.png"></A>). We can then understand what happens to other wavetables by
considering them as superpositions (sums) of sinusoids.
<P>
The accuracy of lookup from a wavetable containing a sinusoid depends
on two factors: the quality of the interpolation scheme, and the period
of the sinusoid. In general, the longer the period of the sinusoid, the more
accurate the result.
<P>
In the case of a synthetic wavetable, we might know its sinusoidal components
from having specified them--in which case the issue becomes one of choosing a
wavetable size appropriately, when calculating the wavetable, to match the
interpolation algorithm and meet the desired standard of accuracy. In the case
of recorded sounds, the accuracy analysis might lead us to adjust the sample
rate of the recording, either at the outset or else by resampling later.
<P>
Interpolation error for a sinusoidal wavetable can have two components: first,
the continuous signal (the theoretical result of reading the wavetable
continuously in time, as if the output sample rate were infinite) might
not be a pure sinusoid; and second, the amplitude might be wrong. (It
is possible to get phase errors as well, but only through
carelessness.)
<P>
In this treatment we'll only consider polynomial interpolation schemes such as
rounding, linear interpolation, and cubic interpolation. These schemes amount
to evaluating polynomials (of degree zero, one, and three, respectively) in the
interstices between points of the wavetable. The idea is that, for any index
<IMG
WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img243.png"
ALT="$x$">, we choose a nearby reference point <IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img244.png"
ALT="${x_0}$">, and let the output be calculated
by some polynomial:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{y_{\mathrm{INT}}}(x) =
{a_0} + {a_1} (x - {x_0}) + {a_2} {{({x - x_0})}^ 2 } + \cdots +
{a_n} {{({x - x_0})}^ n }
\end{displaymath}
-->
<IMG
WIDTH="421" HEIGHT="28" BORDER="0"
SRC="img245.png"
ALT="\begin{displaymath}
{y_{\mathrm{INT}}}(x) =
{a_0} + {a_1} (x - {x_0}) + {a_2} {{({x - x_0})}^ 2 } + \cdots +
{a_n} {{({x - x_0})}^ n }
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Usually we choose the polynomial which passes through the <IMG
WIDTH="40" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img246.png"
ALT="$n+1$"> nearest
points of the wavetable. For 1-point interpolation (a zero-degree polynomial)
this means letting <IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img96.png"
ALT="$a_0$"> equal the nearest point of the wavetable. For
two-point interpolation, we draw a line segment between the two points of the
wavetable on either side of the desired point <IMG
WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img243.png"
ALT="$x$">. We can let <IMG
WIDTH="19" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img247.png"
ALT="$x_0$"> be the
closest integer to the left of <IMG
WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img243.png"
ALT="$x$"> (which we write as <!-- MATH
$\lfloor x \rfloor$
-->
<IMG
WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img248.png"
ALT="$\lfloor x \rfloor$">) and then the
formula for linear interpolation is:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{y_{\mathrm{INT}}}(x) =
y[{x_0}] + (y[{x_0} + 1]- y[{x_0}]) \cdot (x - {x_0})
\end{displaymath}
-->
<IMG
WIDTH="321" HEIGHT="28" BORDER="0"
SRC="img249.png"
ALT="\begin{displaymath}
{y_{\mathrm{INT}}}(x) =
y[{x_0}] + (y[{x_0} + 1]- y[{x_0}]) \cdot (x - {x_0})
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
which is a polynomial, as in the previous formula, with
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{a_0} = y[{x_0}]
\end{displaymath}
-->
<IMG
WIDTH="67" HEIGHT="28" BORDER="0"
SRC="img250.png"
ALT="\begin{displaymath}
{a_0} = y[{x_0}]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{a_1} = y[{x_0} + 1]- y[{x_0}]
\end{displaymath}
-->
<IMG
WIDTH="148" HEIGHT="28" BORDER="0"
SRC="img251.png"
ALT="\begin{displaymath}
{a_1} = y[{x_0} + 1]- y[{x_0}]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
In general, you can fit exactly one polynomial of degree <IMG
WIDTH="40" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img252.png"
ALT="$n-1$"> through any
<IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img75.png"
ALT="$n$"> points as long as their <IMG
WIDTH="12" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img243.png"
ALT="$x$"> values are all different.
