02/04/1439 Chapter 3 - - PDF document

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02/04/1439 Chapter 3 Fourier Series Le Lectures es on on Signals & system ems Eng nginee eering A signal can be represented as a linear combination of basic

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02/04/1439 1

Designed and Presented by

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

Email : eaymanelshenawy@yahoo.com

Le Lectures es on

  • n Signals & system

ems Eng nginee eering

Chapter 3 Fourier Series Representations of Periodic Signals

2

Chapter 3 Fourier Series

  • A signal can be represented as a linear combination of

basic signals.

  • The response of LTI to any input consisting of linear

combination of basic signals is the linear combination of the individual responses to each of the basic signals.

  • Convolution Sum & Convolution Integral represent a

signal as linear combination of shifted impulses.

  • Fourier series and transform uses a complex exponential

signals with different frequencies will be used instead of shifted impulses ( Delayed or advanced).

3

Jean Baptiste Joseph Fourier, born in 1768, in France. 1807,periodic signal could be represented by sinusoidal series. 1829,Dirichlet provided precise conditions. 1960s,Cooley and Tukey discovered fast Fourier transform. Chapter 3 Fourier Series

4

Chapter 3 Fourier Series

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5

Chapter 3 Fourier Series

  • Importance of complex exponentials in LTI

system:

  • The response of an LTI system to a complex

exponential input is the same complex exponential with only a change in amplitude.

  • The complex amplitude factor H(s) or H(z) is a

function of the complex variable s or z.

6

Chapter 3 Fourier Series The Response of LTI Systems to Complex Exponentials LTI

 

t y

st

e

 

t h

  • 1. Continuous-time system

   d h t x t h t x t y

  

   ) ( ) ( ) ( * ) ( ) (

   

  

d h e

t s

) (

) (

   

  

d h e e

s st

) ( ) ( ) ( ) ( s H t x s H est  

st

e

 

st

e s H Eigen function

   

st

H s h t e dt

  

  ——Eigenvalue

7

Chapter 3 Fourier Series The Response of LTI Systems to Complex Exponentials LTI

 

n h

 

n y

n

z

  • 2. Discrete-time system

  

  

k

k h k n x n h n x n y ] [ ] [ ] [ * ] [ ] [

   

k k n

n h z ] [

   

k k n

n h z z ] [ ) ( ] [ ) ( z H n x z H zn  

Eigen function

n

z

 

n

z z H ——Eigenvalue

   

n n

z n h z H

   

8

Chapter 3 Fourier Series For the continuous LTI systems consider the input : From the eignfunction property, the response to each part is:

And from the superposition property:

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9

(3) Input as a combination of Complex Exponentials Continuous time LTI system:

        

 

  N k t s k k N k t s k

k k

e s H a t y e a t x

1 1

) ( ) ( ) (

Discrete time LTI system:

        

 

  N k n k k k N k n k k

z z H a n y z a n x

1 1

) ( ] [ ] [

Chapter 3 Fourier Series

10

Chapter 3 Fourier Series Example 3.1 Consider an LTI system : X(t)

   

3 y t x t  

   

2

1

j t

x t e  Impulse response

𝐼 𝑡 = 𝑓−3𝑡

11

Chapter 3 Fourier Series Example 3.1 Consider an LTI system : X(t)

   

3 y t x t  

   

2 cos4 cos7 x t t t  

       

cos4 3 cos7 3 3 y t t t x t       Impulse response: X(t) 𝑰 𝒕 = 𝒇−𝟒𝒕

𝑰 𝒕 𝒇𝒌𝟓𝒖 𝑰 𝒕 𝒇𝒌𝟖𝒖

𝒇−𝒌𝟐𝟑𝒇𝒌𝟓𝒖 𝟐 𝟑 𝒇−𝒌𝟐𝟑𝒇𝒌𝟓𝒖 𝒇−𝒌𝟑𝟐𝒇𝒌𝟖𝒖 𝟐 𝟑 𝒇−𝒌𝟑𝟐𝒇𝒌𝟖𝒖

𝒛 𝒖 =

𝟐 𝟑 𝒇−𝒌𝟐𝟑𝒇𝒌𝟓𝒖 + 𝟐 𝟑 𝒇−𝒌𝟐𝟑𝒇−𝒌𝟓𝒖+ 𝟐 𝟑 𝒇−𝒌𝟑𝟐𝒇𝒌𝟖𝒖 + 𝟐 𝟑 𝒇−𝒌𝟑𝟐𝒇−𝒌𝟖𝒖

12

Chapter 3 Fourier Series Example : Consider an LTI system for which the input and the impulse response determine the output

 

t t x  2 cos 2 1 1 

   

t u e t h

t 

   t y

 

2 2

1 1 4 4 1 1 2 1 2

j t j t

y t e e j j

 

