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## Impulse responses of recirculating filters

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 (so that the cutoff frequency is radians). The normalizing factor is also . After points, the output diminishes by a factor of

where is Euler's constant, about 2.718. The filter can be said to have a settling time of samples. In the figure, for part (a) and for part (b). In general, the settling time (in samples) is approximately one over the cutoff frequency (in angular units).

Figure 8.22: The impulse response of three elementary recirculating (one-pole) filters, normalized for peak gain 1: (a) low-pass with ; (b) low-pass with ; (c) band-pass (only the real part shown), with and a center frequency 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 has absolute value 0.9 (as in part b), but its argument is set to 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 ? If the pole of a resonant filter has magnitude as above, we have seen in Section 8.2.3 that the bandwidth (call it ) is about , and we see here that the settling time is about . The resonant frequency (call it ) is the argument of the pole, and the period in samples of the ringing is . The number of periods that make up the settling time is thus:

where is the quality 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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