Digital Signal Processing Reference
In-Depth Information
Exercise 7.9: IIR filtering.
Consider a causal IIR filter with the following
transfer function:
z
−
1
1
+
l
g
r
,
y
i
d
.
,
©
,
L
s
H
(
z
)=
z
−
1
0.64
z
−
2
1
−
1.6 cos
(
2
π/
7
)
+
(a) Sketch the pole-zero plot of the filter and the ROC of its transfer func-
tion.
(b) Sketch the magnitude of its frequency response.
(c) Draw at least two different block diagrams which implement the filter
(e.g. direct forms I and II).
(d) Compute the first five values (for
n
=
0,1,...,5) of the signal
y
[
n
]=
h
[
n
]
∗
x
[
n
]
,where
x
[
n
]=
δ
[
n
]+
2
δ
[
n
−
1
]
. Assume zero initial condi-
tions.
Exercise 7.10: Generalized linear phase filters.
Consider the filter
given by
H
(
)=
z
−
1
.
z
1
−
(a) Show that
H
(
z
)
is a generalized linear phase filter, i.e. that it can be
written as
H
e
(
j
ω
)
e
−j
(
ω
d−
α
)
e
j
ω
)=
H
(
(
Give the corresponding group delay
d
and the phase factor
α
.
(b) What type of filter is it (I, II, III or IV)? Explain.
(c) Give the expression of
h
[
n
]
and show that it satisfies
sin
ω
(
)+
α
=
h
[
n
]
n
−
d
0
n
for all
ω
.
(d) More generally, show that any generalized linear phase filter
h
[
n
]
must
satisfy
sin
ω
(
)+
α
=
h
[
n
]
n
−
d
0
n
for all
. The above expression is, thus, a necessary condition for a
filter to be generalized linear phase.
ω
