This preview shows pages 1–3. Sign up to view the full content.
University of Minnesota
Dept. of Electrical and Computer Engineering
EE3015 – Signals and Systems
Discussion Session #12: Laplace Transform
1.
The following differential equation represents the dynamics of an LTI system:
).
(
3
)
(
2
)
(
)
(
2
2
t
x
t
y
dt
t
dy
dt
t
y
d
=

+
Where
)
(
3
)
(
2
t
u
e
t
x
t

=
is the input and
)
(
t
y
is the output, with initial conditions:
1
)
0
(
=

y
and
1
)
(
0
=

=
t
dt
t
dy
Find the natural and forced responses.
The Unilateral Laplace Transform of the equation above is given by:
)
(
3
)
(
2
)
0
(
)
(
)
(
)
0
(
)
(
0
2
s
X
s
Y
y
s
sY
dt
t
dy
sy
s
Y
s
t
=


+



=


We group terms that multiply the input
)
(
s
X
and the initial conditions:
( )
)
0
(
)
(
)
0
(
)
(
3
2
)
(
0
2

=

+
+
+
=

+

y
dt
t
dy
sy
s
X
s
s
s
Y
t
2
)
0
(
)
(
)
0
(
2
)
(
3
)
(
2
0
2

+
+
+
+

+
=

=


s
s
y
dt
t
dy
sy
s
s
s
X
s
Y
t
Forced resp.
Natural resp.
)
(
s
Y
f
)
(
s
Y
n
The Laplace transform of
)
(
s
X
is given by
)
2
(
3
)
(
+
=
s
s
X
. The forced response is given by
2
2
)
2
(
2
1
)
2
)(
1
(
9
)
(
+
+
+
+

=
+

=
s
C
s
B
s
A
s
s
s
Y
f
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document The residues
A, B, C
may be determined by placing the terms in the expansion over a common
denominator:
)
1
(
)
2
(
)
1
(
)
1
(
)
2
(
)
1
)(
2
(
)
1
(
)
2
(
)
2
(
)
2
(
2
1
2
2
2
2
2

+

+

+

+
+

+
+
=
+
+
+
+

s
s
s
C
s
s
s
s
B
s
s
s
A
s
C
s
B
s
A
Equating the coefficient of each power of s in the numerator on the right hand side of this
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 12/20/2009 for the course EE 3015 taught by Professor Staff during the Fall '08 term at Minnesota.
 Fall '08
 Staff

Click to edit the document details