The impulse-invariant design is usually not performed directly in the form of (6.3.5).
In practice, the transfer function of an analog filter H(s) is first expanded into a partial-
fraction form
Hs
P
i1
c
i
s s
i
,
6:3:6
where s Às
i
is the pole of H(s), and c
i
is the residue of the pole at Às
i
. Note that we
have assumed there are no multiple poles. Taking the inverse Laplace transform of
(6.3.6) yields
ht
P
i1
c
i
e
Às
i
t
, t ! 0,
6:3:7
which is the impulse response of the analog filter H(s).
The impulse response samples are obtained by setting t equal to nT. From (6.3.5) and
(6.3.7), we have
hn
P
i1
c
i
e
Às
i
nT
, n ! 0,
6:3:8
The z-transform of the sampled impulse response is given by
Hz
I
n0
hnz
Àn
P
i1
c
i
I
n0
e
Às
i
T
z
À1
n
P
i1
c
i
1 À e
Às
i
T
z
À1
:
6:3:9
The impulse response of H(z) is obtained by taking the inverse z-transform of (6.3.9).
Therefore the filter described in (6.3.9) has an impulse response equivalent to the
sampled impulse response of the analog filter H(s) defined in (6.3.6). Comparing
(6.3.6) with (6.3.9), the parameters of H(z) may be obtained directly from H(s) without
bothering to evaluate h(t) or h(n).
The magnitude response of the digital filter will be scaled by f
s
1=T due to the
sampling operation. Scaling the magnitude response of the digital filter to approximate
magnitude response of the analog filter requires the multiplication of H(z) by T.
The transfer function of the impulse-invariant digital filter given in (6.3.9) is modified
as
Hz T
P
i1
c
i
1 À e
Às
i
T
z
À1
:
6:3:10
The frequency variable ! for the digital filter bears a linear relationship to that for the
analog filter within the operating range of the digital filter. This means that when !
varies from 0 to p around the unit circle in the z-plane, V varies from 0 to p=T along the
jV-axis in the s-plane. Recall that ! VT as given in (3.1.7). Thus critical frequencies
DESIGN OF IIR FILTERS
257