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Lecture 5 (Chapter 3)

Signals gnals & S & Systems ems

Fourier Series (Part I)

Adapted from: Lecture notes from MIT

• Dr. Hamid R. Rabiee

Fall 2013

Lecture 5 (Chapter 3)

Transformation

 General form:

Sharif University of Technology, Department of Computer Engineering, Signals & Systems

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( ) ( )

i i i

x t a t 

 

 

Coefficient Basis Function

Lecture 5 (Chapter 3)

Sharif University of Technology, Department of Computer Engineering, Signals & Systems

3

Desirable Characteristics of a Set of “Basic” Signals

 a) We can represent large and useful classes of signals using these building blocks.  b) The response of LTI systems to these basic signals is particularly simple , useful and insightful.  Previous focus: Unit samples and impulses  Focus now: Eigen functions of all LTI systems

Lecture 5 (Chapter 3)

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) (t

k



System

) (t

k k



Eigen value Eigen function

Eigenfunction in → Same function out with a “gain”

Lecture 5 (Chapter 3)

Sharif University of Technology, Department of Computer Engineering, Signals & Systems

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



k k k

t a t x ) ( ) ( 

System





k k k k

t a t y ) ( ) (  

Now the task of finding response of LTI systems is to determine λk From the superposition property of LTI system

Lecture 5 (Chapter 3)

Sharif University of Technology, Department of Computer Engineering, Signals & Systems

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Complex Exponentials as the Eigen functions of any LTI Systems

st

e t x  ) (

h(t)



   

  

 d

e h t y

t s ) (

) ( ) (

st s

e d e h        

   

 



) (

st

e s H ) ( 

Eigen value Eigen function

Lecture 5 (Chapter 3)

Sharif University of Technology, Department of Computer Engineering, Signals & Systems

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Complex Exponentials as the Eigen functions of any LTI Systems

) (t x

h(t)

) (t y

t sk

e

t s k

k

e s H ) (



   

 dt e t h s H

st

) ( ) (

 

  

k k t s k k t s k

k k

e a s H t y e a t x ) ( ) ( ) (

Lecture 5 (Chapter 3)

Sharif University of Technology, Department of Computer Engineering, Signals & Systems

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Complex Exponentials as the Eigen functions of any LTI Systems

n

z n x  ] [

h[n]



   



m m n

z m h n y ] [ ] [

n m m z

z m h        

   

] [

n

z z H ) ( 

Eigen value Eigen function

Lecture 5 (Chapter 3)

Sharif University of Technology, Department of Computer Engineering, Signals & Systems

9

Complex Exponentials as the Eigen functions of any LTI Systems

] [n x

H[n]

] [n y

n k

z

n k k z

z H ) (



   



n

z n h z H ] [ ) (

 

  

k n k k k k n k k

z a z H n y z a n x ) ( ] [ ] [

Lecture 5 (Chapter 3)

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What kinds of signals can we represent as “sums” of complex exponentials?

?

Lecture 5 (Chapter 3)

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What kinds of signals can we represent as “sums” of complex exponentials?

For Now: Focus on restricted sets of complex exponentials CT: s=jω signals of the form ejωt DT: Z=ejω signals of the form ejωn

Lecture 5 (Chapter 3)

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Fourier Series Representation of CT Periodic Signals

x(t) = x(t+T) for all t

x(t) t T 2T

• T
• 2T

… …

 Smallest such T is the fundamental period  is the fundamental frequency

T   2

0 

Lecture 5 (Chapter 3)

Sharif University of Technology, Department of Computer Engineering, Signals & Systems

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Fourier Series Representation of CT Periodic Signals

x(t) = ejωt Periodic with period T

 

     

   

t T jk k t jk k

e a e a t x k

 

 

2

) (

 Periodic with period T  {ak} are the Fourier (series) Coefficients  k=0: DC  |k|=1: First Harmonic  |k|=2: Second Harmonic

Lecture 5 (Chapter 3)

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Question #1: How do we find the Fourier coefficients?

t t t x   8 sin 2 4 cos ) ( : 1 Example  

?

