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3.7
x
(
t
) = 2 +
1
2
cos(
t
+ 45
◦
) + 2 cos(3
t
)

2 sin(4
t
+ 30
◦
)
(a)
x
(
t
) =
+
∞
∑
n
=
∞
X
[
n
]
e
jnω
0
t
ω
0
=??
cos(
t
+ 45
0
)
isperiodicwithT
1
=
2
π
1
= 2
π
cos(3
t
)
isperiodicwithT
2
=
2
π
3
sin(4
t
+ 30
o
)
isperiodicwithT
3
=
2
π
4
=
π/
2
T
=
LCM
(
T
1
,T
2
,T
3
)
= 2
π
The period of
x
(
t
) = 2
π
⇒
ω
0
=
2
π
2
π
= 1
C
0
= 2 (since
C
0
x
(
t
) can be represented in complex exponential form using Euler’s identity,
x
(
t
) = 2 +
1
4
h
e
j
(
ω
0
t
+45
0
)
+
e

j
(
ω
0
t
+45
0
)
i
+
h
e
j
3
ω
0
t
+
e

j
3
ω
0
t
i
+
j
h
e
j
(4
ω
0
t
+30
0
)

e

j
(4
ω
0
t
+30
0
)
i
whereω
0
= 1
⇒
x
(
t
) = 2 +
1
4
e
j
45
0
e
jω
0
t
+
1
4
e

j
45
0
e

jω
0
t
+
e
j
3
ω
0
t
+
e

j
3
ω
0
t
+
je
j
30
o
e
j
4
ω
0
t

je

j
30
o
e

j
4
ω
0
t
⇒
X
[0] = 2
,X
[1] =
1
4
e
j
45
◦
,X
[

1] =
X
[1]
*
=
1
4
e

j
45
0
,X
[3] = 1
,X
[

3] =
X
[3]
*
= 1
,X
[4] =
je
j
30
0
,X
[

4] =
X
[4]
*
=

je

j
30
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This note was uploaded on 04/03/2008 for the course EECS 216 taught by Professor Yagle during the Fall '08 term at University of Michigan.
 Fall '08
 Yagle

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