248 lines
7.8 KiB
HTML
248 lines
7.8 KiB
HTML
|
<!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>Shelving filter</TITLE>
|
||
|
|
<META NAME="description" CONTENT="Shelving filter">
|
||
|
|
<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="node143.html">
|
||
|
|
<LINK REL="previous" HREF="node141.html">
|
||
|
|
<LINK REL="up" HREF="node139.html">
|
||
|
|
<LINK REL="next" HREF="node143.html">
|
||
|
|
</HEAD>
|
||
|
|
|
||
|
|
<BODY >
|
||
|
|
<!--Navigation Panel-->
|
||
|
|
<A NAME="tex2html2702"
|
||
|
|
HREF="node143.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="tex2html2696"
|
||
|
|
HREF="node139.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="tex2html2690"
|
||
|
|
HREF="node141.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="tex2html2698"
|
||
|
|
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="tex2html2700"
|
||
|
|
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="tex2html2703"
|
||
|
|
HREF="node143.html">Band-pass filter</A>
|
||
|
|
<B> Up:</B> <A NAME="tex2html2697"
|
||
|
|
HREF="node139.html">Designing filters</A>
|
||
|
|
<B> Previous:</B> <A NAME="tex2html2691"
|
||
|
|
HREF="node141.html">One-pole, one-zero high-pass filter</A>
|
||
|
|
<B> <A NAME="tex2html2699"
|
||
|
|
HREF="node4.html">Contents</A></B>
|
||
|
|
<B> <A NAME="tex2html2701"
|
||
|
|
HREF="node201.html">Index</A></B>
|
||
|
|
<BR>
|
||
|
|
<BR>
|
||
|
|
<!--End of Navigation Panel-->
|
||
|
|
|
||
|
|
<H2><A NAME="SECTION001233000000000000000"></A>
|
||
|
|
<A NAME="sect8.shelving"></A>
|
||
|
|
<BR>
|
||
|
|
Shelving filter
|
||
|
|
</H2>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
Generalizing the one-zero, one-pole filter above, suppose we place the
|
||
|
|
zero at a point <IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img592.png"
|
||
|
|
ALT="$q$">, a real number close to, but less than, one. The
|
||
|
|
pole, at the point <IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img57.png"
|
||
|
|
ALT="$p$">, is similarly situated, and might be either
|
||
|
|
greater than or less than <IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img592.png"
|
||
|
|
ALT="$q$">, i.e., to the right or left, respectively,
|
||
|
|
but with both <IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img592.png"
|
||
|
|
ALT="$q$"> and <IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img57.png"
|
||
|
|
ALT="$p$"> within the unit circle. This situation is
|
||
|
|
diagrammed in Figure <A HREF="#fig08.14">8.14</A>.
|
||
|
|
|
||
|
|
<P>
|
||
|
|
|
||
|
|
<DIV ALIGN="CENTER"><A NAME="fig08.14"></A><A NAME="10346"></A>
|
||
|
|
<TABLE>
|
||
|
|
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.14:</STRONG>
|
||
|
|
One-pole, one-zero shelving filter: (a) pole-zero diagram; (b)
|
||
|
|
frequency response.</CAPTION>
|
||
|
|
<TR><TD><IMG
|
||
|
|
WIDTH="540" HEIGHT="380" BORDER="0"
|
||
|
|
SRC="img914.png"
|
||
|
|
ALT="\begin{figure}\psfig{file=figs/fig08.14.ps}\end{figure}"></TD></TR>
|
||
|
|
</TABLE>
|
||
|
|
</DIV>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
At points of the circle far from <IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img57.png"
|
||
|
|
ALT="$p$"> and <IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img592.png"
|
||
|
|
ALT="$q$">, the effects of the pole and the
|
||
|
|
zero are nearly inverse (the distances to them are nearly equal), so the filter
|
||
|
|
passes those frequencies nearly unaltered. In the neighborhood of <IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img57.png"
|
||
|
|
ALT="$p$"> and
|
||
|
|
<IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img592.png"
|
||
|
|
ALT="$q$">, on the other hand, the filter will have a gain greater or less than one
|
||
|
|
depending on which of <IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img57.png"
|
||
|
|
ALT="$p$"> or <IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img592.png"
|
||
|
|
ALT="$q$"> is closer to the circle. This configuration
|
||
|
|
therefore acts as a low-frequency shelving filter. (To make a high-frequency
|
||
|
|
shelving filter we do the same thing, only placing <IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img57.png"
|
||
|
|
ALT="$p$"> and <IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img592.png"
|
||
|
|
ALT="$q$"> close to -1
|
||
|
|
instead of 1.)