 

     Try to solve it

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Chapter 3 Fourier Series

  • If the input to an LTI system is represented as a linear

combination of complex exponential signal Then

  • The
  • utput

also can be represented as linear combination of the same complex exponential signal

  • Output component = input component X eignvalue

Fourier Series Representation of CT Periodic Signals

14

Chapter 3 Fourier Series

Fourier Series Representation of CT Periodic Signals

15

(1) General Form

 2 , 1 , , ) (

) / 2 (

      k e e t

t T jk t jk k  

3.3.1 Linear Combinations of Harmonically Related Complex Exponentials The set of harmonically related complex exponentials: Fundamental period: T ( common period )

Chapter 3 Fourier Series

𝑦 𝑢 =

𝑙=−∞ +∞

𝑏𝑙𝑓𝑘𝑙𝑥0𝑢 =

𝑙=−∞ +∞

𝑏𝑙𝑓𝑘𝑙(

2𝜌 𝑈)𝑢

16

So, arbitrary periodic signal can be represented as

t j t j

e e

0 ,

  

: Fundamental or 1st harmonic components

t j t j

e e

2 2

,

  

: 2nd harmonic components

t jN t jN

e e

0 ,

  

: Nth harmonic components ( Fourier series )

Chapter 3 Fourier Series

k

a

——Fourier Series Coefficients Spectral Coefficients

K=1 K=2 K=N K=0 x(t) is constant, DC component

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17

Chapter 3 Fourier Series Example 3.2

 

t jk k k

e a t x

2 3 3 

  

        

  

3 / 1 , 2 / 1 4 / 1 , 1

3 2 1

a a a a

18 19

Chapter 3 Fourier Series Determination of Fourier Series Representation

 

t jk k k

e a t x

  

 

1

jk t k T

a x t e dt T

   

Synthesis equation Analysis equation

k

a ——Fourier Series Coefficients Spectral Coefficients

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𝒚 𝒖 = 𝟐 + 𝐭𝐣𝐨𝝏𝟏𝒖 + 𝟑 𝐝𝐩𝐭 𝝏𝟏𝒖 + 𝒅𝒑𝒕(𝟑𝝏𝟏𝒖 + 𝝆 𝟓)

𝒚 𝒖 = 𝟐 + 𝟐 𝟑𝒌 𝒇𝒌𝝏𝟏𝒖 − 𝒇−𝒌𝝏𝟏𝒖 + 𝒇𝒌𝝏𝟏𝒖 − 𝒇−𝒌𝝏𝟏𝒖 + 𝟐 𝟑 𝒇𝒌(𝟑𝝏𝟏𝒖+

𝝆 𝟓) − 𝒇−𝒌(𝟑𝝏𝟏𝒖+ 𝝆 𝟓)

𝒚 𝒖 = 𝟐 + (𝟐 +

𝟐 𝟑𝒌)𝒇𝒌𝝏𝟏𝒖+ 𝟐 − 𝟐 𝟑𝒌 𝒇−𝒌𝝏𝟏𝒖 + ( 𝟐 𝟑 𝒇𝒌( 𝝆 𝟓))𝒇𝒌(𝟑𝝏𝟏𝒖)+( 𝟐 𝟑 𝒇−𝒌( 𝝆 𝟓))𝒇−𝒌(𝟑𝝏𝟏𝒖)

𝒚 𝒖 = 𝒃𝟏 + 𝒃𝟐𝒇𝒌𝝏𝟏𝒖+𝒃−𝟐𝒇−𝒌𝝏𝟏𝒖 + 𝒃𝟑𝒇𝒌(𝟑𝝏𝟏𝒖)+𝒃−𝟑𝒇−𝒌(𝟑𝝏𝟏𝒖)

𝒃𝟏=1 𝒃𝟐 = 𝟐 + 𝟐 𝟑𝒌 𝒃𝟑=𝟐

𝟑 𝒇𝒌( 𝝆 𝟓) = 𝟑 𝟓 (𝟐 + 𝒌)

𝒃−𝟑=𝟐

𝟑 𝒇−𝒌( 𝝆 𝟓) = 𝟑 𝟓 (𝟐 − 𝒌)

𝒃𝒍=0 , |k|>2 𝒃−𝟐 = 𝟐 − 𝟐 𝟑𝒌

𝒚 𝒖 = 𝟐 + 𝐭𝐣𝐨𝝏𝟏𝒖 + 𝟑 𝐝𝐩𝐭 𝝏𝟏𝒖 + 𝒅𝒑𝒕(𝟑𝝏𝟏𝒖 + 𝝆 𝟓)

𝒃𝟏=1 𝒃𝟐 = 𝟐 + 𝟐 𝟑𝒌 𝒃𝟑=𝟐

𝟑 𝒇𝒌( 𝝆 𝟓) = 𝟑 𝟓 (𝟐 + 𝒌)