<P>
Figure <A HREF="#fig02.11">2.11</A> shows the effect of using linear (two-point) interpolation
to fill in a sinusoid of period 6. At the top are three traces: the original
sinusoid, the linearly-interpolated result of using 6 points per period to
represent the sinusoid, and finally, another sinusoid, of slightly smaller
amplitude, which better matches the six-segment waveform. The error introduced
by replacing the original sinusoid by the linearly interpolated version has
two components: first, a (barely perceptible) change in amplitude, and second,
a (very perceptible) distortion of the wave shape.
<P>
<DIV ALIGN="CENTER"><A NAME="fig02.11"></A><A NAME="2374"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 2.11:</STRONG>
Linear interpolation of a sinusoid: (upper graph) the original sinusoid, the
interpolated sinusoid, and the best sinusoidal fit back to the interpolated
version; (lower graph) the error, rescaled vertically.</CAPTION>
<TR><TD><IMG
WIDTH="488" HEIGHT="514" BORDER="0"
SRC="img253.png"
ALT="\begin{figure}\psfig{file=figs/fig02.11.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
The bottom graph in the figure shows the difference between the interpolated
waveform and the best-fitting sinusoid. This is a residual signal all of
whose energy lies in overtones of the original sinusoid.
As the number of points increases,
the error decreases in magnitude. Since the error is the difference between
a sinusoid and a sequence of approximating line segments, the magnitude of
the error is
roughly proportional to the square of the phase difference between each
pair of points, or in other words, inversely proportional to the square
of the number of points in the wavetable. Put another way, wavetable
error decreases by 12 dB each time the table doubles in size. (This
rule of thumb is only good for tables with 4 or more points.)
<P>
Four-point (cubic) interpolation works similarly. The interpolation formula
is:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
{y_{\mathrm{INT}}}(x) =
\end{displaymath}
-->
<IMG
WIDTH="69" HEIGHT="28" BORDER="0"
SRC="img254.png"
ALT="\begin{displaymath}
{y_{\mathrm{INT}}}(x) =
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
-f (f-1)(f-2)/6 \cdot y[{x_0}-1]
+ (f+1)(f-1)(f-2)/2 \cdot y[{x_0}]
\end{displaymath}
-->
<IMG
WIDTH="435" HEIGHT="28" BORDER="0"
SRC="img255.png"
ALT="\begin{displaymath}
-f (f-1)(f-2)/6 \cdot y[{x_0}-1]
+ (f+1)(f-1)(f-2)/2 \cdot y[{x_0}]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
- (f+1) f (f-2) / 2 \cdot y[{x_0}+1]
+ (f+1) f (f-1) / 6 \cdot y[{x_0}+2]
\end{displaymath}
-->
<IMG
WIDTH="422" HEIGHT="28" BORDER="0"
SRC="img256.png"
ALT="\begin{displaymath}
- (f+1) f (f-2) / 2 \cdot y[{x_0}+1]
+ (f+1) f (f-1) / 6 \cdot y[{x_0}+2]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
where <IMG
WIDTH="79" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img257.png"
ALT="$f = x - {x_0}$"> is the fractional part of the index. For tables
with 4 or more points, doubling the number of points on the table tends
to improve the RMS error by 24 dB. Table 2.1 shows the
calculated RMS error for sinusoids at various periods for 1, 2, and
4 point interpolation. (A slightly different quantity is measured in
[<A
HREF="node202.html#r-moore90">Moo90</A>, p.164]. There, the errors in amplitude and phase are also
added in, yielding slightly more pessimistic results. See also
[<A
HREF="node202.html#r-hartmann87">Har87</A>].)
