UNIVERSITY OF ILLINOIS AT URBANACHAMPAIGN
Department of Electrical and Computer Engineering
ECE 410
Digital Signal Processing
Homework 6 Solutions
Prof. Bresler, Prof. Jones
October 8, 2008
Problem 1
(30 points)
Determine the twosided z transform of each of the following sequences, if it exists. Include
with your answer the region of convergence in the zplane. Also specify, in each case, if the Discrete
Time Fourier Transform of the sequence exists.
(a)
x
[
n
] = 0
.
5
n
u
[
n
]

2
n
u
[

n

1]
(b)
x
[
n
] = 2
n
u
[
n
] + 0
.
5
n
u
[

n

1]
(c)
x
[
n
] = 2
n
u
[
n
] + 2(3
n
)
u
[

n

1]
(d)
x
[
n
] = 2(0
.
5
n
)
u
[
n
+ 2] + 0
.
8
n
u
[
n
]
(e)
x
[
n
] = 0
.
5
n
u
[
n
]

0
.
8
n
u
[

n
]
(f)
x
[
n
] = 2
n
(
u
[
n
]

u
[
n

10])
Solution:
We begin by deriving some elementary ztransforms:
•
w
[
n
] =
a
n
u
[
n
]
⇒
W
(
z
)
=
∞
X
n
=
∞
w
[
n
]
z

n
=
∞
X
n
=0
a
n
z

n
=
1
1

a
z
=
z
z

a
,
ROC:
fl
fl
fl
a
z
fl
fl
fl
<
1
⇔ 
z

>

a

•
Similarly,
v
[
n
] =
b
n
u
[

n

1]
⇒
V
(
z
)
=
∞
X
n
=
∞
v
[
n
]
z

n
=

1
X
n
=
∞
b
n
z

n
=
∞
X
m
=1
b

m
z
m
=
z
b
1

z
b
=

z
z

b
,
ROC:
fl
fl
fl
z
b
fl
fl
fl
<
1
⇔ 
z

<

b

1
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•
Also recall the shift property of ztransforms
y
[
n
] =
x
[
n

n
0
]
⇒
Y
(
z
) =
z

n
0
X
(
z
)
With those preliminaries, and using the linearity of ztransforms:
(a)
x
[
n
] = 0
.
5
n
u
[
n
]

2
n
u
[

n

1]
X
(
z
) =
z
z

0
.
5
+
z
z

2
,
ROC: 0
.
5
<

z

<
2
(b)
x
[
n
] = 2
n
u
[
n
] + 0
.
5
n
u
[

n

1]
X
(
z
) =
z
z

2

z
z

0
.
5
,
ROC:
{
2
<

z
} ∩ {
z

<
0
.
5
}
=
∅
i.e.
, the ztransform of the sequence does not exist.
(c)
x
[
n
] = 2
n
u
[
n
] + 2(3
n
)
u
[

n

1]
X
(
z
) =
z
z

2

2
z
z

3
,
ROC: 2
<

z

<
3
(d)
x
[
n
] = 2(0
.
5
n
)
u
[
n
+ 2] + 0
.
8
n
u
[
n
]
x
[
n
]
=
8(0
.
5
n
+2
)
u
[
n
+ 2] + 0
.
8
n
u
[
n
]
⇒
X
(
z
)
=
8
z
·
z
2
z

0
.
5

z
z

0
.
8
,
ROC:
{
0
.
5
<

z
} ∩ {
0
.
8
<

z
}
= 0
.
8
<

z

(e)
x
[
n
] = 0
.
5
n
u
[
n
]

0
.
8
n
u
[

n
]
x
[
n
]
=
0
.
5
n
u
[
n
]

(0
.
8)0
.
8
n

1
u
[

(
n

1)

1]
⇒
X
(
z
)
=
z
z

0
.
5
+
0
.
8
z

0
.
8
,
ROC: 0
.
5
<

z

<
0
.
8
(f)
x
[
n
] = 2
n
(
u
[
n
]

u
[
n

10])
X
(
z
)
=
∞
X
n
=
∞
x
[
n
]
z

n
=
9
X
n
=0
2
n
z

n
=
10
,
z
= 2
1

2
10
z

10
1

2
z
=
z

9
(
z
10

1024
)
z

2
,
z
6
= 2
Note:
If we use knowledge of the ztransform of 2
n
u
[
n
], and the linearity and shift properties
of ztransforms, we may
think
that the ROC for the ztransform of 2
n
(
u
[
n
]

u
[
n

10]) is

z

>
2. However, that is not the case since
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