<!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>Two recirculating filters for the price of one</TITLE>
<META NAME="description" CONTENT="Two recirculating filters for the price of one">
<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="previous" HREF="node137.html">
<LINK REL="up" HREF="node132.html">
<LINK REL="next" HREF="node139.html">
</HEAD>

<BODY >
<!--Navigation Panel-->
<A NAME="tex2html2635"
  HREF="node139.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
 SRC="next.png"></A> 
<A NAME="tex2html2629"
  HREF="node132.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
 SRC="up.png"></A> 
<A NAME="tex2html2625"
  HREF="node137.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
 SRC="prev.png"></A> 
<A NAME="tex2html2631"
  HREF="node4.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
 SRC="contents.png"></A> 
<A NAME="tex2html2633"
  HREF="node201.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
 SRC="index.png"></A> 
<BR>
<B> Next:</B> <A NAME="tex2html2636"
  HREF="node139.html">Designing filters</A>
<B> Up:</B> <A NAME="tex2html2630"
  HREF="node132.html">Elementary filters</A>
<B> Previous:</B> <A NAME="tex2html2626"
  HREF="node137.html">Real outputs from complex</A>
 &nbsp; <B>  <A NAME="tex2html2632"
  HREF="node4.html">Contents</A></B> 
 &nbsp; <B>  <A NAME="tex2html2634"
  HREF="node201.html">Index</A></B> 
<BR>
<BR>
<!--End of Navigation Panel-->

<H2><A NAME="SECTION001226000000000000000">
Two recirculating filters for the price of one</A>
</H2>

<P>
When pairing recirculating elementary filters, it is possible to avoid
computing one of each pair, as long as the input is real-valued (and so,
the output is as well.)   Supposing the input is a real sinusoid of the form:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
A{Z^n} + \overline{A}{Z^{-n}}
\end{displaymath}
 -->

<IMG
 WIDTH="93" HEIGHT="25" BORDER="0"
 SRC="img900.png"
 ALT="\begin{displaymath}
A{Z^n} + \overline{A}{Z^{-n}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
and we apply a single recirculating filter with coefficient <IMG
 WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
 SRC="img880.png"
 ALT="$P$">.  Letting <IMG
 WIDTH="30" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
 SRC="img503.png"
 ALT="$a[n]$">
denote the real part of the output, we have:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
a[n] =
    \mathrm{re} \left[ {
        {1 \over {1 - {P}{Z^{-1}}}} {A{Z^n} +
        {1 \over {1 - {P}{Z}}}  \overline{A}{Z^{-n}}}
    } \right ]
\end{displaymath}
 -->

<IMG
 WIDTH="298" HEIGHT="45" BORDER="0"
 SRC="img901.png"
 ALT="\begin{displaymath}
a[n] =
\mathrm{re} \left[ {
{1 \over {1 - {P}{Z^{-1}}}} {...
...n} +
{1 \over {1 - {P}{Z}}} \overline{A}{Z^{-n}}}
} \right ]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
=
    \mathrm{re} \left[ {
        {1 \over {1 - {P}{Z^{-1}}}} {A{Z^n}} +
        {1 \over {1 - \overline{P}{Z^{-1}}}}  {A{Z^n}}
    } \right ]
\end{displaymath}
 -->

<IMG
 WIDTH="274" HEIGHT="45" BORDER="0"
 SRC="img902.png"
 ALT="\begin{displaymath}
=
\mathrm{re} \left[ {
{1 \over {1 - {P}{Z^{-1}}}} {A{Z^n}} +
{1 \over {1 - \overline{P}{Z^{-1}}}} {A{Z^n}}
} \right ]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
=
    \mathrm{re} \left[ {
        {{
            2 - 2 \, \mathrm{re} (P) {Z^{-1}}
        } \over {
            (1 - {P}{Z^{-1}}) (1 - {\overline{P}}{Z^{-1}})
        }}
        {A{Z^n}}
    } \right ]
\end{displaymath}
 -->

<IMG
 WIDTH="243" HEIGHT="46" BORDER="0"
 SRC="img903.png"
 ALT="\begin{displaymath}
=
\mathrm{re} \left[ {
{{
2 - 2 \, \mathrm{re} (P) {Z^{-...
...^{-1}}) (1 - {\overline{P}}{Z^{-1}})
}}
{A{Z^n}}
} \right ]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
=
    \mathrm{re} \left[ {
        {{
            1 -  \mathrm{re} (P) {Z^{-1}}
        } \over {
            (1 - {P}{Z^{-1}}) (1 - {\overline{P}}{Z^{-1}})
        }}
        {A{Z^n}}
        +
        {{
            1 - \mathrm{re} (P) {{\overline{Z}}^{-1}}
        } \over {
            (1 - {\overline{P}}{{\overline{Z}}^{-1}}) (1 - {P}{{\overline{Z}}^{-1}})
        }}
        {\overline{A}{{\overline{Z}}^{-n}}}
    } \right ]
\end{displaymath}
 -->

