Solved Laplace Transform Problems in Signals and Systems

1. Dr.K.G.SHANTHI
Professor/ECE
shanthiece@rmkcet.ac.in
RMK College of Engineering and Technology

2. 2
1)
Solution
2)
Solution
2
Impulse signal
L[δ(t)]
Step signal
L[u(t)]

3. 3
3)Determine the Laplace Transform of the following continuous time
signal and their ROC
0
)
(
)
( 
= t
t
u
A
t
x
dt
e
t
u
A
s
X
t
x
L t
s
−


−

=
= )
(
)
(
]
)
(
[
dt
e
A
s
X t
s
−


=
0
)
(
 
0
.
.
0
)
( s
s
t
s
e
e
s
A
s
e
A
s
X −

−

−
−
−







−
=
 
s
A
s
X
s
A
s
X
=
−
−
=
)
(
1
0
)
(
s-Plane
1
0
ROC as all points in s-plane to the right of line passing through 0
=


4. 4
4)
Solution
5)
Solution
4
Constant
Exponentialsignal

5. 6.Determine the Laplace Transform of the following continuous time
signal and their ROC
0
)
(
)
( 
= t
t
u
t
t
x
dt
e
t
u
t
s
X
t
x
L t
s
−


−

=
= )
(
)
(
]
)
(
[
dt
e
t
s
X
t
x
L t
s
−


=
=
0
)
(
]
)
(
[  
−
= du
v
v
u
dv
u
s
e
v
dt
e
dv
dt
e
dv
t
s
t
s
t
s
−
=
=
=
−
−
−


( )
0
2
0
0
.
.
1
)
(
1
.
0
)
(
s
s
t
s
s
s
e
e
s
s
X
s
e
s
s
e
s
e
s
X
−

−

−
−

−
−
−
=






−
+






−
−
−

=
2
1
)
(
s
s
X =

 −
−






−
−
−
= 
0
0
)
( dt
s
e
s
e
t
s
X
t
s
t
s
0 1
s-Plane
ROC as all points in s-plane to the right of line passing through 0
=


6. 6
7.Determine the Laplace Transform of the following
continuous time signal and their ROC
t
a
e
t
x
−
=
)
(
dt
e
e
dt
e
e st
at
st
at

 
−
−

−
−
+
=
0
0
( ) ( )
dt
e
dt
e t
a
s
t
a
s

 
−
+
−

+
−
+
=
0
0
( )
( )
( )
( )
0
0
)
(

−
+
−

+
−






+
−
+






+
−
=
a
s
e
a
s
e
s
X
t
a
s
t
a
s
( ) ( )





+
−
−
+






+
−
−
=
−
−
a
s
e
e
a
s
e
e 0
0
( ) ( ) ( ) ( ) 2
2
2
1
1
1
1
)
(
a
s
a
a
s
a
s
a
s
a
s
s
X
−
−
=
−
−
+
=
+
−
+
+
=
s-Plane
      dt
e
e
dt
e
e
dt
e
t
x
t
x
L st
t
a
st
at
st


 
−
−
−
−

−
−


−
−
+
=
=
0
)
(
0
)
(
)
(

7. 7
8.Determine the Laplace Transform of the following continuous time signal
)
(
sin
)
( t
u
t
A
t
x 
=

8. 8
9.Determine the Laplace Transform of the following continuous time signal
)
(
sin
)
( t
tu
e
t
x at

= −
dt
e
j
e
e
e
s
X st
t
j
t
j
at −
 
−

−
 






 −
=
0
2
)
(
  dt
e
t
tu
e
dt
e
t
x
t
x
L st
at
st




−
−
−


−
−

=
= )
(
sin
)
(
)
(






−
= −


−
−
−


−

 dt
e
e
e
dt
e
e
e
j
st
t
j
at
st
t
j
at
0
0
2
1
( ) ( )
dt
e
j
dt
e
j
t
j
a
s
t
j
a
s




+
+
−


−
+
−
−
=
0
0
2
1
2
1
( )
( )
( )
( )


+
+
−


−
+
−







+
+
−
−







−
+
−
=
0
0
2
1
2
1
)
(
j
a
s
e
j
j
a
s
e
j
s
X
t
j
a
s
t
j
a
s
( ) ( )






+
+
−
−
−







−
+
−
−
=
−
−
j
a
s
e
e
j
j
a
s
e
e
j
0
0
2
1
2
1
( ) ( )






+
+
−

−
+
−
−
=
j
a
s
j
a
s
j
1
1
2
1
𝑋(𝑠) =
1
2𝑗
2𝑗Ω
𝑠 + 𝑎 2 + Ω2 =
Ω
𝑠 + 𝑎 2 + Ω2

9. 9