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271 lines
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<H2><A NAME="SECTION0012310000000000000000">
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Impulse responses of recirculating filters</A>
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</H2>
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<P>
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In Section <A HREF="node109.html#sect7.recirculatingcomb">7.4</A> we analyzed the impulse response of a
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recirculating comb filter, of which the one-pole low-pass filter is a special
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case. Figure <A HREF="#fig08.22">8.22</A> shows the result for two low-pass filters and
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one complex one-pole resonant filter. All are elementary recirculating filters
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as introduced in Section <A HREF="node135.html#sect8.recirculating">8.2.3</A>. Each is normalized to have
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unit maximum gain.
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<P>
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In the case of a low-pass filter, the impulse response gets longer (and
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lower) as the pole gets closer to one. Suppose the pole is at a point <IMG
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WIDTH="56" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img975.png"
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ALT="$1-1/n$">
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(so that the cutoff frequency is <IMG
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WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img309.png"
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ALT="$1/n$"> radians). The normalizing factor is
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also <IMG
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WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img309.png"
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ALT="$1/n$">. After <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$"> points, the output diminishes by a factor of
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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{ {\left ( 1-{1\over n} \right ) } ^ n } \approx {1\over e}
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\end{displaymath}
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-->
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<IMG
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WIDTH="101" HEIGHT="46" BORDER="0"
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SRC="img976.png"
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ALT="\begin{displaymath}
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{ {\left ( 1-{1\over n} \right ) } ^ n } \approx {1\over e}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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where <IMG
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WIDTH="10" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img977.png"
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ALT="$e$"> is Euler's constant, about
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2.718. The filter can be said to have a
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<A NAME="10483"></A><I>settling time</I> of <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$"> samples. In the figure, <IMG
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WIDTH="42" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img978.png"
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ALT="$n=5$"> for part (a) and
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<IMG
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WIDTH="50" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img164.png"
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ALT="$n=10$"> for part (b). In general, the settling time (in samples) is approximately one
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over the cutoff frequency (in angular units).
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<P>
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<DIV ALIGN="CENTER"><A NAME="fig08.22"></A><A NAME="10487"></A>
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<TABLE>
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<CAPTION ALIGN="BOTTOM"><STRONG>Figure 8.22:</STRONG>
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The impulse response of three elementary recirculating (one-pole)
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filters,
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normalized for peak gain 1: (a) low-pass with <IMG
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WIDTH="57" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img53.png"
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ALT="$P=0.8$">; (b) low-pass with
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<IMG
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WIDTH="57" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img54.png"
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ALT="$P=0.9$">; (c) band-pass (only the real part shown), with <IMG
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WIDTH="66" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img55.png"
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ALT="$\vert P\vert=0.9$"> and a center frequency of <IMG
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WIDTH="44" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img56.png"
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ALT="$2\pi /10$">.</CAPTION>
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<TR><TD><IMG
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WIDTH="441" HEIGHT="491" BORDER="0"
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SRC="img979.png"
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ALT="\begin{figure}\psfig{file=figs/fig08.22.ps}\end{figure}"></TD></TR>
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</TABLE>
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</DIV>
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<P>
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The situation gets more interesting when we look at a resonant one-pole filter,
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that is, one whose pole lies off the real axis. In part (c) of the figure,
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the pole <IMG
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WIDTH="15" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img880.png"
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ALT="$P$"> has absolute value 0.9 (as in part b), but its argument is
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set to <IMG
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WIDTH="44" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img56.png"
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ALT="$2\pi /10$"> radians. We get the same settling time as in part (b), but
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the output rings at the resonant frequency (and so at a period of 10 samples
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in this example).
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<P>
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A natural question to ask is, how many periods of ringing do we get before the
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filter decays to strength <IMG
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WIDTH="26" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img980.png"
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ALT="$1/e$">? If the pole of a resonant filter has magnitude
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<IMG
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WIDTH="56" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img975.png"
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ALT="$1-1/n$"> as above, we have seen in Section <A HREF="node135.html#sect8.recirculating">8.2.3</A> that the
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bandwidth (call it <IMG
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WIDTH="10" HEIGHT="14" ALIGN="BOTTOM" BORDER="0"
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SRC="img21.png"
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ALT="$b$">) is about <IMG
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WIDTH="28" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img309.png"
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ALT="$1/n$">, and we see here that the settling time
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is about <IMG
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WIDTH="13" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img75.png"
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ALT="$n$">. The resonant frequency (call it <IMG
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WIDTH="14" HEIGHT="13" ALIGN="BOTTOM" BORDER="0"
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SRC="img27.png"
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ALT="$\omega $">) is the argument of the
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pole, and the period in samples of the ringing is
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<IMG
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WIDTH="39" HEIGHT="32" ALIGN="MIDDLE" BORDER="0"
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SRC="img138.png"
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ALT="$2 \pi / \omega$">. The number of periods that make up the settling time is thus:
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<BR><P></P>
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<DIV ALIGN="CENTER">
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<!-- MATH
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\begin{displaymath}
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{{n} \over {2\pi/\omega}} = {{1} \over {2\pi}} {{\omega} \over {b}}
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= {{q} \over {2\pi}}
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\end{displaymath}
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-->
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<IMG
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WIDTH="136" HEIGHT="42" BORDER="0"
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SRC="img981.png"
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ALT="\begin{displaymath}
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{{n} \over {2\pi/\omega}} = {{1} \over {2\pi}} {{\omega} \over {b}}
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= {{q} \over {2\pi}}
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\end{displaymath}">
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</DIV>
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<BR CLEAR="ALL">
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<P></P>
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where <IMG
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WIDTH="11" HEIGHT="29" ALIGN="MIDDLE" BORDER="0"
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SRC="img592.png"
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ALT="$q$"> is the
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<A NAME="10499"></A><I>quality</I> of the filter, defined as the center frequency divided by
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bandwidth. Resonant filters are often specified in terms of the center
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frequency and ``q" in place of bandwidth.
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<P>
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<ADDRESS>
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Miller Puckette
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2006-12-30
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