2022-04-12 22:02:59 -03:00
|
|
|
<!DOCTYPE html>
|
2022-04-12 21:54:18 -03:00
|
|
|
|
|
|
|
|
<!--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>
|
2022-04-12 22:02:59 -03:00
|
|
|
|
|
|
|
|
<meta name="viewport" content="width=device-width, initial-scale=1.0">
|
|
|
|
|
|
|
|
|
|
|
2022-04-12 21:54:18 -03:00
|
|
|
<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-->
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html1002"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node32.html">
|
|
|
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="next.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html996"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node26.html">
|
|
|
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="up.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html990"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node30.html">
|
|
|
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="prev.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html998"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node4.html">
|
|
|
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="contents.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html1000"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node201.html">
|
|
|
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="index.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<BR>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Next:</B> <A ID="tex2html1003"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node32.html">Examples</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Up:</B> <A ID="tex2html997"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node26.html">Wavetables and samplers</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Previous:</B> <A ID="tex2html991"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node30.html">Timbre stretching</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> <A ID="tex2html999"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node4.html">Contents</A></B>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> <A ID="tex2html1001"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node201.html">Index</A></B>
|
|
|
|
|
<BR>
|
|
|
|
|
<BR>
|
|
|
|
|
<!--End of Navigation Panel-->
|
|
|
|
|
|
2022-04-12 23:32:40 -03:00
|
|
|
<H1><A ID="SECTION00650000000000000000"></A>
|
|
|
|
|
<A ID="sect2.interpolation"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<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
|
2022-04-12 23:17:03 -03:00
|
|
|
is the difference between the result of the wavetable lookup and the "ideal"
|
2022-04-12 21:54:18 -03:00
|
|
|
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="[*]"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="crossref.png"></A>). We can then understand what happens to other wavetables by
|
2022-04-12 21:54:18 -03:00
|
|
|
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>
|
|
|
|
|
|
2022-04-12 23:32:40 -03:00
|
|
|
<DIV ALIGN="CENTER"><A ID="fig02.11"></A><A ID="2374"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<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>
|
2022-04-12 23:32:40 -03:00
|
|
|
<DIV ALIGN="CENTER"><A ID="2392"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<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>
|
|
|
|
|
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tab02.1"></A></TD></TR>
|
2022-04-12 21:54:18 -03:00
|
|
|
</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-->
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html1002"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node32.html">
|
|
|
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="next.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html996"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node26.html">
|
|
|
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="up.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html990"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node30.html">
|
|
|
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="prev.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html998"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node4.html">
|
|
|
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="contents.png"></A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<A ID="tex2html1000"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node201.html">
|
|
|
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
2022-04-12 22:02:59 -03:00
|
|
|
SRC="index.png"></A>
|
2022-04-12 21:54:18 -03:00
|
|
|
<BR>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Next:</B> <A ID="tex2html1003"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node32.html">Examples</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Up:</B> <A ID="tex2html997"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node26.html">Wavetables and samplers</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> Previous:</B> <A ID="tex2html991"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node30.html">Timbre stretching</A>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> <A ID="tex2html999"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node4.html">Contents</A></B>
|
2022-04-12 23:32:40 -03:00
|
|
|
<B> <A ID="tex2html1001"
|
2022-04-12 21:54:18 -03:00
|
|
|
HREF="node201.html">Index</A></B>
|
|
|
|
|
<!--End of Navigation Panel-->
|
|
|
|
|
<ADDRESS>
|
|
|
|
|
Miller Puckette
|
|
|
|
|
2006-12-30
|
|
|
|
|
</ADDRESS>
|
|
|
|
|
</BODY>
|
|
|
|
|
</HTML>
|
