176 lines
5.5 KiB
HTML
176 lines
5.5 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>Over-sampling</TITLE>
|
|
<META NAME="description" CONTENT="Over-sampling">
|
|
<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="node195.html">
|
|
<LINK REL="previous" HREF="node193.html">
|
|
<LINK REL="up" HREF="node193.html">
|
|
<LINK REL="next" HREF="node195.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html3471"
|
|
HREF="node195.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A NAME="tex2html3465"
|
|
HREF="node193.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A NAME="tex2html3459"
|
|
HREF="node193.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A NAME="tex2html3467"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A NAME="tex2html3469"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html3472"
|
|
HREF="node195.html">Sneaky triangle waves</A>
|
|
<B> Up:</B> <A NAME="tex2html3466"
|
|
HREF="node193.html">Predicting and controlling foldover</A>
|
|
<B> Previous:</B> <A NAME="tex2html3460"
|
|
HREF="node193.html">Predicting and controlling foldover</A>
|
|
<B> <A NAME="tex2html3468"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html3470"
|
|
HREF="node201.html">Index</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H2><A NAME="SECTION001441000000000000000">
|
|
Over-sampling</A>
|
|
</H2>
|
|
|
|
<P>
|
|
As a first line of defense against foldover, we can synthesize the waveform at
|
|
a much higher sample rate, apply a low-pass filter whose cutoff frequency is
|
|
set to the Nyquist frequency (for the original sample rate), then
|
|
down-sample. For example, in the above scenario (44100 sample rate, 440 Hertz
|
|
tone) we could generate the sawtooth at a sample rate of <!-- MATH
|
|
$16\cdot 44100 =
|
|
705600$
|
|
-->
|
|
<IMG
|
|
WIDTH="139" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
|
SRC="img1366.png"
|
|
ALT="$16\cdot 44100 =
|
|
705600$"> Hertz. We need only worry about frequencies in excess of
|
|
<!-- MATH
|
|
$705600-20000=685600$
|
|
-->
|
|
<IMG
|
|
WIDTH="179" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1367.png"
|
|
ALT="$705600-20000=685600$"> Hertz (so that they fold over into audible frequencies;
|
|
foldover to ultrasonic frequencies normally won't concern us) so the first
|
|
problematic partial is <!-- MATH
|
|
$685600/440=1558$
|
|
-->
|
|
<IMG
|
|
WIDTH="136" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img1368.png"
|
|
ALT="$685600/440=1558$">, whose amplitude is -64dB relative
|
|
to that of the fundamental.
|
|
|
|
<P>
|
|
|
|
<DIV ALIGN="CENTER"><A NAME="fig10.08"></A><A NAME="14592"></A>
|
|
<TABLE>
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 10.8:</STRONG>
|
|
Magnitude spectrum of a triangle wave with <IMG
|
|
WIDTH="90" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img71.png"
|
|
ALT="$M/N=0.03$">. The two
|
|
line segments show <IMG
|
|
WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img72.png"
|
|
ALT="$1/k$"> and <IMG
|
|
WIDTH="35" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img73.png"
|
|
ALT="$1/{k^2}$"> behavior at low and high frequencies.</CAPTION>
|
|
<TR><TD><IMG
|
|
WIDTH="480" HEIGHT="344" BORDER="0"
|
|
SRC="img1369.png"
|
|
ALT="\begin{figure}\psfig{file=figs/fig10.08.ps}\end{figure}"></TD></TR>
|
|
</TABLE>
|
|
</DIV>
|
|
|
|
<P>
|
|
This attenuation degrades by 6 dB for every octave the fundamental
|
|
is raised, so that a 10 kHz. sawtooth only enjoys a 37 dB drop from the
|
|
fundamental to the loudest foldover partial. On the other hand, raising the
|
|
sample rate by an additional factor of two reduces foldover by the same
|
|
amount. If we really wish to get 60 decibels of foldover rejection--all the
|
|
way up to a 10 kHz. fundamental--we will have to over-sample by a factor of 256,
|
|
to a sample rate of about 11 million Hertz.
|
|
|
|
<P>
|
|
<HR>
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html3471"
|
|
HREF="node195.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A NAME="tex2html3465"
|
|
HREF="node193.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A NAME="tex2html3459"
|
|
HREF="node193.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A NAME="tex2html3467"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A NAME="tex2html3469"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html3472"
|
|
HREF="node195.html">Sneaky triangle waves</A>
|
|
<B> Up:</B> <A NAME="tex2html3466"
|
|
HREF="node193.html">Predicting and controlling foldover</A>
|
|
<B> Previous:</B> <A NAME="tex2html3460"
|
|
HREF="node193.html">Predicting and controlling foldover</A>
|
|
<B> <A NAME="tex2html3468"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html3470"
|
|
HREF="node201.html">Index</A></B>
|
|
<!--End of Navigation Panel-->
|
|
<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|