UNIVERSITY OF ILLINOIS AT URBANACHAMPAIGN
Department of Electrical and Computer Engineering
ECE 410
Digital Signal Processing
Quiz 2
Thursday, September 18, 2008
Student Name:
Prof. Bresler, Prof. Jones
INSTRUCTIONS
•
You may not use any calculators, cell phones, earphones (or other forms of electronic media)
for this quiz. You may use one side of one sheet of handwritten notes.
•
Show all your work to get full credit for your answers.
•
When you are asked to “
calculate
”, “
determine
or “
find
”, this means providing closedform
expressions, without summation or integral expressions.
Problem 1
(20 points)
Let
X
d
(
ω
) =
(
0

ω
 ≤
ω
c
1
ω
c
<

ω

< π
where 0
< ω
c
< π
. Determine
x
[
n
], the inverse DTFT of
X
d
(
ω
). (Specify
x
[
n
] for all
n
∈
Z
and
show your work step by step for complete credit.)
Solution:
x
[
n
] =
1
2
π
Z
π

π
X
d
(
ω
)
·
e
jωn
dω
=
1
2
π
•Z

ω
c

π
e
jωn
dω
+
Z
π
ω
c
e
jωn
dω
‚
Now for
n
= 0, we have
x
[0] =
2(
π

ω
c
)
2
π
= 1

ω
c
π
(1)
And, for
n
6
= 0, we compute
x
[
n
]
=
1
2
πjn
e
jωn
fl
fl

ω
c

π
+
1
2
πjn
e
jωn
fl
fl
π
ω
c
=
1
2
πjn
h
e

jω
c
n

»
»
»
e

jπn
+
'
''
e
jπn

e
jω
c
n
i
=

ω
c
π
sinc(
ω
c
n
)
,
n
6
= 0
(2)
Combining (1) and (2) we have
x
[
n
] =
δ
[
n
]

ω
c
π
sinc(
ω
c
n
)
1
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Alternative approach 1:
Recall the modulating property of DTFT, using which we can show that
x
[
n
] = 2
y
[
n
]
·
cos
π
+
ω
c
2
n
¶
where
Y
d
(
ω
) =
(
1
,

ω
 ≤
π

ω
c
2
0
,
otherwise
↔
y
[
n
] =
π

ω
c
2
π
sinc
π

ω
c
2
n
¶
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 Fall '09
 Digital Signal Processing, Signal Processing, DFT, DFT indices, pt DFT

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