[PDF] - 12/20/2017 Lectures on Signals & systems Engineering Designed PDF Document

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Designed and Presented by

Dr. Ayman Elshenawy Elsefy
Dept. of Systems & Computer Eng. Al-Azhar University

Email : eaymanelshenawy@yahoo.com

Lectures on Signals & systems Engineering

Chapter 9 Laplace Transform Applications

2

Chapter 9 The Laplace Transform Chapter 9 The Laplace Transform

𝒚 𝒖 = 𝒇−𝒃𝒖𝒗 𝒖 , 𝒃 > 𝟏, 𝒃 ∈ 𝑺

𝒀 𝑻 =

−∞ +∞

𝒚 𝒖 𝒇−𝒕𝒖𝒆𝒖 =

−∞ +∞

𝒇−𝒃𝒖𝒗 𝒖 𝒇−𝒕𝒖𝒆𝒖 =

𝟏 +∞

𝒇−𝒃𝒖 𝒇−𝒕𝒖𝒆𝒖 =

𝟏 +∞

𝒇−(𝒕+𝒃)𝒖 𝒆𝒖 = −𝟐 𝒕 + 𝒃 𝒇− 𝒕+𝒃 ∗∞ − 𝒇− 𝒕+𝒃 ∗𝟏 = −𝟐 𝒕 + 𝒃 𝟏 − 𝟐 = 𝟐 𝒕 + 𝒃 , 𝑺𝒇 𝒕 > −𝒃

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Chapter 9 The Laplace Transform

𝒚 𝒖 = 𝒇−𝒃𝒖𝒗 −𝒖 , 𝒃 > 𝟏 𝒀 𝑻 =

−∞ +∞

𝒚 𝒖 𝒇−𝒕𝒖𝒆𝒖 =

−∞ +∞

𝒇−𝒃𝒖𝒗 −𝒖 𝒇−𝒕𝒖𝒆𝒖 =

−∞ 𝟏

𝒇−𝒃𝒖 𝒇−𝒕𝒖𝒆𝒖 =

−∞ 𝟏

𝒇−(𝒕+𝒃)𝒖 𝒆𝒖 = −𝟐 𝒕 + 𝒃 𝒇− 𝒕+𝒃 ∗𝟏 − 𝒇− 𝒕+𝒃 ∗−∞ = −𝟐 𝒕 + 𝒃 𝟐 − 𝟏 = −𝟐 𝒕 + 𝒃 , 𝑺𝒇 𝒕 < −𝒃

Chapter 9 The Laplace Transform Chapter 9 The Laplace Transform Chapter 9 The Laplace Transform

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Zeroes and Poles of rational Laplace transform 𝒀 𝒕 = ) 𝑶(𝒕 ) 𝑬(𝒕 Zeroes of the Laplace transform  N(s) = 0 Poles of the Laplace transform  D(s) = 0 𝒒𝟐 = 𝟏 then 𝐘 𝒒𝟐 = ∞ If 𝒜𝟐 is a zero, then 𝐘 𝒜𝟐 = 𝟏 Zeroes and Poles of rational Laplace transform Region of Convergence ROC Region of Convergence ROC

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Region of Convergence ROC

14

Chapter 9 The Laplace Transform Basic Laplace Pairs

 

t x

 

s X

Poles ROC

 

t 

1

none

 

  s Re s 1

 

t u

 

Re  s

 

t u  

 

Re  s s 1

 

t u e at



 

a s   Re

 

t u e at  



 

a s   Re a s  1 a s  1  s  s a s   a s  

15

Chapter 9 The Laplace Transform

 

s t u

L

1    

  0

 s Re

 

1  L t 

 

  s Re

1.

 

s t u

L

1  

  0

 s Re

2.

 

a s t u e

L at

    



1

 

a s   Re

 

a s t u e

L at

  



1

 

a s   Re

3.

