1
EE102 Spring 200910
Lee
Systems and Signals
Homework #2
Due: Tuesday, April 20, 2010 at 5 PM.
1. State whether the following systems are linear or nonlinear; time invariant or time variant;
and why.
(a)
y
(
t
) =
x
(
t
) sin(
ω
t
+
φ
)
(b)
y
(
t
) =
x
(
t
)
x
(
t

1)
(c)
y
(
t
) = 1 +
x
(
t
)
(d)
y
(
t
) = cos(
ω
t
+
x
(
t
))
(e)
y
(
t
) =
t
∞
x
(
τ
)
d
τ
(f)
y
(
t
) =
t/
2
∞
x
(
τ
)
d
τ
2. A periodic signal
x
(
t
)
, with a period
T
, is applied to a linear, timeinvariant system
H
.
Show that the output
y
(
t
)
y
(
t
) =
H
(
x
(
t
))
is also periodic, with period
T
.
3.
Sample and hold system.
A sample and hold (S/H) system, with sample time
h
, is described
by
y
(
t
) =
x
(
h t/h
)
, where
a
denotes the largest integer that is less than or equal to
a
.
Sketch an input and corresponding output signal for a S/H, to illustrate that you under
stand what it does.
Is a S/H system linear?
4. Consider a system that takes a signal
x
(
t
)
and returns the even part of
x
(
t
)
as it’s output
x
e
(
t
) =
H
(
x
(
t
))
where
x
e
(
t
)
is the even part of
x
(
t
)
. Is this system linear? Is it time invariant?
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5. Write the following signals as a combination (sums or products) of scaled and shifted unit
rectangles rect
(
t
)
and unit triangles
Δ
(
t
)
.
2
1
1
0
2
1
2
2
1
x(t)
a)
2
1
1
0
2
1
2
2
1
t
x(t)
b)
6.
System Equations from Block Diagrams
Find the differential equation that correspond to this
block diagram.
Hint:
Label all of the intermediate signals, and write equations for each.
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 Spring '09
 Levan
 matlab, Complex Numbers, Complex number

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