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ECEN 314
Quiz # 2
Georghiades
November 3, 2009
Note:
Quiz is openbook, 15 minutes long.
1. Let
x
(
t
) =
tu
(
t
) and
y
(
t
) =
e

t
u
(
t
)
.
Compute the Laplace transform of the following function:
z
(
t
) =
x
(
t
)
*
y
(
t
)
.
2. Compute the inverse Laplace transform,
h
(
t
), of
H
(
s
) =
1
s
2
+ 2
s
+ 2
.
Solution
1. Convolution in the timedomain is multiplication in the frequencydomain. Thus,
Z
(
s
) =
X
(
s
)
Y
(
s
)
.
We have
X
(
s
) =

d
ds
1
s
=
1
s
2
,
and
Y
(
s
) =
1
s
+ 1
.
Thus,
Z
(
s
) =
1
s
2
(
s
+ 1)
.
To compute
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Unformatted text preview: z ( t ), we perform partialfraction expansion: Z ( s ) = 1 s 2 + 2 s + 2 = A s + B s 2 + C s + 1 =1 s + 1 s 2 + 1 s + 1 . Thus, z ( t ) = (1 + t + et ) u ( t ) . 2. We have H ( s ) = 1 s 2 + 2 s + 2 = 1 ( s + 1) 2 + 1 , and therefore from the frequency translation property h ( t ) = et sin( t ) u ( t ) . 1...
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This note was uploaded on 01/01/2012 for the course ECEN 314 taught by Professor Halverson during the Spring '08 term at Texas A&M.
 Spring '08
 HALVERSON
 Laplace

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