Digital Signal Processing Reference
In-Depth Information
•
f
is a vector of frequency points, specified in the range 0
≤
f
≤
1,
1.
•
H
is a vector containing the desired magnitude response at the points
specified in the vector
f
.
which corresponds to the digital frequency limit 0
≤
ω
≤
•
N
is the order of the filter.
Exercise 2: Conversion of analog to digital filters
There are two important methods of conversion of classical analog filter
response
H
(
s
) to corresponding digital filter response
H
(
z
):
the impulse invari-
ance method
and the
bilinear transformation.
The analog transfer function is given by
bs
m
+
b
s
m
−
1
+ …+
bs b
m
m
−
1
1
0
Hs
()
=
n
n
−
1
as
+
a
s
+ …
a
1
ssa
+
n
n
−
1
0
and the digital transfer function is given by
bz
−
m
+
b
z
−−
(
m
1
)
+ …
bz
−
1
+
b
m
m
−
1
1
0
Hz
()
=
−
n
−−−
(
n
1
)
−
1
az
+
a
z
+…
az
+
a
n
n
−
1
1
0
Transform the following second-order cascade lowpass analog filter into
digital filters (impulse invariance and bilinear methods):
Ω
Ω
2
c
Hs
()
=
2
⎣
s
+
⎦
c
where
c
is the 3 dB cutoff frequency of the analog filter.
Design the digital filter cutoff frequency at
Ω
ω
=
π
/
2 rad./sec., and sampling
c
frequency
f
s
= 10 Hz. Plot the magnitude response
H
(
e
j
ω
)
, -
π
≤
ω
≤
π
for both
bilinear and impulse invariance transformation.
Note:
The MATLAB commands are
bilinear
(for bilinear transformation) and
impinvar
(for impulse invariance). Type
help bilinear
or
help impinvar
,
after the MATLAB prompt >> for instructions on usage.
Exercise 3: Design of FIR filters using windowing method
Design a digital windowed bandpass FIR filter of order 7 with the following
desired frequency response:
()
=
j
ω
He
13
,
π
≤
w
≤
23
π
d
=
0
,otherwiseintheperiod (
−π π
,)
