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+<H2><A NAME="SECTION0012310000000000000000">
+Impulse responses of recirculating filters</A>
+</H2>
+
+<P>
+In Section <A HREF="node109.html#sect7.recirculatingcomb">7.4</A> we analyzed the impulse response of a
+recirculating comb filter, of which the one-pole low-pass filter is a special
+case.  Figure <A HREF="#fig08.22">8.22</A> shows the result for two low-pass filters and
+one complex one-pole resonant filter.  All are elementary recirculating filters
+as introduced in Section <A HREF="node135.html#sect8.recirculating">8.2.3</A>. Each is normalized to have
+unit maximum gain.
+
+<P>
+In the case of a low-pass filter, the impulse response gets longer (and
+lower) as the pole gets closer to one.  Suppose the pole is at a point <IMG
+ WIDTH="56" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ SRC="img975.png"
+ ALT="$1-1/n$">
+(so that the cutoff frequency is <IMG
+ WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ SRC="img309.png"
+ ALT="$1/n$"> radians).  The normalizing factor is
+also <IMG
+ WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ SRC="img309.png"
+ ALT="$1/n$">.  After <IMG
+ WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
+ SRC="img75.png"
+ ALT="$n$"> points, the output diminishes by a factor of
+<BR><P></P>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ \begin{displaymath}
+{ {\left ( 1-{1\over n} \right ) } ^ n } \approx {1\over e}
+\end{displaymath}
+ -->
+
+<IMG
+ WIDTH="101" HEIGHT="46" BORDER="0"
+ SRC="img976.png"
+ ALT="\begin{displaymath}
+{ {\left ( 1-{1\over n} \right ) } ^ n } \approx {1\over e}
+\end{displaymath}">
+</DIV>
+<BR CLEAR="ALL">
+<P></P>
+where <IMG
+ WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
+ SRC="img977.png"
+ ALT="$e$"> is Euler's constant, about
+2.718.  The filter can be said to have a 
+<A NAME="10483"></A><I>settling time</I> of <IMG
+ WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
+ SRC="img75.png"
+ ALT="$n$"> samples.  In the figure, <IMG
+ WIDTH="42" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
+ SRC="img978.png"
+ ALT="$n=5$"> for part (a) and
+<IMG
+ WIDTH="50" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
+ SRC="img164.png"
+ ALT="$n=10$"> for part (b).  In general, the settling time (in samples) is approximately one
+over the cutoff frequency (in angular units).
+
+<P>
+
+<DIV ALIGN="CENTER"><A NAME="fig08.22"></A><A NAME="10487"></A>
+<TABLE>
+<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.22:</STRONG>
+The impulse response of three elementary recirculating (one-pole)
+filters,
+normalized for peak gain 1:  (a) low-pass with <IMG
+ WIDTH="57" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="img53.png"
+ ALT="$P=0.8$">; (b) low-pass with
+<IMG
+ WIDTH="57" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="img54.png"
+ ALT="$P=0.9$">; (c) band-pass (only the real part shown), with <IMG
+ WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ SRC="img55.png"
+ ALT="$\vert P\vert=0.9$"> and a center frequency of <IMG
+ WIDTH="44" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ SRC="img56.png"
+ ALT="$2\pi /10$">.</CAPTION>
+<TR><TD><IMG
+ WIDTH="441" HEIGHT="491" BORDER="0"
+ SRC="img979.png"
+ ALT="\begin{figure}\psfig{file=figs/fig08.22.ps}\end{figure}"></TD></TR>
+</TABLE>
+</DIV>
+
+<P>
+The situation gets more interesting when we look at a resonant one-pole filter,
+that is, one whose pole lies off the real axis.  In part (c) of the figure,
+the pole <IMG
+ WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="img880.png"
+ ALT="$P$"> has absolute value 0.9 (as in part b), but its argument is
+set to <IMG
+ WIDTH="44" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ SRC="img56.png"
+ ALT="$2\pi /10$"> radians.  We get the same settling time as in part (b), but
+the output rings at the resonant frequency (and so at a period of 10 samples
+in this example).
+
+<P>
+A natural question to ask is, how many periods of ringing do we get before the
+filter decays to strength <IMG
+ WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ SRC="img980.png"
+ ALT="$1/e$">?  If the pole of a resonant filter has magnitude
+<IMG
+ WIDTH="56" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ SRC="img975.png"
+ ALT="$1-1/n$"> as above, we have seen in Section <A HREF="node135.html#sect8.recirculating">8.2.3</A> that the
+bandwidth (call it <IMG
+ WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
+ SRC="img21.png"
+ ALT="$b$">) is about <IMG
+ WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ SRC="img309.png"
+ ALT="$1/n$">, and we see here that the settling time
+is about <IMG
+ WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
+ SRC="img75.png"
+ ALT="$n$">. The resonant frequency (call it <IMG
+ WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
+ SRC="img27.png"
+ ALT="$\omega $">) is the argument of the
+pole, and the period in samples of the ringing is
+<IMG
+ WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
+ SRC="img138.png"
+ ALT="$2 \pi / \omega$">.  The number of periods that make up the settling time is thus:
+<BR><P></P>
+<DIV ALIGN="CENTER">
+<!-- MATH
+ \begin{displaymath}
+{{n} \over {2\pi/\omega}} = {{1} \over {2\pi}} {{\omega} \over {b}}
+        = {{q} \over {2\pi}}
+\end{displaymath}
+ -->
+
+<IMG
+ WIDTH="136" HEIGHT="42" BORDER="0"
+ SRC="img981.png"
+ ALT="\begin{displaymath}
+{{n} \over {2\pi/\omega}} = {{1} \over {2\pi}} {{\omega} \over {b}}
+= {{q} \over {2\pi}}
+\end{displaymath}">
+</DIV>
+<BR CLEAR="ALL">
+<P></P>
+where <IMG
+ WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
+ SRC="img592.png"
+ ALT="$q$"> is the 
+<A NAME="10499"></A><I>quality</I>  of the filter, defined as the center frequency divided by
+bandwidth.  Resonant filters are often specified in terms of the center
+frequency and ``q" in place of bandwidth.
+
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