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<H2><A ID="SECTION001222000000000000000"></A>
<A ID="sect8.secondform"></A>
<BR>
Non-recirculating filter, second form
</H2>
<P>
Sometimes we will need a variant of the filter above, shown in Figure
<A HREF="#fig08.10">8.10</A>, called the
<I>elementary non-recirculating filter, second form</I>.
Instead of multiplying the delay output by <IMG
WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$Q$"> we multiply the direct signal
by its
<A ID="10166"></A><I>complex conjugate</I>
<IMG
WIDTH="16" HEIGHT="36" ALIGN="MIDDLE" BORDER="0"
SRC="img872.png"
ALT="$\overline{Q}$">. If
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
A = a+bi = r \cdot (\cos(\alpha) + i \sin(\alpha))
\end{displaymath}
-->
<IMG
WIDTH="238" HEIGHT="28" BORDER="0"
SRC="img873.png"
ALT="\begin{displaymath}
A = a+bi = r \cdot (\cos(\alpha) + i \sin(\alpha))
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
is any complex number, its complex conjugate is defined as:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
\overline{A} = a-bi = r \cdot (\cos(\alpha) - i \sin(\alpha))
\end{displaymath}
-->
<IMG
WIDTH="238" HEIGHT="28" BORDER="0"
SRC="img874.png"
ALT="\begin{displaymath}
\overline{A} = a-bi = r \cdot (\cos(\alpha) - i \sin(\alpha))
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Graphically this reflects points of the complex plane up and down across the
real axis. The transfer function of the new filter is
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
H(Z) = \overline{Q} - {Z^{-1}}
\end{displaymath}
-->
<IMG
WIDTH="120" HEIGHT="28" BORDER="0"
SRC="img875.png"
ALT="\begin{displaymath}
H(Z) = \overline{Q} - {Z^{-1}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
This gives rise to the same frequency response as before since
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
|\overline{Q} - {Z^{-1}}| = |Q - \overline{Z^{-1}}| = |Q- Z|
\end{displaymath}
-->
<IMG
WIDTH="232" HEIGHT="28" BORDER="0"
SRC="img876.png"
ALT="\begin{displaymath}
\vert\overline{Q} - {Z^{-1}}\vert = \vert Q - \overline{Z^{-1}}\vert = \vert Q- Z\vert
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Here we use the fact that <!-- MATH
$\overline{Z} = {Z^{-1}}$
-->
<IMG
WIDTH="65" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img877.png"
ALT="$\overline{Z} = {Z^{-1}}$">, for any unit complex
number <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img20.png"
ALT="$Z$">, as can be verified by writing out <IMG
WIDTH="27" HEIGHT="17" ALIGN="BOTTOM" BORDER="0"
SRC="img878.png"
ALT="$Z\overline{Z}$"> in either
polar or rectangular form.
<P>
Although the two forms of the elementary non-recirculating filter have the same
frequency response, their phase responses are different; this will
occasionally lead us to prefer the second form.
<P>
<DIV ALIGN="CENTER"><A ID="fig08.10"></A><A ID="10180"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.10:</STRONG>
The elementary non-recirculating filter, second form.</CAPTION>
<TR><TD><IMG
WIDTH="121" HEIGHT="168" BORDER="0"
SRC="img879.png"
ALT="\begin{figure}\psfig{file=figs/fig08.10.ps}\end{figure}"></TD></TR>
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<ADDRESS>
Miller Puckette
2006-12-30
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