Signals & Systems The Continuous-Time Fourier Transform Adapted - - PowerPoint PPT Presentation

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

The Continuous-Time Fourier Transform

Adapted From: Lecture Notes From MIT

• Dr. Hamid R. Rabiee

Fall 2013

Signals & Systems

Lecture 8 (Chapter 4)

• Derivation of the CT Fourier Transform pair
• Examples of Fourier Transform
• Fourier Transforms of Periodic Signals
• Properties of the CT Fourier Transform

Content

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Fourier’s Derivation of the CTFT

• x(t) - an aperiodic signal
• view it as the limit of a periodic signal as T → ∞
• For a periodic signal, the harmonic

components are spaced ω0 = 2π/T apart ...

• As T → ∞, ω0 → 0, and harmonic components are spaced

closer and closer in frequency Fourier series Fourier integral

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Motivating Example: Square wave

T1 kept fixed and T increases

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Motivating Example: Square wave

Discrete frequency points become denser in

ω as T increases

T1 kept fixed and T increases

5 1

2sin( )

k

k T a k T   

1

2sin( )

k

T Ta   

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

So, on with the Derivation ...

For simplicity, assume x(t) has a finite duration.

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( ), 2 2 ( ) , | | 2 T T x t t x t T periodic t            

, ( ) ( ) AsT x t x t for allt  

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Derivation (Continued…)

in this interval If we defined

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2 ( ) ( )

jk t k k

x t a e T



 

 

 



2 2 2 2

1 1 ( ) ( )

T T jk t jk t k T T

a x t e dt x t e dt T T

     

 

 

( ) ( ) x t x t 

1 ( )

jk t

x t e dt T

   





( ) ( )

j t

X jw x t e dt

   

 

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Derivation (Continued…)

Thus, for As

We can get the CT fourier transform pairs 8

2 2 T T t   

1 ( ) ( ) ( ) 1 ( ) 2

jk t k jk t k

x t x t X jk e T X jk e

 

   

   

  

 

, T w dw   

 

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

For What Kinds of Signals Can We Do This?

(1) It works also even if x(t) is infinite duration, but satisfies: a) Finite energy

In this case, there is zero energy in the error

b) Dirichlet conditions

Then

c) By allowing impulses in x(t)or in X(jω), we can represent even more signals E.g. It allows us to consider FT for periodic signals

9 2

| ( ) | x t dt

 

 



2

| ( ) | e t dt

 





1 ( ) ( ) ( ) 2

j t

e t x t X j e dt



 

 

 



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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Example #1

Synthesis equation for δ(t)

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

( ) ( ) ( ) ( ) ( ) 1 1 ( ) 2 ( ) ( ) ( ) ( ) ( )

j t j t j t j t

a x t t X j t e dt t e d b x t t t X j t t e dt e

   

   

         

              

  

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Example #1

Synthesis equation for δ(t)

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( ) ( ) ( ) ( ) ( ) 1 1 ( ) 2 ( ) ( ) ( ) ( ) ( )

j t j t j t j t

a x t t X j t e dt t e d b x t t t X j t t e dt e

   

   

         

              

  

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Example #2: Exponential Function

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( )

( ) ( ), ( ) ( ) 1 1 ( )

at j t at j t a j t

x t e u t a X j x t e dt e e dt e a j a j

  

  

        

        

 

∞

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Example #2: Exponential Function

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( )

( ) ( ), ( ) ( ) 1 1 ( )

at j t at j t a j t

x t e u t a X j x t e dt e e dt e a j a j

  

  

        

        

 

∞

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

Lecture 8 (Chapter 4)

Example #2 (continued)

Even symmetry Odd symmetry

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Example #3: A Square Pulse in the Time-domain

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Example #3: A Square Pulse in the Time-domain

Note the inverse relation between the two widths ⇒ Uncertainty principle

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1 1

1

2sin ( )

T j t T

T X j e dt



  

 

 



( ) X j d  

 



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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Exampl1e #4:

17

2

( )

at

x t e 

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Exampl1e #4:

(Pulse width in t)•(Pulse width in ω) ⇒ ∆t•∆ω ~ (1/a1/2)•(a1/2) = 1 Uncertainty Principle! Cannot make both ∆t and ∆ω arbitrarily small.

Also a Gaussian!

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2

( )

at

x t e 

2 2 2 2 2 2 2

[ ( ) ] ( ) 2 2 ( ) 2 4 4

( ) [ ].

at j t j j a t j t a a a a j a t a a a

X j e e dt e dt e dt e e a

      

 

               

   

  

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

CT Fourier Transforms of Periodic Signals

— periodic in t with frequency ωo — All the energy is concentrated in

• ne frequency — ωo

Suppose That is More generally

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( ) ( ) 1 1 ( ) ( ) 2 2

j t j t

X j x t e dw e 



      

 

       



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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Example #5:

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

𝑦 𝑢 = cos 𝜕0𝑢

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Example #5:

“Line spectrum”

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1 1 ( ) cos 2 2 ( ) ) )

j t j t

x t t e e X jw

 

  

  

       

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Example #6:

(Sampling function) x(t)

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( ) ( )

n

x t t nT

 

  



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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Example #6:

(Sampling function)

Same function in the frequency-domain!

x(t)

(period in t) T⇔ (period in ω) 2π/T Inverse relationship again!

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( ) ( )

n

x t t nT

 

  



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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Properties of the CTFT

1) Linearity 2) Time Shifting 3)FT magnitude unchanged Proof:

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( ) ( ) ( ) ( ) ax t by t aX j bY j     

( ) ( )

j t

x t t e X j







 

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

Properties (Continued)

5) Conjugate Symmetry Even Odd Even Odd 4)Linear change in FT phase

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

Lecture 8 (Chapter 4) Lecture 8 (Chapter 4)

6) Time-Scaling E.g. a > 1 → at > t compressed in time ⇔ stretched in frequency

Properties (Continued)

a) x(t) real and even E.g. If a= -1 Eq(1) Real & even With Eq(1)

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1 ( ) ( ) | | x at X j a a  

( ) ( ) x t X j    

*

( ) ( ) ( ) ( ) ( ) x t x t X j X j X j        

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

Lecture 8 (Chapter 4)

b) x(t) real and odd c) For real x(t)

Properties (Continued)

With Eq(1)

Purely imaginary

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( ) Re{ ( )} Im{ ( )} ( ) { ( )} { ( )} X j X j j X j x t Ev x t Odd x t         

*

( ) ( ) ( ) ( ) ( ) x t x t X j X j X j            

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