Fourier Transform ,LAPLACE TRANSFORM,ROC and its Properties

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

2. 2
Fourier series
Fourier Transform
Laplace Transforms

3. 3
Fourier
Series
Fourier
Transform
CTF
S
DTF
S
CTF
T
DTF
T
Fourier Transform can be used for Periodic signal also
 If the input signal x(t)

4. 4
 Mathematical transformation employed to transform signals between
time (or spatial) domain and frequency domain.
 Fourier method of representing non-periodic signals as a function of
frequency
 Fundamental period T tend to infinity
 F.T Analysis : Break the signal or functions into simpler constituent
parts
 
      dt
e
t
x
j
X
t
x
F t
j
 






    



d
e
j
X
t
x t
j





2
1
Analysis Equation:
Synthesis Equation
 F.T Synthesis : Reassemble a signal from its constituent parts
Fourier transform Pair : Analysis+
Synthesis

5.      


 j
X
j
j
X
j
X i
r 

   

 j
X
j
Xr of
part
Real

   

 j
X
j
Xi of
part
Imaginary

   
   
 2
2


 j
X
j
X
j
X i
r 

or
     


 j
X
j
X
j
X *

   

 j
X
of
Conjugate
j
X 
*
The X(jω) is a complex function of ω. Hence it can be expressed as
The magnitude of X(jω) is called Magnitude Spectrum.

6. 6
The phase of X(jω) is called Phase Spectrum
The phase spectrum can be written as
The magnitude and phase spectrum together
is called frequency spectrum
 
 
 








 



j
X
j
X
j
X
r
i
1
tan

7. 7
Fourier Transform does not exist for some signals.
For example    
t
u
e
t
x t
2

 Fourier Transform for x(t)does
not exists because it is not
absolutely integrable
 Existence of Fourier Transform-The Dirichlet Conditions
should be satisfied
 Signal should have finite number of maxima and minima
 Signal should have finite number of discontinuities
 Signal should be absolutely integrable
 






dt
t
x

8. 8
 It is used to transform a time domain to complex
frequency domain signal (s-domain)
 Two Sided Laplace transform (or) Bilateral Laplace transform
 Let 𝑥(𝑡) be a continuous time signal defined for all values of 𝑡.
 Let 𝑋(𝑆) be Laplace transform of 𝑥(𝑡)(non-causal signal ).
 One sided Laplace transform (or) Unilateral Laplace transform
 Let 𝑥(𝑡) be a continuous time signal defined for 𝑡≥0 (ie If 𝑥(𝑡) is
causal) then,
 
      dt
e
t
x
s
X
t
x
L t
s







 
      dt
e
t
x
s
X
t
x
L t
s





0
Complex variable, S= σ+ jω

9. 9
 Inverse Laplace transform
 (S-domain signal 𝑋(𝑆) Time domain signal x(t) )
   
s
X
t
x
 Laplace transformX(s) and Inverse Laplace transform x(t)
are called Laplace Transform Pair and can be expressed as
 
      ds
s
X
j
t
x
s
X
L
j
s
j
s













2
1
1

10. 10
Not absolutely integrable Absolutely integrable for σ>2
   
t
u
e
t
x t
2

   
   
t
u
e
e
t
x
t
u
e
e
e
t
x
t
t
t
t
t
)
2
(
2










converges

11. 11
The Laplace transform of a signal is given by
 The range of ‘s’ (σ) for which the Laplace transform converges
(Finite) is called region of convergence
  dt
e
t
x t
s





Complex variable, S= σ+ jω
Re(s)
- ∞ 0 ∞
jω
σ
LHS RHS
Img(s)
S plane

12. 12
The zeros are found by setting the numerator polynomial to Zero
The zeros of the transform X(s)are the values of s for which the
Transform is Zero
The Poles are found by setting the Denominator polynomial to Zero.
The Poles of the transform X(s)are the values of s for which the
Transform is infinite.
 
)
(
)
(
s
D
s
N
s
X 

13.     0

 
a
where
t
u
e
t
x
Let t
a
Now Laplace transform of x(t) is given by,
 
      dt
e
t
x
s
X
t
x
L t
s







  dt
e
t
u
e
s
X t
s
t
a 





 )
(
dt
e
e st
at





0
 
dt
e t
a
s





0
 
 












0
a
s
e t
a
s
   
a
s
a
s
e
e
s
X













1
)
(
0

Since the given signal is
right sided signal or
causal signal then,
a
ROC 


:
s-Plane
Case i: Causal Signal or Right sided Signal
a
a
s
a
s
o
a
s









)
Re(

14. Now Laplace transform of x(t) is given by,
 
      dt
e
t
x
s
X
t
x
L t
s







a



:
ROC
Case ii: Non causal Signal or Left sided Signal
    0
Let 


 
a
where
t
u
e
t
x t
a
  dt
e
t
u
e
s
X t
s
t
a 






  )
(
dt
e
e st
at






0
 
dt
e t
s
a






0
  0











a
s
e t
s
a
 
a
s
s
X


1
   
 






 s
a
s
a
e
e
a
s
0
.
1









b
a
cx
b
a
cx
e
e
 
dt
e t
s
a




0
a
a
s
a
s
a
s









)
Re(
0

15. 15
a



:
ROC
Non causal Signal or Left sided Signal

16. 16
Case iii: Two sided Signal
Let      
t
u
e
t
u
e
t
x t
b
t
a


 

     
t
x
t
x
t
x 2
1 

   
t
u
e
t
x t
a


1
   
t
u
e
t
x t
b

 
2
and
 
s
X
Find 1
  dt
e
t
u
e
s
X t
s
t
a 





 )
(
dt
e
e st
at





0
 
dt
e t
a
s





0

17. 17
dt
e
e st
bt





0
 
dt
e t
s
b





0
 
   
 



















 s
b
s
b
t
s
b
e
e
b
s
b
s
e 0
.
0
1









b
a
cx
b
a
cx
e
e
 
s
X
Find 2
 
 












0
a
s
e t
a
s
   
a
s
a
s
e
e
s
X













1
)
(
0
  dt
e
t
u
e
s
X t
s
t
b 





  )
(
2
 
b
s
s
X



1
2
a
is
ROC 


b
is
ROC 


dt
e t
b
s





0
)
(

18. 18
Therefore ROC of X(s) is the region between
two lines passing through poles –a and –b
that is
b
a 


 
s-Plane
ROC of a Two sided Signal

19. 19
Property 1
The ROC of X(s) consists of parallel strips to the imaginary axis.
Property 2
The ROC of Laplace transform does not include any pole of X(s)

20. 20
Property 3
If x(t) is right sided or causal signal
,the ROC of X(s) extends to the
right of the right most poles and no
pole is located inside the ROC.
Property 4
If x(t) is left sided or non causal
signal ,the ROC of X(s) extends to
the left of the left most poles and
no pole is located inside the ROC.
   
t
u
e
t
x
E t
a


g
 
a
s
s
X


1
)
( a
ROC 


:
   
t
u
e
t
x t
a


 
Eg
 
a
s
s
X


1
a



:
ROC

21. 21
Property 5
If x(t) is two sided signal the ROC of X(s) is a
strip in the s-plane bounded by poles and no
pole is located inside the ROC.
s-Plane
Property 7
Impulse function is the only function for
which the ROC is the entire plane.
Property 6
The ROC of the sum of two or more signals
is equal to the intersection of the ROCs of
those signals.