Digital Signal Processing Reference
In-Depth Information
The above M
ATLAB
code results in the following values for the coefficients
of
H
(
z
):
numz = [0 0.4023 0]
and
denumz = [1 -0.8475 0.2786],
which correspond to the following transfer function:
H
(
z
)
=
0
.
4023
z
z
2
−
0
.
8475
z
+
0
.
2786
.
The above expression is the same as the one obtained analytically, and it is
included in row 1 of Table 16.1.
16.2.2 Look-up table
Examples 16.1 and 16.2 present direct methods to compute the impulse response
h
[
k
], or correspondingly the transfer function
H
(
z
), of the DT filter by sampling
the impulse response
h
(
t
) of an analog filter. The process can be simplified
further in cases where the transfer function
H
(
s
) of the analog filter is a rational
function. In such cases, the transfer function
H
(
s
) can be expressed in terms of
partial fractions as follows:
H
(
s
)
=
N
k
r
s
+ α
r
,
(16.21)
r
=
1
where
k
r
is the coefficient of the
r
th partial fraction. Applying the impulse
invariance transformation, Eq. (16.12), the transfer function
H
(
z
) of the digital
filter is given by
H
(
z
)
=
N
k
r
z
z
−
e
−α
r
T
.
(16.22)
r
=
1
Table 16.2 lists a number of commonly occurring s-domain terms and the
equivalent representation in the z-domain. We now list the steps involved in
the design of digital IIR filters using the impulse invariance transformation.
16.2.3 IIR filter design using impulse invariance transformation
The steps involved in designing IIR filters using the impulse invariance trans-
formation are as follows.
= ω
T
, transform the specifications of the digital filter from
the DT frequency
Ω
domain to the CT frequency
ω
domain. For convenience,
we choose
T
Step 1
Using
Ω
=
1.
Step 2
Using the analog filter techniques (see Chapter 7), design an analog
filter
H
(
s
) based on the transformed specifications obtained in step 1.
Step 3
Using the impulse invariance transformation specified in Eq. (16.12),
1
s
+ α
T
1
−
e
−α
T
z
−
1
zT
z
−
e
−α
T
im
pulse invaria
nce
←−
−→
or
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