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UNIVERSITY OF ILLINOIS AT URBANACHAMPAIGN
Department of Electrical and Computer Engineering
ECE 410
Digital Signal Processing
Homework 4 Solution
Prof. Bresler, Prof. Jones
September 24, 2008
Problem 1
(10 points)
Complete the following signal ﬂow diagram for 8point, radix2 decimationintime FFT algorithm,
and use it to compute the DFT of the 8pt sequence
x
[
n
] =
{
1
,

4
,

2
,

3
,
1
,
4
,
2
,
3
}
by hand
.
Specify the value of the input sequence, output sequence, and the signal values after each stage,
i.e.
, at all the points marked by the solid circles in the diagram.
Note:
W
0
2
= 1;
W
0
4
= 1;
W
1
4
=

j
;
W
0
8
= 1;
W
1
8
=
1
√
2

j
1
√
2
;
W
2
8
=

j
;
W
3
8
=

1
√
2

j
1
√
2
Refer Figure 1.
Problem 2
(10 points)
The sequence
x
[
n
] = cos
±
π
8
n
²
,
∞
< n <
∞
was obtained by sampling a continuoustime signal
x
c
(
t
) = cos(Ω
0
t
)
,
∞
< t <
∞
at a sampling rate of 1024 samples/sec. What are three possible values of Ω
0
that can result in the
same sequence of
x
[
n
]?
Since
w
= Ω
T
and
T
=
1
1024
secs, the signal frequency is
Ω
0
=
±
π
8
+ 2
πn
²
1
T
=
±
π
8
+ 2
πn
²
1024
= 128
π
+ 2048
πn
1
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where n is any integer.
Since cos function is symmetric, the following can be answers:
Ω
0
=

128
π
+ 2048
πn.
Therefore, the possible three signal frequencies could be
Ω
0
= 128
π,
2176
π,
4224
π,
when
n
= 0
,
1
,
2.
Problem 3
(20 points)
The continuoustime signal
x
a
(
t
) = cos(600
πt
)
is sampled with a sampling period
T
to obtain a discretetime signal
x
d
[
n
] =
x
a
(
nT
)
.
(a) Compute and sketch the magnitude of the continuoustime Fourier transform of
x
a
(
t
) and the
discretetime Fourier transform of
x
d
[
n
] for
T
= 1
ms
.
Continuoustime Fourier transform(CTFT) of
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