283 lines
8.0 KiB
HTML
283 lines
8.0 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>Real outputs from complex filters</TITLE>
|
||
|
|
<META NAME="description" CONTENT="Real outputs from complex 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="node138.html">
|
||
|
|
<LINK REL="previous" HREF="node136.html">
|
||
|
|
<LINK REL="up" HREF="node132.html">
|
||
|
|
<LINK REL="next" HREF="node138.html">
|
||
|
|
</HEAD>
|
||
|
|
|
||
|
|
<BODY >
|
||
|
|
<!--Navigation Panel-->
|
||
|
|
<A NAME="tex2html2623"
|
||
|
|
HREF="node138.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="tex2html2617"
|
||
|
|
HREF="node132.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="tex2html2611"
|
||
|
|
HREF="node136.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="tex2html2619"
|
||
|
|
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="tex2html2621"
|
||
|
|
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="tex2html2624"
|
||
|
|
HREF="node138.html">Two recirculating filters for</A>
|
||
|
|
<B> Up:</B> <A NAME="tex2html2618"
|
||
|
|
HREF="node132.html">Elementary filters</A>
|
||
|
|
<B> Previous:</B> <A NAME="tex2html2612"
|
||
|
|
HREF="node136.html">Compound filters</A>
|
||
|
|
<B> <A NAME="tex2html2620"
|
||
|
|
HREF="node4.html">Contents</A></B>
|
||
|
|
<B> <A NAME="tex2html2622"
|
||
|
|
HREF="node201.html">Index</A></B>
|
||
|
|
<BR>
|
||
|
|
<BR>
|
||
|
|
<!--End of Navigation Panel-->
|
||
|
|
|
||
|
|
<H2><A NAME="SECTION001225000000000000000">
|
||
|
|
Real outputs from complex filters</A>
|
||
|
|
</H2>
|
||
|
|
|
||
|
|
<P>
|
||
|
|
In most applications, we start with a real-valued signal to filter and we need
|
||
|
|
a real-valued output, but in general, a compound filter with a transfer
|
||
|
|
function as above will give a complex-valued output. However, we can
|
||
|
|
construct filters with non-real-valued coefficients which nonetheless give
|
||
|
|
real-valued outputs, so that the analysis that we carry out using complex
|
||
|
|
numbers can be used to predict, explain, and control real-valued output
|
||
|
|
signals. We do this by pairing each elementary filter (with coefficient
|
||
|
|
<IMG
|
||
|
|
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img880.png"
|
||
|
|
ALT="$P$"> or <IMG
|
||
|
|
WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img42.png"
|
||
|
|
ALT="$Q$">) with another having as its coefficient the complex conjugate
|
||
|
|
<IMG
|
||
|
|
WIDTH="15" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
|
||
|
|
SRC="img890.png"
|
||
|
|
ALT="$\overline{P}$"> or <IMG
|
||
|
|
WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img872.png"
|
||
|
|
ALT="$\overline{Q}$">.
