2. Lecture Fourier Transform Outline - - PowerPoint PPT Presentation

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2 lecture fourier transform outline

2. Lecture Fourier Transform Outline - - PowerPoint PPT Presentation

Apr 02, 2024 147 likes •565 views

2. Lecture Fourier Transform Outline

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SLIDE 1

2. Lecture

Fourier Transform

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SLIDE 2

Outline

2

slide-3

SLIDE 3

Example: Fourier transform

3

slide-4

SLIDE 4

Introduction

!

"

4

" # $

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SLIDE 5

Definition of the Fourier transform

$ # % & ' ( (

5
# ( )*

$ % ' ( (

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SLIDE 6

Definition of the Fourier transform

$ + * $ # & * , #% +

6

+ ( - %*

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SLIDE 7

Definition of the Fourier transform

+ + ' ( #

7

(

( )
$

*

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SLIDE 8

Definition of the Fourier transform

$ # ' ( (

$ %

8

$ % %*

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SLIDE 9

Fourier transform example

( ( (

./
A

f(x)

9

(
. )/

(

. /

(

X

x

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SLIDE 10

Fourier transform example

+ # %0 * (

A

X x f(x)

10

(

(
|F(x)|

u

1/X

AX

2/X 3/X

3/X -2/X -1/X

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SLIDE 11

Fourier transform example

1+' ( ( 2 + # ( -

11

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SLIDE 12

Discrete-time Fourier transform

$ $$ ./ $ ./ ** $$ %

12

** $$ %

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SLIDE 13

Discrete-time Fourier transform

&' $ $$ ./ % (

./

13

, + % # ( -

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SLIDE 14

Discrete-time Fourier transform

% +

14

(

# (

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SLIDE 15

Discrete-time Fourier transform

3 , $$

15

, $$ 45#

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SLIDE 16

Discrete-time Fourier transform

1+' $ $$ ./ % (

./ ( .2/ ( )

16

$

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SLIDE 17

Discrete-time Fourier transform

1+' ./ ( ./ ) , $$ %

17

(

./ (
(
(

) ) ( )

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SLIDE 18

Discrete-time Fourier transform

$ $$ (

# #

2 0.6

18

3
2
1

1 2 3 0.5 1 1.5 ω/π Magnitude

3
2
1

1 2 3

0.4
0.2

0.2 0.4 ω/π Phase in radians

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SLIDE 19

Discrete-time Fourier transform

$ $$ ./ , #

19

(

./

(

./ (
./ (

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SLIDE 20

Inverse discrete-time FT

$ % & ' ./ ( )

20

( )

j

X e ω [ ] x n

DTFT IDTFT

time domain frequency domain

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SLIDE 21

Discrete Fourier transform

$ $$ ./ $ $ #

21

./ $ $ $$ +

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SLIDE 22

Discrete Fourier transform

&' ./ ( (

./

2 )

22

6.5/ "

* $ 6.5/ $ ./

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SLIDE 23

Discrete Fourier transform

./ (

./ 2 )

DFT

23

[ ] X k [ ] x n

IDFT

time domain frequency domain

./ ( )

./ 2 )

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SLIDE 24

DFT Computation Using MATLAB

./ ( 7)7 2 )8

10

24

0.2 0.4 0.6 0.8 1 2 4 6 8 Normalized angular frequency Magnitude

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SLIDE 25

Matrix relations

$ $ & ./ (

./ 2 )

+ +

25

( # ( ..2/.)/!!!!!. )// ( ..2/.)/!!!!!. )//

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SLIDE 26

Matrix relations

$ + % (

)

) ) ) )

)
*

* * * * * * * * *** * * *

26
*

* * * * * ** * * )

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SLIDE 27

Matrix relations

$ % $ + + ( #

,$ + )

) ) )

27
( )
)

) ) ) )

)
*

* * * * * * * * *** * * * )

"'

(

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SLIDE 28

Computational complexity

+ + $ ./ (

./ ( ./ 2 )

28

1 # 5 + $ # % ) # + " $ + +

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SLIDE 29

FFT

$ $ $ $ , + + 9

29

$ "# $ " : % ' ! $ # $* ! # $ $ * ! $ $ ***

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SLIDE 30

FFT

!% ' # #* 87 ; 82 * )2< ; )222 * =3>=

30

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SLIDE 31

Inverse FFT

$ = #= % , ./ + ? # % ./ # %

31

% ./ # % ./

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SLIDE 32

Speedup of the FFT

32
4@ $ 0 A)B 5 )*

$ $ # )2C2*

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SLIDE 33

Properties of the Fourier transform

! 4 ! !

33

! %

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SLIDE 34

Properties of the Fourier transform

! ( 4

34
-

! )

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SLIDE 35

Properties of the Fourier transform

! ( * $

35

%* $ ( ( - 1 % $ ( ) % % 2 % % 2 ( )

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SLIDE 36

Properties of the Fourier transform

%' % % %* %

36

%

$ & ?

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SLIDE 37

The 2D discrete Fourier transform

$ ) & ( ' ( ) (

&

( 2 ) !!! () ' ( 2 ) !!! )

37

( 2 ) !!! () ' ( 2 ) !!! ) + ( 2 ) !!! ( ) & ( 2!) !!! ) ' ( & (

'

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SLIDE 38

The 2D discrete Fourier transform

$ # ) $

38
3 0 ( #

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SLIDE 39

The 2D discrete Fourier transform

) & '

& - &&

( 2 ) !!! () & ( 2 ) !!! )*

39

$

' ( ) ( ' ( ) &

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SLIDE 40

The 2D discrete Fourier transform

+ % & * $ ' %0

40

' (

.

' - '/

$ & ' ' (

'

'

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SLIDE 41

Scaling terms

$ )( $ $ % $ # , $ %

41

, $ % D#% 5# # , E # %9 ' (

& (

) (