In this slide Laplace transform is described, i.e conversion between time domain and frequency domain. Also zeros and poles are described.

1. LAPLACE TRANSFORM
CREATED BY:
BASSIT A. GOLRA
23:51:49 1

2. Laplace Transform
• Definition:
It takes function of a positive real
variable t (time) to a function of a complex
variable s ( complex frequency).
• Irrespective of whether stable or unstable
system
Source:
https://en.wikipedia.org/wiki/Laplace_transform
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3. 23:51:49 3

4. Laplace Continued …
• For continues time we use Laplace transform.
• For discrete time we use Z-Transform
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5. Laplace Continued …
• There are two types of Laplace transform:
i) Unilateral Laplace transform
ii) Bilateral Laplace transform
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6. Unilateral Laplace
• Unilateral Laplace Transform limits from zero
to positive infinity ( 0 to ∞ )
Bilateral Laplace
 Bilateral Laplace transform limits from minus
infinity to positive infinity ( -∞ to ∞ )
• Note: we will discuss Unilateral transform only
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7. Mathematical definition:
• Laplace transform equals to:
• The parameter s is the complex
number frequency:
and
 Where σ is real part and ω is imaginary part
23:51:49 7

8. 8/13
Example 9.1: Laplace Transform
• Consider the signal
• The Fourier transform X(jw) converges for a>0:
• The Laplace transform is:
• which is the Fourier Transform of e-(s+a)tu(t)
• Or
• If a is negative or zero, the Laplace Transform still exists
)()( tuetx at

0,
1
)()(
0


 





a
aj
dteedtetuejX tjattjat
w
w ww











0
)(
0
)(
)()(
dtee
dtedtetuesX
tjta
tasstat
ws
0,
)(
1
)( 

 a
ja
jX s
ws
ws
as
as
sXtue
L
at



}Re{,
1
)()(

9. EE-2027 SaS, L12 9/13
• The Region Of Convergence (ROC) of the Laplace transform
is the set of values for s (=s+jw) for which the Fourier
transform of x(t)e-st converges (exists).
• The ROC is generally displayed by drawing separating
line/curve in the complex plane, as illustrated below for
Examples 9.1 and 9.2, respectively.
• The shaded regions denote the ROC for the Laplace
transform
Region of Convergence
Re
Im
-a
as }Re{

10. Graph for example 9.1:
23:51:49 10
as }Re{

11. Example 9.2: Laplace Transform
• Consider the signal
• The Laplace transform is:
• Convergence requires that Re{s+a}<0 or Re{s}<-a.
• The Laplace transform expression is identical to Example 9.1
• (similar but different signals), however the regions of convergence of
s are mutually exclusive (non-intersecting).
• For a Laplace transform, we need both the expression and the
Region Of Convergence (ROC).
)()( tuetx at
 
as
dte
dttueesX
tas
stat











1
)()(
0
)(
Re
Im
-a

12. Graph for example 9.2:
12
as }Re{

13. Example 9.3: Laplace Transform
• Consider a signal that is the sum of two real exponentials:
• The Laplace transform is then:
• Using Example 1, each expression can be evaluated as:
• The ROC associated with these terms are Re{s}>-1 and Re{s}>-2. Therefore,
both will converge for Re{s}>-1, and the Laplace transform:
)(2)(3)( 2
tuetuetx tt 

 













dtetuedtetue
dtetuetuesX
sttstt
sttt
)(2)(3
)(2)(3)(
2
2
1
2
2
3
)(




ss
sX
23
1
)( 2



ss
s
sX

14. Example 9.3 Graph
23:51:49 14

15. Properties of ROC of Laplace
Transform
• ROC contains strip lines parallel
to jω axis in s-plane.
• If x(t) is absolutely integral and it is of finite duration,
then ROC is entire s-plane.
• If x(t) is a right sided sequence then ROC : Re{s} > σo.
• If x(t) is a left sided sequence then ROC : Re{s} < σo.
• If x(t) is a two sided sequence then ROC is the
combination of two regions.
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16. Common transform pairs
23:51:49 16
( )f t ( ) [ ( )]F s L f t
1 or ( )u t 1
s
T-1
t
e  1
s 
T-2
sin tw
2 2
s
w
w
T-3
cos tw
2 2
s
s w
T-4
sint
e t
w
2 2
( )s
w
 w 
T-5*
cost
e t
w
2 2
( )
s
s

 w

 
T-6*
t
2
1
s
T-7
n
t 1
!
n
n
s 
T-8
t n
e t
1
!
( )n
n
s  

T-9
( )t 1 T-10
*Use when roots are complex.