<P>
<BR><P></P>
<DIV ALIGN="CENTER"><A NAME="2392"></A>
<TABLE>
<CAPTION><STRONG>Table 2.1:</STRONG>
RMS error for table lookup using 1, 2, and 4 point interpolation
at various table sizes.</CAPTION>
<TR><TD><TABLE CELLPADDING=3 BORDER="1">
<TR><TD ALIGN="RIGHT">period</TD>
<TD ALIGN="CENTER" COLSPAN=3>interpolation points</TD>
</TR>
<TR><TD ALIGN="RIGHT"> </TD>
<TD ALIGN="RIGHT">1</TD>
<TD ALIGN="RIGHT">2</TD>
<TD ALIGN="RIGHT">4</TD>
</TR>
<TR><TD ALIGN="RIGHT">2</TD>
<TD ALIGN="RIGHT">-1.2</TD>
<TD ALIGN="RIGHT">-17.1</TD>
<TD ALIGN="RIGHT">-20.2</TD>
</TR>
<TR><TD ALIGN="RIGHT">3</TD>
<TD ALIGN="RIGHT">-2.0</TD>
<TD ALIGN="RIGHT">-11.9</TD>
<TD ALIGN="RIGHT">-15.5</TD>
</TR>
<TR><TD ALIGN="RIGHT">4</TD>
<TD ALIGN="RIGHT">-4.2</TD>
<TD ALIGN="RIGHT">-17.1</TD>
<TD ALIGN="RIGHT">-24.8</TD>
</TR>
<TR><TD ALIGN="RIGHT">8</TD>
<TD ALIGN="RIGHT">-10.0</TD>
<TD ALIGN="RIGHT">-29.6</TD>
<TD ALIGN="RIGHT">-48.4</TD>
</TR>
<TR><TD ALIGN="RIGHT">16</TD>
<TD ALIGN="RIGHT">-15.9</TD>
<TD ALIGN="RIGHT">-41.8</TD>
<TD ALIGN="RIGHT">-72.5</TD>
</TR>
<TR><TD ALIGN="RIGHT">32</TD>
<TD ALIGN="RIGHT">-21.9</TD>
<TD ALIGN="RIGHT">-53.8</TD>
<TD ALIGN="RIGHT">-96.5</TD>
</TR>
<TR><TD ALIGN="RIGHT">64</TD>
<TD ALIGN="RIGHT">-27.9</TD>
<TD ALIGN="RIGHT">-65.9</TD>
<TD ALIGN="RIGHT">-120.6</TD>
</TR>
<TR><TD ALIGN="RIGHT">128</TD>
<TD ALIGN="RIGHT">-34.0</TD>
<TD ALIGN="RIGHT">-77.9</TD>
<TD ALIGN="RIGHT">-144.7</TD>
</TR>
</TABLE>
<A NAME="tab02.1"></A></TD></TR>
</TABLE>
</DIV><P></P>
<BR>
<P>
The allowable input domain for table lookup depends on the number of points of
interpolation. In general, when using <IMG
WIDTH="12" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img58.png"
ALT="$k$">-point interpolation into a table
with <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$N$"> points, the input may range over an interval of <IMG
WIDTH="73" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img258.png"
ALT="$N + 1 - k$"> points.
If <IMG
WIDTH="41" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img259.png"
ALT="$k=1$"> (i.e., no interpolation at all), the domain is 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="$N$">
(including the endpoint at <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img179.png"
ALT="$0$"> but excluding the one at <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$N$">) assuming input
values are truncated (as is done for non-interpolated table lookup in Pd). The
domain is from -<IMG
WIDTH="27" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img98.png"
ALT="$1/2$"> to <IMG
WIDTH="61" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img260.png"
ALT="$N-1/2$"> if, instead, we round the input to the nearest
integer instead of interpolating. In either case, the domain stretches over a
length of <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$N$"> points.
<P>
For two-point interpolation, the input must lie between the first and last
points, that is, between <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img179.png"
ALT="$0$"> and <IMG
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img171.png"
ALT="$N-1$">. So the <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$N$"> points suffice to define
the function over a domain of length <IMG
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img171.png"
ALT="$N-1$">. For four-point interpolation,
we cannot get values for inputs between 0 and 1 (not having the required
two points to the left of the input) and neither can we for the space between
the last two points (<IMG
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img261.png"
ALT="$N-2$"> and <IMG
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img171.png"
ALT="$N-1$">). So in this case the domain reaches from
<IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img262.png"
ALT="$1$"> to <IMG
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img261.png"
ALT="$N-2$"> and has length <IMG
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img263.png"
ALT="$N-3$">.
