40
D. Rocchesso: Sound Processing
damped oscillator
A
P
B
P
0
0
Figure 18: Graphic construction of the bandwidth. P
0
is the pole. P
0
P 1 - r.
The transfer function (39) can be expanded in partial fractions as
H(z) =
G
(1 - re
j
0
z
-1
)(1 - re
-j
0
z
-1
)
=
G/(1 - e
-j2
0
)
1 - re
j
0
z
-1
-
Ge
-j2
0
/(1 - e
-j2
0
)
1 - re
-j
0
z
-1
,
(45)
and each addendum is the Z transform of a causal complex exponential se-
quence. By manipulating the two sequences algebraically and expressing the
sine function as the difference of complex exponentials we can obtain the ana-
lytic expression of the impulse response
h(n) =
Gr
n
sin
0
sin (
0
n +
0
) .
(46)
The impulse response is depicted in fig. 19, which shows that a resonant filter can
be interpreted in the time domain as a damped oscillator with a characteristic
frequency that corresponds to the phase of the poles in the complex plane.
0
50
100
-0.1
0
0.1
time [samples]
h
Figure 19: Impulse response of a second-order resonant filter
7
The reader is invited to work out the expression (46).