Problem 1
Suppose that a discretetime signal
[ ]
x n
is given by the formula:
[ ]
2.2cos(0.3
/3)
x n
n
π
π
=

and that it was obtained by sampling a continuoustime signal
0
( )
cos(2
)
x t
A
f t
π
φ
=
+
at a sampling rate of
6000samples/sec
s
f
=
. Determine three different continuoustime
signals that could have produced
[ ]
x n
. These continuoustime signals should all have
a frequency less than
8kHz
. Write the mathematical formula for all three.
Solution:
[ ]
2.2cos(0.3
/3)
x n
n
π
π
=

Compare to
0
0
(
)
cos(2
)
2
0.3 ,
s
s
s
n
n
x
A
f
f
f
f
or
or
f
π
ϕ
π
π
π
π
π
π
=
+
⇒
=
0.3
+ 2 ,
0.3
 2
Solve:
0
0
2
0.3
0.3
(
)
6000
0.15
900Hz
2
( )
2.2cos(1800
/3)
s
s
f
f
f
f
x t
t
π
π
π
π
=
⇒
=
=
×
=
→
=

Then
0
0
2
2.3
2.3
(
)
6900Hz
2
( )
2.2cos(2
(6900)
/3)
s
s
f
f
f
f
x t
t
π
π
π
π
=
⇒
=
=
→
=

Finally
0
0
2
1.7
1.7
(
)
5100Hz
2
( )
2.2cos(2
( 5100)
/3)
( )
2.2cos(2
(5100)
/3)
s
s
f
f
f
f
x t
t
x t
t
π
π
π
π
π
π

= 
⇒
=
= 
=


→
=
+
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200
600
200
600
f(Hz)
2
3
8
j
e
π

2
1
8
j
e
π
2
3
8
j
e
π
2
1
8
j
e
π

Problem 2
(a) Draw the spectrum of
)
400
(
sin
)
(
3
t
t
x
π
=
. Label the frequencies and complex
amplitudes of each component.
(b) Determine the minimum sampling rate that can be used to sample
)
(
t
x
without
any aliasing. Sketch the spectrum of the corresponding discrete time signal.
Solution:
(a)
(
29
t
j
t
j
t
j
t
j
t
j
t
j
e
e
e
e
j
j
e
e
t
x
π
π
π
π
π
π
1200
400
400
1200
3
400
400
3
3
8
1
2
)
(




+


=

=
(b)
z
s
high
s
H
f
f
f
1200
2
≥
⇒
≥
For
z
s
H
f
1200
=
,
s
f
T
s
s
)
1200
/
1
(
/
1
=
=
Then
(
29
(
29
(
29
3
/
3
/
3
/
3
/
1200
400
400
1200
3
3
8
3
3
8
1
3
3
8
1
)
(
]
[
n
j
n
j
n
j
n
j
n
j
n
j
nT
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 Fall '10
 ShuHungLeung
 Digital Signal Processing, Signal Processing, Cos, sampling rate

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