Digital Filters
45
(a)
(b)
0
1
2
3
-3
-2.5
-2
-1.5
-1
-0.5
0
frequency [rad/sample]
phase [rad]
0
1
2
3
0
5
10
15
20
frequency [rad/sample]
phase delay [samples]
Figure 22: Phase of the frequency response (a) and phase delay (b) for a first-
order allpass filter. Pole in p
1
= 0.9 (solid line) and pole in p
1
= -0.9 (dashed
line)
(a)
(b)
0
1
2
3
-6
-5
-4
-3
-2
-1
0
frequency [rad/sample]
phase [rad]
0
1
2
3
0
5
10
15
frequency [rad/sample]
phase delay [samples]
Figure 23: Phase of the frequency response (a) and phase delay (b) for a second-
order allpass filter. Poles in p
1,2
= 0.9 ± i0.2 (solid line) and p
1,2
= -0.9 ± i0.2
(dashed line)
The reader might think that the allpass filters are like open doors for audio
signals, since the phase shifts are barely distinguishable by the human hearing
system. Actually, this is true only for stationary signals, i.e., signals formed
by stable sinusoidal components. Real-world sounds are made of transients at
least as much as they are made of stationary components, and the transient
response of allpass filters can be characterized according to what we showed
in sec. 2.2. During transients, the phase response plays an important role for
perception, and in this sense the allpass filters can modify the sound signals
appreciably. For instance, very-high-order allpass filters are used to construct
artificial reverberators. These filters usually have a long time constant, so that
the effects of their phase response are mainly perceived in the time domain in
the form of a reverberation tail.
The importance of allpass filters becomes readily evident when they are
inserted into complex computational structures, typically to construct filters
whose properties should be easy to control. We will see an example of this use