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Chapter 9 The Laplace Transform Example 9.13

      

1 Re 2 1 1

2

     s s s s X

   

1 Re 1 1

1

    s s s X   j

1  2 

  j

1 

     

2 1

    s s X s X s X

 

2 Re   s

   

2t

x t e u t



   j

2 

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Chapter 9 The Laplace Transform §9.5.2 Time Shifting

   

L st

e s X t t x



  

   

s X t x  L R Roc  R Roc 

Example

   

kT t t x

k

 

 



 

Re s 

 

1 1

sT

X s e     j

pole-zero plot T j  2 T j  2 

18

   

s X t x  L

Chapter 9 The Laplace Transform §9.5.3 Shifting in s-Domain

   

L

s s X e t x

t s

   R Roc 

 

Re s R Roc  

ROC

  j

2

r

 

2 1

Re r s r  

1

r   j

     

2 1

Re Re Re s r s s r    

 

1

Re s r 

 

2

Re s r 

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Chapter 9 The Laplace Transform §9.5.6 Convolution Property

   

s X t x

L 1 1

 

1

R Roc 

   

s X t x

L 2 2

 

2

R Roc 

       

s X s X t x t x

L 2 1 2 1

  

2 1

R R Roc  

   

2 Re 2 1

1

     s s s s X

   

1 Re 1 2

2

     s s s s X

    1

2 1

 s X s X

 

  s Re

     

t t x t x    

2 1

Convolution Property

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Chapter 9 The Laplace Transform Example

           

?

2 1 3 2 2 1

    



t x t x t u e t x t u e t x

t t

       

5 1 5 1

3 2 2 1

t u e t u e t x t x

t t

   



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Chapter 9 The Laplace Transform §9.5.7 Differentiation in the Time Domain

   

s X t x  L R Roc 

R Roc 

   

s sX dt t dx  L 1 t

 

t x



2 4 6 8

Example Determine

 

s X

     

 

 

2 1 2 2 2

1 2 Re 1

s s s

e e X s X s X s s s e

  

     

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§9.5.8 Differentiation in the s-Domain Chapter 9 The Laplace Transform

   

s X t x  L R Roc  R Roc 

   

ds s dX t tx   

L

   

2

1 a s t u te

L at

  



 

a s   Re

   

3 2

1 2 1 a s t u e t

L at

  



 

a s   Re

24

Chapter 9 The Laplace Transform §9.7 Analysis and Characterization of LTI Systems Using the Laplace Transform

 

t y

 

t h

 

s H

 

s Y

 

t x

 

s X      

t h t x t y  

     

s H s X s Y 

 

s H

——System Function or Transfer Function

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Causality of LTI Stability of LTI

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Chapter 9 The Laplace Transform

    

2 1 1     s s s s H   j 2 1    j 2 1    j 2 1 

   

2 Re a  s

Causal , unstable system

   

2 Re 1 b   s

noncausal , stable system

   

1 Re c   s

anticausal , unstable system

30

Transfer Function of the system

§9.7.3 LTI Systems Characterized by Linear Constant-Coefficient Differential Equations

     

t x t y dt t dy  3

   

k k M k k k k N k k

dt t x d b dt t y d a

 

 

 ROC

k k N k k k M k

s a s b

 

 



     

s X s Y s H 

31

Chapter 9 The Laplace Transform

Example Consider a causal LTI system whose input and output related through an linear constant-coefficient differential equation of the form

 

t x  

y t

       

3 2 y t y t y t x t     

Determine the unit step response of the system.

   

2

1 1 2 2

t t

s t e e u t

 

        

32

Chapter 9 The Laplace Transform

   

LC s L R s LC s H / 1 / / 1

2

  

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Chapter 9 The Laplace Transform Example 9.25

Consider an LTI system with input , Output . (a) Determine the system function. (b) Justify the properties of the system. (c) Determine the differential equation of the system.

   

t u e t x

t 3 



  

  

t u e e t y

t t 2   

       

3 Re 1 1 2 s H s s

s

s     

         

3 2 3 y t y t y t x t x t       

34

Chapter 9 The Laplace Transform Example Consider a causal LTI system ,

       

t bu t u e t h dt t dh

t

  

4

2 . 2

       

           t

e

t y t

e

t x

t t

6 1 . 1

2 2

b——unknown constant Determine the system function and b.

 

s H      

2 Re 4 H s s s s   

System Functions for Interconnections of LTI Systems System Functions for Interconnections of LTI Systems

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RC Circuit System Model – Case 1 RC Circuit System Model – Case 1 ( System Function & Impulse Response)

RC Circuit System Model – Case 1

RC Circuit System Model – Case 2 RC Circuit System Model – Case 2 ( System Function & Impulse Response)

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Example: Example:

Realization of Transfer Function Realization of Transfer Function – First Order System

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