<IMG
 WIDTH="474" HEIGHT="54" BORDER="0"
 SRC="img904.png"
 ALT="\begin{displaymath}
=
\mathrm{re} \left[ {
{{
1 - \mathrm{re} (P) {Z^{-1}}
...
...}}^{-1}})
}}
{\overline{A}{{\overline{Z}}^{-n}}}
} \right ]
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
(In the second step we used the fact that you can conjugate all or part of an
expression, without changing the result, if you're just going to take the real
part anyway.  The fourth step did the same thing backward.)  Comparing the input
to the output, we see that the effect of passing a real signal through a complex
one-pole filter, then taking the real part, is equivalent to passing the signal
through a two-pole, one-zero filter with transfer function equal to:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
{H_{\mathrm{re}}}(Z) = {{
            1 - \mathrm{re} (P) {Z^{-1}}
        } \over {
            (1 - {P}{Z^{-1}}) (1 - {\overline{P}}{Z^{-1}})
        }}
\end{displaymath}
 -->

<IMG
 WIDTH="235" HEIGHT="46" BORDER="0"
 SRC="img905.png"
 ALT="\begin{displaymath}
{H_{\mathrm{re}}}(Z) = {{
1 - \mathrm{re} (P) {Z^{-1}}
} \over {
(1 - {P}{Z^{-1}}) (1 - {\overline{P}}{Z^{-1}})
}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
A similar calculation shows that taking the imaginary part (considered as a real
signal) is equivalent to filtering the input with the transfer function:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
 \begin{displaymath}
{H_{\mathrm{im}}}(Z) = {{
            \mathrm{im} (P) {Z^{-1}}
        } \over {
            (1 - {P}{Z^{-1}}) (1 - {\overline{P}}{Z^{-1}})
        }}
\end{displaymath}
 -->

<IMG
 WIDTH="238" HEIGHT="46" BORDER="0"
 SRC="img906.png"
 ALT="\begin{displaymath}
{H_{\mathrm{im}}}(Z) = {{
\mathrm{im} (P) {Z^{-1}}
} \over {
(1 - {P}{Z^{-1}}) (1 - {\overline{P}}{Z^{-1}})
}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
So taking either the real or imaginary part of a one-pole filter output gives
filters with two conjugate poles.  The two parts can be combined
to synthesize filters with other possible numerators; in other words, with
one complex recirculating filter we can synthesize a filter that acts on
real signals with two (complex conjugate) poles and one (real) zero.

<P>
This technique, known as
<I>partial fractions</I>, may be repeated for
any number of stages in series as long as we compute the appropriate
combination of real and imaginary parts of the output of each stage to
form the (real) input of the next stage.
No similar shortcut seems to exist for non-recirculating filters; for them
it is necessary to compute each member of each complex-conjugate pair
explicitly.  

<P>
<HR>
<!--Navigation Panel-->
<A NAME="tex2html2635"
  HREF="node139.html">
<IMG WIDTH="37" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="next"
 SRC="next.png"></A> 
<A NAME="tex2html2629"
  HREF="node132.html">
<IMG WIDTH="26" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="up"
 SRC="up.png"></A> 
<A NAME="tex2html2625"
  HREF="node137.html">
<IMG WIDTH="63" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="previous"
 SRC="prev.png"></A> 
<A NAME="tex2html2631"
  HREF="node4.html">
<IMG WIDTH="65" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="contents"
 SRC="contents.png"></A> 
<A NAME="tex2html2633"
  HREF="node201.html">
<IMG WIDTH="43" HEIGHT="24" ALIGN="BOTTOM" BORDER="0" ALT="index"
 SRC="index.png"></A> 
<BR>
<B> Next:</B> <A NAME="tex2html2636"
  HREF="node139.html">Designing filters</A>
<B> Up:</B> <A NAME="tex2html2630"
  HREF="node132.html">Elementary filters</A>
<B> Previous:</B> <A NAME="tex2html2626"
  HREF="node137.html">Real outputs from complex</A>
 &nbsp; <B>  <A NAME="tex2html2632"
  HREF="node4.html">Contents</A></B> 
 &nbsp; <B>  <A NAME="tex2html2634"
  HREF="node201.html">Index</A></B> 
<!--End of Navigation Panel-->
<ADDRESS>
Miller Puckette
2006-12-30
</ADDRESS>
</BODY>
</HTML>