|
||
|
|
|
||
|
|
<P>
|
||
|
|
For example, putting two non-recirculating filters, with coefficients <IMG
|
||
|
|
WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img42.png"
|
||
|
|
ALT="$Q$"> and
|
||
|
|
<IMG
|
||
|
|
WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img872.png"
|
||
|
|
ALT="$\overline{Q}$">, in series gives a transfer function equal to:
|
||
|
|
<BR><P></P>
|
||
|
|
<DIV ALIGN="CENTER">
|
||
|
|
<!-- MATH
|
||
|
|
\begin{displaymath}
|
||
|
|
H(Z) = (1 - {Q}{Z^{-1}}) \cdot (1 - \overline{Q}{Z^{-1}})
|
||
|
|
\end{displaymath}
|
||
|
|
-->
|
||
|
|
|
||
|
|
<IMG
|
||
|
|
WIDTH="233" HEIGHT="28" BORDER="0"
|
||
|
|
SRC="img891.png"
|
||
|
|
ALT="\begin{displaymath}
|
||
|
|
H(Z) = (1 - {Q}{Z^{-1}}) \cdot (1 - \overline{Q}{Z^{-1}})
|
||
|
|
\end{displaymath}">
|
||
|
|
</DIV>
|
||
|
|
<BR CLEAR="ALL">
|
||
|
|
<P></P>
|
||
|
|
which has the property that:
|
||
|
|
<BR><P></P>
|
||
|
|
<DIV ALIGN="CENTER">
|
||
|
|
<!-- MATH
|
||
|
|
\begin{displaymath}
|
||
|
|
H(\overline{Z}) = \overline{H(Z)}
|
||
|
|
\end{displaymath}
|
||
|
|
-->
|
||
|
|
|
||
|
|
<IMG
|
||
|
|
WIDTH="99" HEIGHT="28" BORDER="0"
|
||
|
|
SRC="img892.png"
|
||
|
|
ALT="\begin{displaymath}
|
||
|
|
H(\overline{Z}) = \overline{H(Z)}
|
||
|
|
\end{displaymath}">
|
||
|
|
</DIV>
|
||
|
|
<BR CLEAR="ALL">
|
||
|
|
<P></P>
|
||
|
|
Now if we put any real-valued sinusoid:
|
||
|
|
<BR><P></P>
|
||
|
|
<DIV ALIGN="CENTER">
|
||
|
|
<!-- MATH
|
||
|
|
\begin{displaymath}
|
||
|
|
{X_n} = 2 \, \mathrm{re}(A{Z^n}) = A{Z^n} + \overline{A}
|
||
|
|
{{\overline{Z}}^n}
|
||
|
|
\end{displaymath}
|
||
|
|
-->
|
||
|
|
|
||
|
|
<IMG
|
||
|
|
WIDTH="217" HEIGHT="28" BORDER="0"
|
||
|
|
SRC="img893.png"
|
||
|
|
ALT="\begin{displaymath}
|
||
|
|
{X_n} = 2 \, \mathrm{re}(A{Z^n}) = A{Z^n} + \overline{A}
|
||
|
|
{{\overline{Z}}^n}
|
||
|
|
\end{displaymath}">
|
||
|
|
</DIV>
|
||
|
|
<BR CLEAR="ALL">
|
||
|
|
<P></P>
|
||
|
|
we get out:
|
||
|
|
<BR><P></P>
|
||
|
|
<DIV ALIGN="CENTER">
|
||
|
|
<!-- MATH
|
||
|
|
\begin{displaymath}
|
||
|
|
A \cdot H(Z) \cdot {Z^n} +
|
||
|
|
\overline{A} \cdot \overline{H(Z)} \cdot {{\overline{Z}}^n}
|
||
|
|
\end{displaymath}
|
||
|
|
-->
|
||
|
|
|
||
|
|
<IMG
|
||
|
|
WIDTH="207" HEIGHT="28" BORDER="0"
|
||
|
|
SRC="img894.png"
|
||
|
|
ALT="\begin{displaymath}
|
||
|
|
A \cdot H(Z) \cdot {Z^n} +
|
||
|
|
\overline{A} \cdot \overline{H(Z)} \cdot {{\overline{Z}}^n}
|
||
|
|
\end{displaymath}">
|
||
|
|
</DIV>
|
||
|
|
<BR CLEAR="ALL">
|
||
|
|
<P></P>
|
||
|
|
which, by inspection, is another real sinusoid.
