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ECEN 314
Solution to Quiz # 2
Georghiades
October 17, 2008
Note:
Quiz is open-book, open notes, 15 minutes long. Problems carry equal weight. The use of
calculators is NOT allowed.
1. Find the inverse Laplace transform of the following functions:
(a)
X
(
s
) =
(
s
+1)
(
s
+2)(
s
+4)
.
(b)
X
(
s
) =
(
s
+4)
(
s
+2)
2
+4
Solution
(a) Using partial fraction expansion:
X
(
s
) =
-
1
/
2
(
s
+ 2)
+
3
/
2
(
s
+ 4)
.
Therefore,
x
(
t
) =
bracketleftbigg
-
1
3
e
-
2
t
+
3
2
e
-
4
t
bracketrightbigg
u
(
t
)
.
(b) We have
X
(
s
) =
(
s
+ 4)
(
s
+ 2)
2
+ 4
=
(
s
+ 2)
(
s
+ 2)
2
+ 4
+
2
(
s
+ 2)
2
+ 4
.
Using the frequency translation property:
x
(
t
) =
bracketleftBig
e
-
2
t
cos(2
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Unformatted text preview: ) ² u ( t ) . 2. A linear, time-invariant system has the following transfer function: H ( s ) = ( s + 1) ( s + 2)( s + 3) . Find the steady-state response when the input is x ( t ) = 2cos( t + π/ 6) . Solution The output will be another sinusoidal signal at the same frequency as the input ( ω = 1). We have H ( jω ) = ( j + 1) ( j + 2)( j + 3) = 1 5 . Therefore: y ss ( t ) = 2 5 cos( t + π/ 6) . 1...
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- HALVERSON
- Laplace, #, 15 minutes
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