134 lines
3.6 KiB
HTML
134 lines
3.6 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>Compound filters</TITLE>
|
|
<META NAME="description" CONTENT="Compound filters">
|
|
<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="node137.html">
|
|
<LINK REL="previous" HREF="node135.html">
|
|
<LINK REL="up" HREF="node132.html">
|
|
<LINK REL="next" HREF="node137.html">
|
|
</HEAD>
|
|
|
|
<BODY >
|
|
<!--Navigation Panel-->
|
|
<A NAME="tex2html2609"
|
|
HREF="node137.html">
|
|
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
|
|
SRC="next.png"></A>
|
|
<A NAME="tex2html2603"
|
|
HREF="node132.html">
|
|
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
|
|
SRC="up.png"></A>
|
|
<A NAME="tex2html2597"
|
|
HREF="node135.html">
|
|
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
|
|
SRC="prev.png"></A>
|
|
<A NAME="tex2html2605"
|
|
HREF="node4.html">
|
|
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
|
|
SRC="contents.png"></A>
|
|
<A NAME="tex2html2607"
|
|
HREF="node201.html">
|
|
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
|
|
SRC="index.png"></A>
|
|
<BR>
|
|
<B> Next:</B> <A NAME="tex2html2610"
|
|
HREF="node137.html">Real outputs from complex</A>
|
|
<B> Up:</B> <A NAME="tex2html2604"
|
|
HREF="node132.html">Elementary filters</A>
|
|
<B> Previous:</B> <A NAME="tex2html2598"
|
|
HREF="node135.html">Elementary recirculating filter</A>
|
|
<B> <A NAME="tex2html2606"
|
|
HREF="node4.html">Contents</A></B>
|
|
<B> <A NAME="tex2html2608"
|
|
HREF="node201.html">Index</A></B>
|
|
<BR>
|
|
<BR>
|
|
<!--End of Navigation Panel-->
|
|
|
|
<H2><A NAME="SECTION001224000000000000000">
|
|
Compound filters</A>
|
|
</H2>
|
|
|
|
<P>
|
|
We can use the recirculating and non-recirculating filters developed here to
|
|
create a
|
|
<A NAME="10201"></A><A NAME="10202"></A><I>compound filter</I> by putting several elementary ones in series. If the parameters
|
|
of the non-recirculating ones (of the first type) are <!-- MATH
|
|
${Q_1}, \ldots, {Q_j}$
|
|
-->
|
|
<IMG
|
|
WIDTH="77" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img887.png"
|
|
ALT="${Q_1}, \ldots, {Q_j}$">
|
|
and those of the recirculating ones are <!-- MATH
|
|
${P_1}, \ldots, {P_k}$
|
|
-->
|
|
<IMG
|
|
WIDTH="74" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
|
SRC="img888.png"
|
|
ALT="${P_1}, \ldots, {P_k}$">, then putting
|
|
them all in series, in any order, will give the transfer function:
|
|
<BR><P></P>
|
|
<DIV ALIGN="CENTER">
|
|
<!-- MATH
|
|
\begin{displaymath}
|
|
H(Z) = {
|
|
{
|
|
(1 - {Q_1}{Z^{-1}}) \cdots (1 - {Q_j}{Z^{-1}})
|
|
} \over {
|
|
(1 - {P_1}{Z^{-1}}) \cdots (1 - {P_k}{Z^{-1}})
|
|
}
|
|
}
|
|
\end{displaymath}
|
|
-->
|
|
|
|
<IMG
|
|
WIDTH="263" HEIGHT="45" BORDER="0"
|
|
SRC="img889.png"
|
|
ALT="\begin{displaymath}
|
|
H(Z) = {
|
|
{
|
|
(1 - {Q_1}{Z^{-1}}) \cdots (1 - {Q_j}{Z^{-1}})
|
|
} \over {
|
|
(1 - {P_1}{Z^{-1}}) \cdots (1 - {P_k}{Z^{-1}})
|
|
}
|
|
}
|
|
\end{displaymath}">
|
|
</DIV>
|
|
<BR CLEAR="ALL">
|
|
<P></P>
|
|
The frequency response of the resulting compound filter is the product of
|
|
those of
|
|
the elementary ones. (One could also combine elementary filters by adding
|
|
their outputs, or making more complicated networks of them; but for most
|
|
purposes the series configuration is the easiest one to work with.)
|
|
|
|
<P>
|
|
<BR><HR>
|
|
<ADDRESS>
|
|
Miller Puckette
|
|
2006-12-30
|
|
</ADDRESS>
|
|
</BODY>
|
|
</HTML>
|