Digital Signal Processing I (ECE 410, Summer II 2009)
University of Illinois at UrbanaChampaign
Electrical and Computer Engineering
Midterm I
J. W. Choi and P. Linden
1. (a) Noncausal since the output depends on the future input.
(b) Timeinvariant since
x
[
n

n
0
]
↔
∑
k
=1
h
[
k
]
x
[
n

n
0

k
] =
y
[
n

n
0
].
(c) Not stable since the inner most poles are
±
1
10
j
and hence the ROC does not include the unit
circle.
2. (a)
X
d
(0) =
∞
X
n
=
∞
x
[
n
]
e

jωn

ω
=0
=
∞
X
n
=
∞
x
[
n
]
=
∞
X
n
=0
±
1
√
2
¶
2
k
+
±

1
4
+
1
4
+
3
4
+
5
4
¶
=
1
1

1
2
+ 2
= 4
(b)
X
d
(
π
) =
∞
X
n
=
∞
x
[
n
]
e

jωn

ω
=
π
=
∞
X
n
=
∞
x
[
n
](

1)
n
=
∞
X
n
=0
±
1
√
2
¶
2
k

±

1
4
+
1
4
+
3
4
+
5
4
¶
= 2

2 = 0
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View Full Document(c) Using Parseval’s theorem
∞
X
n
=
∞

x
[
n
]

2
=
1
2
π
Z
π

π

X
d
(
ω
)

2
dω
∴
Z
π

π

X
d
(
ω
)

2
dω
= 2
π
∞
X
n
=
∞

x
[
n
]

2
= 2
π
∞
X
n
=
∞
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
±
1
√
2
¶
2
n
ﬂ
ﬂ
ﬂ
ﬂ
ﬂ
2
+
±
1
16
+
1
16
+
9
16
+
25
16
¶
= 2
π
ˆ
∞
X
n
=0
±
1
4
¶
n
+
1
16
9
4
#
= 2
π
±
3
4
+
9
4
¶
= 2
π
×
16 + 27
12
= 2
π
×
43
12
=
43
6
π
(d)
Z
π

π
Y
d
(
ω
)
dω
= 2
πy
[0]
(1)
To obtain
y
[0], we express the output
y
[
n
] as
y
[
n
] =
h
[
n
]
*
x
[
n
]
=
∞
X
k
=
∞
x
[
k
]
h
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 Spring '08
 Staff
 Digital Signal Processing, Signal Processing, Impulse response, k=0, J. W. Choi

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