1
EE 483 – Introduction to Digital Signal Processing
Spring 2009
Homework #7 Solutions
7.1
+
+
+
+
+
=
+
+
+
+
=
-
-
-
-
-
-
-
-
2
1
2
1
2
1
2
1
4
4
2
5
5
2
4
1
2
1
4
2
5
5
.
4
2
5
.
4
)
(
z
z
z
z
z
z
z
z
z
H
(
29
)
(
)
(
2
1
1
0
z
A
z
A
+
=
, where
1
)
(
0
=
z
A
and
2
1
2
1
1
4
2
5
5
2
4
)
(
-
-
-
-
+
+
+
+
=
z
z
z
z
z
A
are stable allpass transfer functions.
Now
H
4
(
e
j
ϖ
)
2
=
1
4
{
A
0
(
e
j
ϖ
)
A
0
*
(
e
j
ϖ
)
+
A
1
(
e
j
ϖ
)
A
0
*
(
e
j
ϖ
)
+
A
0
(
e
j
ϖ
)
A
1
*
(
e
j
ϖ
)
+
A
1
(
e
j
ϖ
)
A
1
*
(
e
j
ϖ
)
}
. Since
)
(
0
z
A
and
)
(
1
z
A
are allpass functions,
A
0
(
e
j
ϖ
)
=
e
j
φ
0
(
ϖ
)
and
A
1
(
e
j
ϖ
)
=
e
j
φ
1
(
ϖ
)
.
Therefore,
H
4
(
e
j
ϖ
)
2
=
1
4
2
+
e
j
(
φ
0
(
ϖ
)
-
φ
1
(
ϖ
))
+
e
-
j
(
φ
0
(
ϖ
)
-
φ
1
(
ϖ
))
{
}
≤
1
as maximum values of
e
j
(
φ
0
(
ϖ
)
-
φ
1
(
ϖ
))
and
e
-
j
(
φ
0
(
ϖ
)
-
φ
1
(
ϖ
))
are
.
1
H
4
(
z
)
is stable since
)
(
0
z
A
and
)
(
1
z
A
are
stable allpass functions.
Hence,
H
4
(
z
)
is BR.
7.2
. Labeling
1
V
the output variable of the top multiplier connected to input
1
X
we then analyze
Figure P7.10(b) and obtain
2
1
1
1
2
1
1
1
),
(
X
z
V
Y
X
z
X
k
V
m
-
-
+
=
-
=
,
)
1
(
2
2
X
k
X
k
m
m
+
-
=
and
.
)
1
(
1
2
1
1
1
2
X
k
X
k
z
X
V
Y
m
m
+
+
=
+
=
-
Hence, the transfer matrix of the two-pair is given by
τ
(
.
1
1
1
1
-
+
-
=
-
-
z
k
k
z
k
k
m
m
m
m
Using Eq. (7.128b) we then arrive at the chain matrix
Γ
.
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