Lecture 5 (Chapter 3)

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Question #1: How do we find the Fourier coefficients?

t t t x   8 sin 2 4 cos ) ( : 1 Example  

] [ 2 2 ] [ 2 1 ) (

8 8 4 4 Realtion s Euler' t j t j t j t j

e e j e e t x

     

  



2 1 4 2 2 4            T

Lecture 5 (Chapter 3)

Sharif University of Technology, Department of Computer Engineering, Signals & Systems

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Fourier Series Representation of CT Periodic Signals

For real periodic signals, there are two other commonly used forms for CT Fourier series:

1

( ) [ cos sin ]

k k k

x t a k t k t    

 

  



1

( ) [ cos( )]

k k k

x t a k t  

 

   



• r

 Because of the Eigen function property of ejωt, we will usually use the complex exponential form in this course  A consequence of this is that we need to include terms for both positive and negative frequencies:

t jk t jk

e e

0 ,

  

Lecture 5 (Chapter 3)

Sharif University of Technology, Department of Computer Engineering, Signals & Systems

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Question #1: How do we find the Fourier coefficients?

( )

jk t k k

x t a e

  

 

( )

jn t jk t jn t k k T T

x t e dt a e e dt

      

      

  

( ) j k n t k k T

a e dt

   

      

 

] [ , ,

) (

n k T n k n k T dt e

T t n k j

       



 





Here denotes integral over interval

• f length T (one period).



T

Lecture 5 (Chapter 3)

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Question #1: How do we find the Fourier coefficients?

( )

( ) . [ ]

jn t j k n t k k T T

x t e dt a e dt a T k n

 



      

        

   

( )

jn t n T

x t e dt a T

 





Equation Analysis ) ( 1 Equation Synthesis ) (

 

   

 

T t jk k t jk k

dt e t x T a e a t x

 

Lecture 5 (Chapter 3)

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Example #2: Periodic Square Wave

t T/2 T

• T/2
• T

… … x(t) T1

• T1

Lecture 5 (Chapter 3)

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Example #2: Periodic Square Wave

t T/2 T

• T/2
• T

… … x(t) T1

• T1

For k = 0





 

2 2 1

2 ) ( 1

T T

T T dt t x T a

DC component is just the average

Lecture 5 (Chapter 3)

Sharif University of Technology, Department of Computer Engineering, Signals & Systems

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Example #2: Periodic Square Wave

For k ≠ 0

1 1

2 2

1 1 ( )

T T jk t jk t T k T

a x t e dt e dt T T

     

 

 

1 1

1

sin 1 |

jk t T T

k T e jk T k



  

 

  

2 ( ) T   

Lecture 5 (Chapter 3)

Sharif University of Technology, Department of Computer Engineering, Signals & Systems

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Convergence of CT Fourier Series

 How can the Fourier series for the square wave possibly make sense?  The key is: What do we mean by ?  One useful notion for engineers

 There is no energy in the difference

 Just need x(t) to have finite energy per period.

( ) ( )

jk t k

e t x t a e

  

  ( )

jk t k

x t a e

  

 





T

dt t e | ) ( |

2



 

T

dt t x

2

| ) ( |

Lecture 5 (Chapter 3)

Sharif University of Technology, Department of Computer Engineering, Signals & Systems

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Dirichlet Conditions  The Dirichlet conditions are sufficient conditions for a real- valued, periodic function f(x) to be equal to the sum of its Fourier series at each point where f is continuous.  The behaviour of the Fourier series at points of discontinuity is determined as well, by these conditions.  These conditions are named after Johann Peter Gustav Lejeune Dirichlet.

Lecture 5 (Chapter 3)

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Dirichlet Conditions

 Condition 1: x(t) is absolutely integrable over one period, i. e.  Condition 2: In a finite time interval, x(t) has a finite number

• f maxima and minima.

Ex. An example that violates Condition 2.  Condition 3: In a finite time interval, x(t) has only a finite number of discontinuities. Ex. An example that violates Condition 3.



 

T

dt t x | ) ( |

1 ) 2 sin( ) (    t t t x 

Lecture 5 (Chapter 3)

Sharif University of Technology, Department of Computer Engineering, Signals & Systems

25

Dirichlet Conditions

 Dirchlet Conditions are met for the signals we will encounter in the real world. Then:

 The Fourier series = x(t) at points where x(t) is continuous.  The Fourier series = “midpoint” at points of discontinuity.

 Still, convergence has some interesting characteristics:

 As N→ ∞, xN(t) exhibits Gibbs’ phenomenon at points of discontinuity.

( )

N jk t N k k N

x t a e

 

 

Applet 15

Lecture 5 (Chapter 3)

Gibbs Phenomenon

 Fourier sums overshoot at a jump discontinuity  This overshoot does not die out as the frequency increases.

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5 N  25 N  125 N 

     

1 1 sin sin 3 sin ( 1) 3 1

N

S f x x x N x N      