Digital Signal Processing Reference
In-Depth Information
h
[
k
]
0.144
0.134
0.12
k
W
023 4
1
5
6
7
8
9101112
13 14
15
16 17 18 19 20
−
p
−0.5
p
0.5
p
p
0
(a)
(b)
<
H
(
W
)
20 log
10
(|
H
(
W
)| )
−
p
−0.5
p
0
−20
0
0.5
p
−40
−60
W
W
p
p
−
p
0
−
p
0
−0.5
p
0.5
p
−0.5
p
0.5
p
(c)
(d)
Fig. 14.11. IIR filter in Example
14.3. (a) Impulse response
h
[
k
];
(b) magnitude spectrum
H
(
Ω
); (c) phase spectrum
<
H
(
Ω
); (d) magnitude spectrum
H
(
Ω
) in decibels.
Using Heaviside's partial fraction formula the coefficients of the partial fractions
k
1
and
k
2
are given by
0
.
12
(
z
−
0
.
8)(
z
−
0
.
4)
0
.
12
z
−
0
.
4
k
1
=
(
z
−
0
.
8)
=
=
0
.
3
z
=
0
.
8
z
=
0
.
8
and
0
.
12
(
z
−
0
.
8)(
z
−
0
.
4)
0
.
12
z
−
0
.
8
k
2
=
(
z
−
0
.
4)
=
=−
0
.
3
.
z
=
0
.
4
z
=
0
.
4
The partial fraction expansion of
H
(
z
) is therefore given by
+
−
0
.
3
z
z
−
0
.
4
Taking the inverse z-transform of
H
(
z
) yields
0
.
3
z
z
−
0
.
8
H
(
z
)
=
h
[
k
]
=
0
.
3[(0
.
8)
k
−
(0
.
4)
k
]
u
[
k
]
.
which is plotted in Fig. 14.11(a). Note that the IIR filter has infinite length, as
expected.
The Fourier transfer function of the IIR filter is obtained by substituting
z
=
exp(j
Ω
):
−
j
Ω
1
−
1
.
2e
−
j
Ω
+
0
.
32e
−
j2
Ω
.
The magnitude spectrum of the IIR filter is plotted in Figs. 14.11(b) and (d).
Since the gain of the filter is unity at low frequencies (around
Ω
≈
0) and
close to zero at high frequencies (around
Ω
≈ π
), the impulse response
h
[
k
]
0
.
12e
H
(
Ω
)
=
Search WWH ::
Custom Search