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1
EE 483 – Introduction to Digital Signal Processing
Spring 2009
Midterm Examination – February 25, 2009
1.
(10 Points)
Consider the following finite-length sequences:
(i)
h
[
n
],
±
M
²
n
²
N
,
and (ii)
g
[
n
],
K
±
n
±
N
,
where
M
,
N
and
K
are positive integers with
K
<
N
. Define
(a)
y
1
[
n
]
=
h
[
n
]O
*
h
[
n
],
(b)
y
2
[
n
]
=
h
[
n
]O
*
g
[
n
],
(c)
y
3
[
n
]
=
g
[
n
]O
*
g
[
n
].
What are the lengths of each of the above convolved sequences? What are the ranges of the
index
n
for which each of the above convolved sequences are defined?
2.
(15 Points)
Let
X
(
e
j
±
)
denote the DTFT of a discrete-time sequence
x
[
n
].
(a) Express the DTFT of
(
±
1)
n
x
[
n
]
in terms of
X
(
e
j
)
.
(b) Express
(
n
=
±²
²
³
±
1)
n
x
[
n
]
in terms of
X
(
e
j
)
.
(c) Let
Y
(
e
j
)
=
X
(
e
j
)
+
X
(
e
j
(
²
³
)
).
Show that
Y
(
e
j
)
depends only on the even-indexed
samples
x
[2
m
]
and not on the odd-indexed samples
x
[2
m
+
1].
3.

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