44
D. Rocchesso: Sound Processing
pole-zero couple
allpass filter
(a)
(b)
0
1
2
3
4
0
1
2
frequency [rad/sample]
magnitude
0
1
2
3
4
-8
-6
-4
-2
0
2
frequency [rad/sample]
phase [rad]
Figure 21: Frequency response (magnitude and phase) of an IIR filter with two
poles (r = 0.95) and two zeros. The notch filter (dashed line) has the zeros with
magnitude 1.0. The boost filter (solid line) has the zeros with magnitude 0.9.
z
i
= 1/p
i
in the transfer function. In other words, we form the pole-zero couple
H
i
(z) =
z
-1
- p
i
1 - p
i
z
-1
,
(48)
which places the pole and the zero on reciprocal points about the unit circum-
ference and along tha same radius that links them to the origin. Moving along
the circumference we can realize that the vectors drawn from the pole and the
zero have lengths that keep a constant ratio. A more accurate analysis can be
done using the frequency response of this pole-zero couple, which is written as
H
i
() =
e
-j
- p
i
1 - p
i
e
-j
= e
-j
1 - p
i
e
j
1 - p
i
e
-j
.
(49)
It is clear that numerator and denominator of the fraction in the last member
of (49) are complex conjugate one to each other, thus meaning that the rational
function has unit magnitude at any frequency. Therefore, the couple (49) is
the fundamental block for the construction of an allpass filter, whose frequency
response is obtained by multiplication of blocks such as (49).
The allpass filters are systems that leave all frequency component magni-
tudes unaltered. Stationary sinusoidal input signals can only be subject to phase
delays, with no modification in magnitude. The phase response and phase delay
of the fundamental pole-zero couple are depicted in fig. 22 for values of pole set
to p
1
= 0.9 and p
1
= -0.9. A second-order allpass filter with real coefficients is
obtained by multiplication of two allpass pole-zero couples, where the poles are
the conjugate of each other. Fig. 23 shows the phase response and the phase de-
lay of a second order allpass filter with poles in p
1
= 0.9+i0.2 and p
2
= 0.9-i0.2
(solid line) and in p
1
= -0.9 + i0.2 and p
2
= -0.9 - i0.2 (dashed line). It can
be shown that the phase response of any allpass filter is always negative and
monotonically decreasing [65]. The group and phase delays are always functions
that take positive values. This fact allows us to think about allpass filters as
media where signals propagate with a frequency-dependent delay, without being
subject to any absorption or amplification.