17. Table of selected Laplace Transforms
23:51:49 17
1
1
)()()sin()(
1
)()()cos()(
1
)()()(
1
)()()(
2
2








s
sFtuttf
s
s
sFtuttf
as
sFtuetf
s
sFtutf
at

18. Inverse Laplace transform
• The signal x(t) is said to be the inverse Laplace
transform of X(s). It can be shown that
• where c is a constant chosen to ensure the
convergence of the first
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19. Inverse Laplace Transforms:
• When a differential equation is solved by Laplace
transforms, the solution is obtained as a function
of the variable s. The inverse transform must be
formed in order to determine the time response.
• The simplest forms are those that can be
recognized within the tables and a few of those
will now be considered.
• If inverse Laplace transform is complicated we
use partial fraction
23:51:49 20

20. Example. Determine the inverse
transform of the function below.
2
5 12 8
( )
3
F s
s s s
  

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3
( ) 5 12 8 t
f t t e
  

21. Example. Determine the inverse
transform of the function below.
2
200
( )
100
V s
s


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2 2
10
( ) 20
(10)
V s
s
 
  
 
( ) 20sin10v t t

22. Example. Determine the inverse
transform of the function below.
2
8 4
( )
6 13
s
V s
s s


 
23:51:49 23
When the denominator contains a quadratic, check the roots. If they
are real, a partial fraction expansion will be required. If they are
complex, the table may be used. In this case, the roots are
1,2 3 2s i  
2
2 2 2
2
2 2
6 13
6 (3) 13 (3)
6 9 4
( 3) (2)
s s
s s
s s
s
 
    
   
  

23. 2 2 2 2
2 2 2 2
8( 3) 4 24
( )
( 3) (2) ( 3) (2)
8( 3) 10(2)
( 3) (2) ( 3) (2)
s
V s
s s
s
s s
 
 
   

 
   
23:51:49 24
3 3
( ) 8 cos2 10 sin 2t t
v t e t e t 
 

24. Properties of Laplace Transforms
• Linearity
• Scaling in time
• Time shift
• “frequency” or s-plane shift
• Multiplication by tn
• Integration
• Differentiation
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25. Linearity Property
23:51:49 26
With ROC containing R1∩R2

26. Time-Scaling Property
• Time domain scaling ⇔
frequency domain scaling
Example:
23:51:49 27
  






a
s
a
atL F
1
}{f
  22
4
2
)()2sin(
w
w
w


s
tutL
With ROC R1=R/a

27. Time Shifting
• If F (s) is the Laplace Transforms of f (t), then
• Example:
23:51:49 28
  )()()( sFeatuatfL as

as
e
tueL
s
ta




10
)10(
)}10({
With ROC =R

28. Frequency Shift
• If F (s) is the Laplace Transforms of f (t), then
23:51:49 29
)()}({ asFtfeL at

With ROC =R + Re{a}.

29. Multiplication by tn
• Eq.
• Example:
23:51:49 30
)()1()}({ sF
ds
d
tftL n
n
nn

1
!
)
1
()1(
)}({



n
n
n
n
n
s
n
sds
d
tutL
With ROC = R

30. Convolution Property
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31. Differentiation Property
23:51:49 32
• Eq.
• With ROC = R

32. Integration Property
• Eq.
• With ROC= R ∩ Re{s}
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33. Properties of Laplace Transform
23:51:49 34

34. Analysis of LTI System using Laplace
Transform
• Causality
• The ROC associated with the system function
for a causal system is a right half plane.
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35. Stability
• An LTI system is stable if and only if the ROC of
its function H(s) includes the jῳ-axis.
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36. Thanks!!
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