1
EE102 Spring 200910
Lee
Systems and Signals
Homework #1
Due: Tuesday April 13, 2010 at 5 PM.
1. Find the even and odd decomposition of this signal:
1
2
0
1
2
t
1
2
x
(
t
)
2. Given the signal
x
(
t
)
shown below
2
1
0
1
2
1
1
t
x
(
t
)
draw the following signals:
(a)
x
(2(1

t
))
(b)
x
(
t
2

1
)
3. In class we showed that any signal can be written as the sum of an evan and odd compo
nent,
x
(
t
) =
x
e
(
t
) +
x
o
(
t
)
.
Show that the energy of x(t) is the sum of the energies of the even and odd components
∞
∞
x
2
(
t
)
dt
=
∞
∞
x
2
e
(
t
)
dt
+
∞
∞
x
2
o
(
t
)
dt.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
4.
Periodic Signals
(a) Assume that the signal
x
(
t
)
is periodic with period
T
0
, and that
x
(
t
)
is odd (
i.e.
x
(
t
) =

x
(

t
)
). What is the value of
x
(
T
0
)
?
(b) Two continuoustime sequences
x
1
(
t
)
and
x
2
(
t
)
are periodic with periods
T
1
and
T
2
.
Find values of
T
1
and
T
2
such that
x
1
(
t
)
+
x
2
(
t
)
is aperiodic.
5.
Power and Energy Signals
Plot these signals, and classify them as energy or power signals. Support your classification
with an explicit calculation or an argument. In each case
∞
< t <
∞
.
(a)
x
(
t
) =
e

2

t

(b)
x
(
t
) =
1
√
t
t
≥
1
0
t <
1
(c)
x
(
t
) =
0
t
≥
0
e
t
t <
0
(d)
x
(
t
) =
e


t

6.
Review of Complex Numbers
(a) Simplify the following expression
e
i
(
ω
t
+
φ
)
1 +
j
(1

j
)
and leave the result in polar form.
(b) Simplify
(cos
ω
t
+
j
sin
ω
t
) (cos 2
ω
t

j
sin 2
ω
t
)
and leave the result in polar form.
(c) Use the relationship
cos
θ
=
1
2
e
j
θ
+
e

j
θ
to determine how many frequency components there are in the signal
s
(
t
) = (cos
ω
t
)
2
.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '09
 Levan
 matlab, Oscilloscope, Lissajous Curve

Click to edit the document details