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<H2><A NAME="SECTION001221000000000000000"></A>
<A NAME="sect8.nonrecirculating"></A>
<BR>
Elementary non-recirculating filter
</H2>
<P>
The non-recirculating comb
filter may be generalized to yield the design shown in Figure <A HREF="#fig08.07">8.7</A>.
This is the
<A NAME="10133"></A><A NAME="10134"></A><I>elementary non-recirculating filter</I>,
of the first form. Its single, complex-valued parameter <IMG
WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$Q$"> controls the
complex gain of the delayed signal subtracted from the original one.
<P>
<DIV ALIGN="CENTER"><A NAME="fig08.07"></A><A NAME="10680"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.7:</STRONG>
A delay network with a single-sample delay and a complex
gain <IMG
WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$Q$">. This is the non-recirculating elementary filter, first form. Compare
the non-recirculating comb filter shown in Figure <A HREF="node108.html#fig07.03">7.3</A>,
which corresponds to choosing <IMG
WIDTH="57" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img43.png"
ALT="$Q=-1$"> here.</CAPTION>
<TR><TD><IMG
WIDTH="90" HEIGHT="210" BORDER="0"
SRC="img860.png"
ALT="\begin{figure}\psfig{file=figs/fig08.07.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
To find its frequency response, as in Chapter 7 we feed the delay network
a complex sinusoid <!-- MATH
$1, Z, {Z^2}, \ldots$
-->
<IMG
WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img664.png"
ALT="$1, Z, {Z^2}, \ldots$"> whose frequency is <!-- MATH
$\omega=\arg(Z)$
-->
<IMG
WIDTH="81" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img861.png"
ALT="$\omega=\arg(Z)$">.
The <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img75.png"
ALT="$n$">th sample of the input is <IMG
WIDTH="24" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img653.png"
ALT="$Z^n$"> and that of the output
is
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
(1 - Q{Z^{-1}}){Z^n}
\end{displaymath}
-->
<IMG
WIDTH="99" HEIGHT="28" BORDER="0"
SRC="img862.png"
ALT="\begin{displaymath}
(1 - Q{Z^{-1}}){Z^n}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
so the transfer function is
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
H(Z) = 1 - Q{Z^{-1}}
\end{displaymath}
-->
<IMG
WIDTH="128" HEIGHT="28" BORDER="0"
SRC="img863.png"
ALT="\begin{displaymath}
H(Z) = 1 - Q{Z^{-1}}
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
This can be analyzed graphically as shown in Figure <A HREF="#fig08.08">8.8</A>.
The real numbers
<IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img467.png"
ALT="$r$"> and <IMG
WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img7.png"
ALT="$\alpha $"> are the magnitude and argument of the complex number <IMG
WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$Q$">:
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
Q = r \cdot (\cos(\alpha) + i \sin(\alpha))
\end{displaymath}
-->
<IMG
WIDTH="178" HEIGHT="28" BORDER="0"
SRC="img864.png"
ALT="\begin{displaymath}
Q = r \cdot (\cos(\alpha) + i \sin(\alpha))
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
The gain of the filter is the distance from the point <IMG
WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$Q$"> to the point <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img20.png"
ALT="$Z$">
in the complex plane. Analytically we can see this because
<BR><P></P>
<DIV ALIGN="CENTER">
<!-- MATH
\begin{displaymath}
|1 - Q{Z^{-1}}| = |Z||1 - Q{Z^{-1}}| = |Q - Z|
\end{displaymath}
-->
<IMG
WIDTH="268" HEIGHT="28" BORDER="0"
SRC="img865.png"
ALT="\begin{displaymath}
\vert 1 - Q{Z^{-1}}\vert = \vert Z\vert\vert 1 - Q{Z^{-1}}\vert = \vert Q - Z\vert
\end{displaymath}">
</DIV>
<BR CLEAR="ALL">
<P></P>
Graphically, the number <IMG
WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img866.png"
ALT="$Q{Z^{-1}}$"> is just the number <IMG
WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$Q$"> rotated backwards
(clockwise) by the angular frequency <IMG
WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img27.png"
ALT="$\omega $"> of the incoming sinusoid. The
value <!-- MATH
$|1 - Q{Z^{-1}}|$
-->
<IMG
WIDTH="81" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img867.png"
ALT="$\vert 1 - Q{Z^{-1}}\vert$"> is the distance from <IMG
WIDTH="45" HEIGHT="34" ALIGN="MIDDLE" BORDER="0"
SRC="img866.png"
ALT="$Q{Z^{-1}}$"> to <IMG
WIDTH="11" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img262.png"
ALT="$1$"> in the complex
plane, which is equal to the distance from <IMG
WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$Q$"> to <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img20.png"
ALT="$Z$">.
<P>
<DIV ALIGN="CENTER"><A NAME="fig08.08"></A><A NAME="10688"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.8:</STRONG>
Diagram for calculating the frequency response of the
non-recirculating elementary filter
(Figure <A HREF="#fig08.07">8.7</A>). The frequency response is given by the length of the
segment connecting <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img20.png"
ALT="$Z$"> to <IMG
WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$Q$"> in the complex plane.</CAPTION>
<TR><TD><IMG
WIDTH="341" HEIGHT="379" BORDER="0"
SRC="img868.png"
ALT="\begin{figure}\psfig{file=figs/fig08.08.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
<P>
As the frequency of the input sweeps from 0 to <IMG
WIDTH="21" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img16.png"
ALT="$2\pi $">, the point <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img20.png"
ALT="$Z$"> travels
couterclockwise around the unit circle. At the point where <!-- MATH
$\omega = \alpha$
-->
<IMG
WIDTH="45" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
SRC="img869.png"
ALT="$\omega = \alpha$">,
the distance is at a minimum, equal to <IMG
WIDTH="38" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
SRC="img870.png"
ALT="$1-r$">. The maximum occurs which <IMG
WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
SRC="img20.png"
ALT="$Z$"> is
at the opposite point of the circle. Figure <A HREF="#fig08.09">8.9</A> shows the transfer
function for three different values of <IMG
WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img44.png"
ALT="$r=\vert Q\vert$">.
<P>
<DIV ALIGN="CENTER"><A NAME="fig08.09"></A><A NAME="10689"></A>
<TABLE>
<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.9:</STRONG>
Frequency response of the elementary non-recirculating filter
Figure <A HREF="#fig08.07">8.7</A>. Three values of <IMG
WIDTH="16" HEIGHT="30" ALIGN="MIDDLE" BORDER="0"
SRC="img42.png"
ALT="$Q$"> are used, all with the
same argument (-2 radians), but with varying absolute value (magnitude) <IMG
WIDTH="53" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
SRC="img44.png"
ALT="$r=\vert Q\vert$">.</CAPTION>
<TR><TD><IMG
WIDTH="323" HEIGHT="212" BORDER="0"
SRC="img871.png"
ALT="\begin{figure}\psfig{file=figs/fig08.09.ps}\end{figure}"></TD></TR>
</TABLE>
</DIV>
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