Digital Filters
23
steady-state response
transient response
The frequency response gives a clear picture of the behavior of a filter
when its inputs are stationary signals, which can be decomposed as constant-
amplitude sinusoids. Therefore, the frequency response represents the steady-
state response of the system. In practice, even signals composed by sinusoids
have to be turned on at a certain instant, thus producing a transient response
that comes before the steady-state. However, the knowledge of the Z transform
of a causal complex sinusoid and the knowledge of the filter transfer function al-
low us to study the overall response analytically. As we show in appendix A.8.3,
the Z transform of causal exponential sequence is
X(z) =
1
1 - e
j
0
z
-1
.
(11)
If we multiply, in the z domain, X(z) by the transfer function H(z) we get
Y (z) = H(z)X(z) =
1
2
(1 + z
-1
)
1
1 - e
j
0
z
-1
=
1
2
1
1 - e
j
0
z
-1
+
1
2
z
-1
1 - e
j
0
z
-1
.
(12)
The second term of the last member of (12) is, by the shift theorem, the trans-
form of a causal complex sinusoidal sequence delayed by one sample. Therefore,
the overall response can be thought of as a sum of two identical sinusoids shifted
by one sample and this turns out to be another sinusoid, but only after the first
sampling instant. The first instant has a different behavior since it is part of
the transient of the response (see fig. 3). It is easy to realize that, for an FIR
filter, the transient lasts for a number of samples that doesn't exceed the order
(memory) of the filter itself. Since an order-N FIR filter has a memory of N
samples, the transient is at most N samples long.
0
10
20
30
40
-1
0
1
samples
Figure 3: Response of an FIR averaging filter to a causal cosine: input and
delayed input (), actual response (×)
2.1.2
The Phase Response
If we filter a sound with a nonlinear-phase filter we alter its time-domain wave
shape. This happens because the different frequency components are subject to
a different delay while being transferred from the input to the output of the
filter. Therefore, a compact wavefront is dispersed during its traversal of the