|
||
|
|
|
||
|
|
<P>
|
||
|
|
To find the parameters of a shelving filter given a desired
|
||
|
|
transition frequency <IMG
|
||
|
|
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img27.png"
|
||
|
|
ALT="$\omega $"> (in angular units) and low-frequency
|
||
|
|
gain <IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img29.png"
|
||
|
|
ALT="$g$">, first we choose an average distance <IMG
|
||
|
|
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img28.png"
|
||
|
|
ALT="$d$">, as pictured in the figure,
|
||
|
|
from the pole and the zero to the edge of the circle. For small values of <IMG
|
||
|
|
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img28.png"
|
||
|
|
ALT="$d$">,
|
||
|
|
the region of influence is about <IMG
|
||
|
|
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img28.png"
|
||
|
|
ALT="$d$"> radians, so
|
||
|
|
simply set <IMG
|
||
|
|
WIDTH="43" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img915.png"
|
||
|
|
ALT="$d = \omega$"> to get the desired transition frequency.
|
||
|
|
|
||
|
|
<P>
|
||
|
|
Then put the pole at <!-- MATH
|
||
|
|
$p = 1 - d / \sqrt{g}$
|
||
|
|
-->
|
||
|
|
<IMG
|
||
|
|
WIDTH="97" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img916.png"
|
||
|
|
ALT="$p = 1 - d / \sqrt{g}$"> and the zero at
|
||
|
|
<!-- MATH
|
||
|
|
$q = 1 - d \sqrt{g}$
|
||
|
|
-->
|
||
|
|
<IMG
|
||
|
|
WIDTH="89" HEIGHT="31" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img917.png"
|
||
|
|
ALT="$q = 1 - d \sqrt{g}$">. The gain at zero frequency is then
|
||
|
|
<BR><P></P>
|
||
|
|
<DIV ALIGN="CENTER">
|
||
|
|
<!-- MATH
|
||
|
|
\begin{displaymath}
|
||
|
|
{{1-q} \over {1-p}} = g
|
||
|
|
\end{displaymath}
|
||
|
|
-->
|
||
|
|
|
||
|
|
<IMG
|
||
|
|
WIDTH="67" HEIGHT="41" BORDER="0"
|
||
|
|
SRC="img918.png"
|
||
|
|
ALT="\begin{displaymath}
|
||
|
|
{{1-q} \over {1-p}} = g
|
||
|
|
\end{displaymath}">
|
||
|
|
</DIV>
|
||
|
|
<BR CLEAR="ALL">
|
||
|
|
<P></P>
|
||
|
|
as desired. For example, in the figure, <IMG
|
||
|
|
WIDTH="11" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img28.png"
|
||
|
|
ALT="$d$"> is 0.25 radians and <IMG
|
||
|
|
WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img29.png"
|
||
|
|
ALT="$g$"> is 2.
|
||
|
|
|
||
|
|
<P>
|
||
|
|
<HR>
|
||
|
|
<!--Navigation Panel-->
|
||
|
|
<A NAME="tex2html2702"
|
||
|
|
HREF="node143.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="tex2html2696"
|
||
|
|
HREF="node139.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="tex2html2690"
|
||
|
|
HREF="node141.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="tex2html2698"
|
||
|
|
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="tex2html2700"
|
||
|
|
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="tex2html2703"
|
||
|
|
HREF="node143.html">Band-pass filter</A>
|
||
|
|
<B> Up:</B> <A NAME="tex2html2697"
|
||
|
|
HREF="node139.html">Designing filters</A>
|
||
|
|
<B> Previous:</B> <A NAME="tex2html2691"
|
||
|
|
HREF="node141.html">One-pole, one-zero high-pass filter</A>
|
||
|
|
<B> <A NAME="tex2html2699"
|
||
|
|
HREF="node4.html">Contents</A></B>
|
||
|
|
<B> <A NAME="tex2html2701"
|
||
|
|
HREF="node201.html">Index</A></B>
|
||
|
|
<!--End of Navigation Panel-->
|
||
|
|
<ADDRESS>
|
||
|
|
Miller Puckette
|
||
|
|
2006-12-30
|
||
|
|
</ADDRESS>
|
||
|
|
</BODY>
|
||
|
|
</HTML>
|