<P>
Periodic waveforms stored in wavetables require special treatment at the
ends of the table. For example, suppose we wish to store a pure sinusoid of
length <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$N$">. For non-interpolating table lookup, it suffices to set, for
example,
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
x[n] = \cos(2 \pi n / N) ,\, n = 0, \ldots, N-1
\end{displaymath}
-->
<IMG
WIDTH="255" HEIGHT="28" BORDER="0"
SRC="img264.png"
ALT="\begin{displaymath}
x[n] = \cos(2 \pi n / N) ,\, n = 0, \ldots, N-1
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
For two-point interpolation, we need <IMG
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img265.png"
ALT="$N+1$"> points:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
x[n] = \cos(2 \pi n / N) ,\, n = 0, \ldots, N
\end{displaymath}
-->
<IMG
WIDTH="228" HEIGHT="28" BORDER="0"
SRC="img266.png"
ALT="\begin{displaymath}
x[n] = \cos(2 \pi n / N) ,\, n = 0, \ldots, N
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
In other words, we must repeat the first (<IMG
WIDTH="42" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img79.png"
ALT="$n=0$">) point at the end, so that the
last segment from <IMG
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img171.png"
ALT="$N-1$"> to <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$N$"> reaches back to the beginning value.
<P>
For four-point interpolation, the cycle must be adjusted to start at the point
<IMG
WIDTH="42" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img267.png"
ALT="$n=1$">, since we can't get properly interpolated values out for inputs less than
one. If, then, one cycle of the wavetable is arranged from <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img262.png"
ALT="$1$"> to <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$N$">, we
must supply extra points for <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img179.png"
ALT="$0$"> (copied from <IMG
WIDTH="18" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img3.png"
ALT="$N$">), and also <IMG
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img265.png"
ALT="$N+1$"> and <IMG
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img268.png"
ALT="$N+2$">,
copied from <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img262.png"
ALT="$1$"> and <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img269.png"
ALT="$2$">, to make a table of length <IMG
WIDTH="45" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img270.png"
ALT="$N+3$">. For the same sinusoid
as above, the table should contain:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
x[n] = \cos(2 \pi (n-1) / N) ,\, n = 0, \ldots, N+2
\end{displaymath}
-->
<IMG
WIDTH="294" HEIGHT="28" BORDER="0"
SRC="img271.png"
ALT="\begin{displaymath}
x[n] = \cos(2 \pi (n-1) / N) ,\, n = 0, \ldots, N+2
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<P>
<HR>
<!--Navigation Panel-->
<A NAME="tex2html1002"
HREF="node32.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
SRC="file:/usr/local/share/lib/latex2html/icons/next.png"></A>
<A NAME="tex2html996"
HREF="node26.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
SRC="file:/usr/local/share/lib/latex2html/icons/up.png"></A>
<A NAME="tex2html990"
HREF="node30.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
SRC="file:/usr/local/share/lib/latex2html/icons/prev.png"></A>
<A NAME="tex2html998"
HREF="node4.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
SRC="file:/usr/local/share/lib/latex2html/icons/contents.png"></A>
<A NAME="tex2html1000"
HREF="node201.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
SRC="file:/usr/local/share/lib/latex2html/icons/index.png"></A>
<BR>
<B> Next:</B> <A NAME="tex2html1003"
HREF="node32.html">Examples</A>
<B> Up:</B> <A NAME="tex2html997"
HREF="node26.html">Wavetables and samplers</A>
<B> Previous:</B> <A NAME="tex2html991"
HREF="node30.html">Timbre stretching</A>
<B> <A NAME="tex2html999"
HREF="node4.html">Contents</A></B>
<B> <A NAME="tex2html1001"
HREF="node201.html">Index</A></B>
<!--End of Navigation Panel-->
<ADDRESS>
Miller Puckette
2006-12-30
</ADDRESS>
</BODY>
</HTML>