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ECE220
Signals and Information
Spring 2008
Homework 1 Solutions
by Tae Eung Sung
Problem 1
(15 pts)
The general representation for a cosine function is as follows:
))
(
cos(
)
2
cos(
)
(
0
0
s
t
t
A
t
f
A
t
x
−
=
+
=
ω
φ
π
where
0
f
(Hz),
0
(rad/s),
A
,
and
s
t
(sec) are the cyclic frequency, radian frequency, amplitude, phase
shift
and time shift, respectively.
1)
438
0
=
f
,
876
0
=
,
8
.
4
=
A
,
128
=
,
112128
1
−
=
s
t
2)
Since
)
2
cos(
)
sin(
−
+
=
+
t
t
, then
))
120
1
(
20
cos(
25
.
0
)
2
3
10
2
cos(
25
.
0
)
(
−
=
−
+
=
t
t
t
x
.
Hence,
10
0
=
f
,
20
0
=
,
25
.
0
=
A
,
6
−
=
and
120
1
=
s
t
3)
Since
2
)
1
)(
1
(
=
+
−
j
j
and
ϑ
2
cos
1
cos
2
2
+
=
, then
))
8192
1
(
8192
cos(
2
)
4096
2
cos(
2
)
8192
cos(
2
)))
4096
2
cos(
1
(
1
(
2
)
(
+
=
+
=
−
=
⋅
+
−
=
t
t
t
t
t
x
.
Hence,
4096
0
=
f
,
8192
0
=
,
2
=
A
,
=
and
8192
1
−
=
s
t
Problem 2
(10 pts)
1
0.8
0.6
0.4
0.2
0
0.2
0.4
0.6
0.8
1
0.5
0
0.5
1
1.5
2
time (sec)
x(t),y(t),z(t)
Plot for Problem 2
x(t)
y(t)
z(t)
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Problem 3
(16 pts)
By Euler’s formula
ϑ
sin
cos
j
e
j
+
=
,
1)
5
3
5
3
5
5
5
3
5
4
5
2
6180
.
1
)
5
cos(
2
)
(
π
j
j
j
j
j
j
j
e
e
e
e
e
e
e
=
=
+
=
+
−
.
2)
)
5054
.
0
(
)
14
15
2
arctan(
2
16
15
4
196
15
2
14
)
15
(
−
−
=
⋅
+
=
−
=
−
j
j
e
e
j
j
.
3)
Since
)
100
2
cos(
50
t
always
takes real value for all
]
2
,
0
[
∈
t
, the imaginary part of it must be 0.
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This note was uploaded on 06/25/2008 for the course ECE 2200 taught by Professor Johnson during the Spring '05 term at Cornell University (Engineering School).
 Spring '05
 JOHNSON

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