|
||
|
|
Here we're using two properties of complex conjugates. First, you can
|
||
|
|
add and multiply them at will:
|
||
|
|
<BR><P></P>
|
||
|
|
<DIV ALIGN="CENTER">
|
||
|
|
<!-- MATH
|
||
|
|
\begin{displaymath}
|
||
|
|
\overline{A+B} = \overline{A} + \overline{B}
|
||
|
|
\end{displaymath}
|
||
|
|
-->
|
||
|
|
|
||
|
|
<IMG
|
||
|
|
WIDTH="110" HEIGHT="25" BORDER="0"
|
||
|
|
SRC="img895.png"
|
||
|
|
ALT="\begin{displaymath}
|
||
|
|
\overline{A+B} = \overline{A} + \overline{B}
|
||
|
|
\end{displaymath}">
|
||
|
|
</DIV>
|
||
|
|
<BR CLEAR="ALL">
|
||
|
|
<P></P>
|
||
|
|
<BR><P></P>
|
||
|
|
<DIV ALIGN="CENTER">
|
||
|
|
<!-- MATH
|
||
|
|
\begin{displaymath}
|
||
|
|
\overline{AB} = \overline{A} \cdot \overline{B}
|
||
|
|
\end{displaymath}
|
||
|
|
-->
|
||
|
|
|
||
|
|
<IMG
|
||
|
|
WIDTH="82" HEIGHT="24" BORDER="0"
|
||
|
|
SRC="img896.png"
|
||
|
|
ALT="\begin{displaymath}
|
||
|
|
\overline{AB} = \overline{A} \cdot \overline{B}
|
||
|
|
\end{displaymath}">
|
||
|
|
</DIV>
|
||
|
|
<BR CLEAR="ALL">
|
||
|
|
<P></P>
|
||
|
|
and second, anything plus its complex conjugate is real, and is in fact
|
||
|
|
twice its real part:
|
||
|
|
<BR><P></P>
|
||
|
|
<DIV ALIGN="CENTER">
|
||
|
|
<!-- MATH
|
||
|
|
\begin{displaymath}
|
||
|
|
A + \overline{A} = 2 \, \mathrm{re} (A)
|
||
|
|
\end{displaymath}
|
||
|
|
-->
|
||
|
|
|
||
|
|
<IMG
|
||
|
|
WIDTH="112" HEIGHT="28" BORDER="0"
|
||
|
|
SRC="img897.png"
|
||
|
|
ALT="\begin{displaymath}
|
||
|
|
A + \overline{A} = 2 \, \mathrm{re} (A)
|
||
|
|
\end{displaymath}">
|
||
|
|
</DIV>
|
||
|
|
<BR CLEAR="ALL">
|
||
|
|
<P></P>
|
||
|
|
This result for two conjugate filters extends to any compound filter; in
|
||
|
|
general, we always get a real-valued output from a real-valued input if we
|
||
|
|
arrange that each coefficient <IMG
|
||
|
|
WIDTH="21" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img898.png"
|
||
|
|
ALT="$Q_i$"> and <IMG
|
||
|
|
WIDTH="19" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
|
||
|
|
SRC="img899.png"
|
||
|
|
ALT="$P_i$"> in the compound filter is
|
||
|
|
either real-valued, or else appears in a pair with its complex conjugate.
|
||
|
|
|
||
|
|
<P>
|
||
|
|
<HR>
|
||
|
|
<!--Navigation Panel-->
|
||
|
|
<A NAME="tex2html2623"
|
||
|
|
HREF="node138.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="tex2html2617"
|
||
|
|
HREF="node132.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="tex2html2611"
|
||
|
|
HREF="node136.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="tex2html2619"
|
||
|
|
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="tex2html2621"
|
||
|
|
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="tex2html2624"
|
||
|
|
HREF="node138.html">Two recirculating filters for</A>
|
||
|
|
<B> Up:</B> <A NAME="tex2html2618"
|
||
|
|
HREF="node132.html">Elementary filters</A>
|
||
|
|
<B> Previous:</B> <A NAME="tex2html2612"
|
||
|
|
HREF="node136.html">Compound filters</A>
|
||
|
|
<B> <A NAME="tex2html2620"
|
||
|
|
HREF="node4.html">Contents</A></B>
|
||
|
|
<B> <A NAME="tex2html2622"
|
||
|
|
HREF="node201.html">Index</A></B>
|
||
|
|
<!--End of Navigation Panel-->
|
||
|
|
<ADDRESS>
|
||
|
|
Miller Puckette
|
||
|
|
2006-12-30
|
||
|
|
</ADDRESS>
|
||
|
|
</BODY>
|
||
|
|
</HTML>
|