𝒃−𝟑=𝟐

𝟑 𝒇−𝒌( 𝝆 𝟓) = 𝟑 𝟓 (𝟐 − 𝒌)

𝒃𝒍=0 , |k|>2 𝒃−𝟐 = 𝟐 − 𝟐 𝟑𝒌 |𝒃𝒍| < 𝒃𝒍 Plots of the magnitude and phase of the Fourier Series of the signal

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25

Chapter 3 Fourier Series

   T T / 2  

 

1

4 a T T 

 

1

8 b T T 

 

1

16 c T T 

26

Chapter 3 Fourier Series Convergence of Fourier series Chapter 3 Fourier Series

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Chapter 3 Fourier Series Chapter 3 Fourier Series Chapter 3 Fourier Series

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Chapter 3 Fourier Series Chapter 3 Fourier Series Chapter 3 Fourier Series

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1 2 3

EE-2027 SaS, L7 39/16

  • Fourier transforms map a time-domain signal into a frequency domain

signal

  • Simple interpretation of the frequency content of signals in the

frequency domain (as opposed to time).

  • Design systems to filter out high or low frequency components.

Analyse systems in frequency domain.

Why is Fourier Theory Important (ii)?

Invariant to high frequency signals

EE-2027 SaS, L7 40/16

Why is Fourier Theory Important (iii)?

  • If

F{x(t)} = X(j)  is the frequency

  • Then F{x’(t)} = jX(j)
  • So solving a differential equation is transformed from a calculus
  • peration in the time domain into an algebraic operation in the

frequency domain (see Laplace transform)

  • Example
  • becomes
  • and is solved for the roots  (N.B. complementary equations):
  • and we take the inverse Fourier transform for those .

3 2

2 2

   y dt dy dt y d

3 2

2

      j ) ( 3 ) ( 2 ) (

2

         j Y j Y j j Y

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dt e t h s H

st

   

 ) ( ) (  

t h

  st

e s H

st

e ] [n h

 

n

z z H

n

z

  

n n

z n h z H ] [ ) (

Chapter 3 Fourier Series and LTI system Eigenfunctions of LTI System System Function

) (  j s 

dt e t h j H

t j

   

 ) ( ) (

 

t h

 

t j

e j H

t j

e 

Frequency Response

  

n n j j

e n h e H

 

] [ ) (

) (

 j

e z 

] [n h

 

n j j

e e H

  n j

e 

Continuous LTI system Discrete LTI system

42

Chapter 3 Fourier Series and LTI Systems Linear Combinations of Eigenfunctions Frequency Response of LTI System

   

dt e t h S H

t s

  

Periodic Signal

k gain k

a jk H a     

" " 0)

(  

) (

) ( ) (

 

jk H j

e jk H jk H

including both amplitude & phase

 

t h

   

t jk k k

e jk H a t y

  

 

t jk k k

e a t x

  

 ) (  j H

must be well defined and finite.

   

dt e t h j H

t j 

   

 𝑻 = 𝒌𝒙

43

Chapter 3 Fourier Series

 

t jk k k

e a t x

2 3 3 

  

        

  

3 / 1 , 2 / 1 4 / 1 , 1

3 2 1

a a a a

Suppose the periodic signal x(t) is the input signal to LTI system with impulse response

𝒊 𝒖 = 𝒇−𝒖𝒗(𝒖)

  • 1. Compute the frequency response 𝑰(𝒌𝒙)

ℎ 𝑘𝑥 =

∞ 𝑓−𝜐 𝑓−𝑘𝑥𝜐𝑒𝜐= ∞ 𝑓−(1+𝑘𝑥)𝜐 𝑒𝜐= −1 1+𝑘𝑥 𝑓−(1+𝑘𝑥)𝜐 = −1 1+𝑘𝑥

𝑓− 1+𝑘𝑥 𝜐 |𝜐=∞ − 𝑓− 1+𝑘𝑥 𝜐 |𝜐=0 =

1 1+𝑘𝑥

H(jw) =

1 1+𝑘𝑥

 

t jk k k

e a t x

2 3 3 

  

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H(jw) =

1 1+𝑘𝑥

 

t jk k k

e a t x

2 3 3 

  

  • 1. Consider a LTI system implemented as the RL circuit shown in figure , a

current source produces an input current x(t), and the system output is considered to be the current y(t) flowing through the inductor:

  • a. find the differential equation relating x(t) to y(t)

Determine the frequency response of this system by considering the

  • utput of the system to inputs of the form 𝒚 𝒖 = 𝒇𝒌𝒙𝒖

Determine the output y(t) if the input 𝒚(𝒖) = 𝒅𝒑𝒕 𝒖

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