Digital Signal Processing Reference
In-Depth Information
⎛
⎜
⎞
⎟
()
=
Ω
j
ω
He
H j
T
(5.5)
c
through the frequency transformation given in Equation 5.2.
Bilinear transformation method
The bilinear transformation of the analog filter system function,
H
(
s
), yields
c
the corresponding digital filter system function,
H
(
z
), and is
obtained through
the transformation:
⎛
⎜
−
1
⎞
⎟
21
1
−
+
z
z
s
=
(5.6)
T
−
1
into Equation 5.6, and equating real and
imaginary parts of the resulting equation, the following frequency transfor-
mation is obtained:
Substituting
s
=
σ
+
j
ω
and
z
=
e
j
ω
2
(
)
Ω=
T
tan
ω
2
(5.7)
5.1.2.2
Analytical Techniques for FIR Filter Design
One of the most widely used methods of FIR digital filter design is the
window method
, which will be briefly explained below, in a series of steps.
Step 1: Specification of the desired filter response
H
d
(
e
j
ω
)
For example, a desired low-pass response is shown in Figure 5.4.
H(e
j
ω
)
1
ω,
rad./sec.
−ω
c
0
ω
c
FIGURE 5.4
Desired low-pass response of digital filter.
