Digital Filters
37
partial fraction expansion
steady-state response
transient response
region of convergence
time constant
dominant pole
elementary resonator
where we have done a partial fraction expansion of Y (z). The second addendum
of the last member of (34) represents the steady-state response, and it is the
product of the Z transform of the causal complex exponential sequence by the
filter frequency response evaluated at the same frequency of the input signal.
The first addendum of the last member of (34) represents the transient response
and it can be represented as a causal exponential sequence:
y
t
(n) = Ap
0
n
,
(35)
where A =
1/2
1-2e
j0
. Since |p
0
| < 1 (i.e., the pole is within the unit circle), the
transient response is doomed to die out for increasing values of n. In general, for
causal systems, the stability condition (29) of chapter 1 is shown to be equivalent
to having all the poles within the unit circle. If the condition is not satisfied,
even if the steady-state response is bounded, the transient will diverge. In terms
of Z transform, a system is stable if the region of convergence is a geometric
ring containing the unit circumference; the system is causal if such ring extends
to infinity out of the circle, and it is anticausal if it extends down to the origin.
It is useful to evaluate the time needed to exhaust the initial transient. We
define the time constant
n
(in samples) of the filter as the time taken by the
exponential sequence p
0
n
to reduce its amplitude to 1% of the initial value. We
have
p
0
n
= 0.01 ,
(36)
and, therefore,
n
=
ln 0.01
ln p
0
,
(37)
where the logarithm can be evaluated in any base. In our example, where p
0
=
1/2, we obtain
n
6.64 samples. The time constant in seconds is obtained
by multiplication of
n
by the sampling rate. This way of evaluating the time
constant corresponds to evaluating the time needed to attenuate the transient
response by 40dB. When we refer to systems for artificial reverberation such
lower threshold of attenuation is moved to 60dB, thus corresponding to 0.1% of
the initial amplitude of the impulse response.
In the case of higher-order IIR filters, we can always do a partial fraction
expansion of the response to a causal exponential sequence, in a way similar to
what has been done in (34), where each addendum but the last one corresponds
to a single complex pole of the transfer function. The transient response of
these systems is, therefore, the superposition of causal complex exponentials,
each corresponding to a complex pole of the transfer function. If the goal is to
estimate the duration of the transient response, the pole that is closest to the
unit circumference is the dominant pole, since its time constant is the longest.
It is customary to define the time constant of the whole system as the constant
associated with the dominant pole.
2.2.2
Higher-Order IIR Filters
The two-pole IIR filter is a very important component of any sound processing
environment. Such filter, which is capable of selecting the frequency components
in a narrow range, can find practical applications as an elementary resonator.