Digital Signal Processing Reference
In-Depth Information
Similarly, the M
ATLAB
code for transforming the fourth-order Butterworth
filter is given by
>>fs=1; %fs=1/T=1
>> nums = [6.2902]; % numerator of the CT filter
>> denums = [1 4.1383 8.5603 10.3791 6.2902];
% denominator of CT filter
>> [numz,denumz] = impinvar (nums,denums,fs);
% coefficients of the DT filter
which returns the following values:
numz = [0 0.3298 0.4276 0.0428]
denumz = [1 -0.4977 0.3961 -0.1197 0.0159].
The transfer function of the fourth-order IIR filter is given by
0
.
3298
z
3
+
0
.
4276
z
2
+
0
.
0428
z
z
4
−
0
.
4977
z
3
+
0
.
3958
z
2
−
0
.
1197
z
+
0
.
0159
.
H
(
z
)
=
The above expression is similar to the one obtained in Example 16.3 for the
fourth-order Butterworth filter.
16.2.4 Limitations of impulse invariance method
As illustrated in Example 16.3, the impulse invariance method introduces alias-
ing while transforming an analog filter to a digital filter. Since the analog fil-
ter is not band-limited, the impulse invariance transformation would always
introduce aliasing in the digital domain. Therefore, a higher-order DT filter is
generally required to satisfy the design constraints. Section 16.3 introduces a
second transformation, known as the bilinear transformation, to eliminate the
effect of aliasing.
16.3 Bilinear trans
formation
The bilinear transformation provides a one-to-one mapping from the s-plane to
the z-plane. The mapping equation is given by
=
k
z
−
1
s
z
+
1
,
(16.23)
where
k
is the normalization constant given by 2
/
T
, where
T
is the sampling
interval. To derive the frequency characteristics of the bilinear transformation,
we substitute
z
=
exp( j
Ω
) and
s
=
j
ω
in Eq. (16.23). The resulting expression
Search WWH ::
Custom Search