In Section 7.4 we analyzed the impulse response of a recirculating comb filter, of which the one-pole low-pass filter is a special case. Figure 8.22 shows the result for two low-pass filters and one complex one-pole resonant filter. All are elementary recirculating filters as introduced in Section 8.2.3. Each is normalized to have unit maximum gain.
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 1ドル-1/n$
(so that the cutoff frequency is 1ドル/n$ radians). The normalizing factor is
also 1ドル/n$. After $n$ points, the output diminishes by a factor of
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 $P$ has absolute value 0.9 (as in part b), but its argument is set to 2ドル\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).
A natural question to ask is, how many periods of ringing do we get before the
filter decays to strength 1ドル/e$? If the pole of a resonant filter has magnitude
1ドル-1/n$ as above, we have seen in Section 8.2.3 that the
bandwidth (call it $b$) is about 1ドル/n,ドル and we see here that the settling time
is about $n$. The resonant frequency (call it $\omega $) is the argument of the
pole, and the period in samples of the ringing is
2ドル \pi / \omega$. The number of periods that make up the settling time is thus:
