Continuous-Time Fourier Transform

1

Content

Introduction

Fourier Integral

Fourier Transform

Properties of Fourier Transform

Convolution

Parseval’s Theorem

2

Continuous-Time Fourier Transform

Introduction

3

The Topic

Fourier

Series

Discrete

Fourier

Transform

Continuous

Fourier

Transform

Fourier

Transform

Continuous

Time

Discrete

Time

Per

iodic

Aper

iod

ic

4

Review of Fourier Series

Deal with continuous-time periodic signals.

Discrete frequency spectra.

A Periodic Signal

T 2T 3T

t

f(t)

5

Two Forms for Fourier Series

T

ntb

T

nta

atf

n

n

n

n

2sin

2cos

2)(

11

0Sinusoidal

Form

Complex

Form:

n

tjn

nectf0)( dtetf

Tc

T

T

tjn

n

2/

2/

0)(1

dttfT

aT

T2/

2/0 )(

2tdtntf

Ta

T

Tn 0

2/

2/cos)(

2

tdtntfT

bT

Tn 0

2/

2/sin)(

2

6

How to Deal with Aperiodic Signal?

A Periodic Signal

T

t

f(t)

If T, what happens?

7

Continuous-Time Fourier Transform

Fourier Integral

8

tjn

n

nT ectf0)(

Fourier Integral

n

tjnT

T

jn

T edefT

002/

2/)(

1

dtetfT

cT

T

tjn

Tn

2/

2/

0)(1

T

20

2

1 0

T

n

tjnT

T

jn

T edef00

0

2/

2/)(

2

1

LetT

20

0 dT

n

tjnT

T

jn

T edef00

2/

2/)(

2

1

1 ( )2

j j tf e d e d

9

dedeftf tjj)(2

1)(

Fourier Integral

F(j)

dtetfjF tj

)()(

dejFtf tj)(2

1)( Synthesis

Analysis

10

Fourier Series vs. Fourier Integral

n

tjn

nectf0)(

Fourier

Series:

Fourier

Integral:

dtetfT

cT

T

tjn

Tn

2/

2/

0)(1

dtetfjF tj

)()(

dejFtf tj)(2

1)(

Period Function

Discrete Spectra

Non-Period

Function

Continuous Spectra11

Continuous-Time Fourier Transform

Fourier Transform

12

Fourier Transform Pair

dtetfjF tj

)()(

dejFtf tj)(2

1)( Synthesis

Analysis

Fourier Transform:

Inverse Fourier Transform:

13

Existence of the Fourier Transform

dttf |)(|

Sufficient Condition:

f(t) is absolutely integrable, i.e.,

14

dtetfjF tj

)()(

Continuous Spectra

)()()( jjFjFjF IR

)(|)(| jejF FR(j)

FI(j)

()

MagnitudePhase

15

Example

1-1

1

t

f(t)

dtetfjF tj

)()( dte tj

1

1

1

1

1

tje

j

)(

jj eej 2sin

2sin 2c f

16

Example

1-1

1

t

f(t)

dtetfjF tj

)()( dte tj

1

1

1

1

1

tje

j

)(

jj eej

sin2

-10 -5 0 5 10-1

0

1

2

3

F(

)-10 -5 0 5 10

0

1

2

3

|F(

)|

-10 -5 0 5 100

2

4arg

[F(

)]

17

Example

dtetfjF tj

)()( dtee tjt

0

t

f(t)

et

dte tj

0

)(

j

1

18

Example

dtetfjF tj

)()( dtee tjt

0

t

f(t)

et

dte tj

0

)(

j

1

-10 -5 0 5 100

0.5

1

|F(j

)|

-10 -5 0 5 10-2

0

2

arg

[F(j

)]

=2

19

Continuous-Time Fourier Transform

Properties of

Fourier Transform

20

Notation

)()( jFtf F

)()]([ jFtfF

)()]([1 tfjF- F

21

Linearity

)()()()( 22112211 jFajFatfatfaF

22

Time Scaling

ajF

aatf

||

1)( F

23

Time Reversal

jFtf F)(

Pf)dtetftf tj

)()]([F dtetft

t

tj

)(

)()( tdetft

t

tj

)()( tdetft

t

tj

dtetft

t

tj

)( dtetft

t

tj

)(

dtetf tj

)( )( jF

24

Time Shifting

0)( 0tj

ejFttf

F

Pf)dtettfttf tj

)()]([ 00F dtettft

t

tj

)( 0

)()( 0)(0

0

0 ttdetftt

tt

ttj

dtetfet

t

tjtj

)(0

dtetfe tjtj

)(0tj

ejF 0)(

25

Frequency Shifting (Modulation)

)()( 00

jFetfj F

Pf)dteetfetf tj

tjtj

00 )(])([F

dtetftj

)( 0)(

)( 0 jF

26

Symmetry Property

)(2)]([ fjtFF

Pf)

dejFtf tj)()(2

dejFtf tj)()(2

dtejtFf tj

)()(2

Interchange symbols and t

)]([ jtFF27

Fourier Transform for Real Functions

If f(t) is a real function, and F(j) = FR(j) + jFI(j)

F(j) = F*(j)

dtetfjF tj

)()(

dtetfjF tj

)()(* )( jF28

Fourier Transform for Real FunctionsFourier Transform for Real Functions

If f(t) is a real function, and F(j) = FR(j) + jFI(j)

F(j) = F*(j)

FR(j) is even, and FI(j) is odd.

FR(j) = FR(j) FI(j) = FI(j)

Magnitude spectrum |F(j)| is even, and

phase spectrum () is odd.

29

Fourier Transform for Real FunctionsFourier Transform for Real Functions

If f(t) is real and even

F(j) is real

If f(t) is real and odd

F(j) is pure imaginary

Pf)

)()( tftf

Pf)

Even

)()( jFjF

)(*)( jFjFReal

)(*)( jFjF

)()( tftf Odd

)()( jFjF

)(*)( jFjFReal

)(*)( jFjF

30

Example:

)()]([ jFtfF ?]cos)([ 0 ttfF

Sol)

))((2

1cos)( 000

tjtjeetfttf

])([2

1])([

2

1]cos)([ 000

tjtjetfetfttf

FFF

)]([2

1)]([

2

100 jFjF

31

Example:

d/2d/2

1

t

wd(t)

d/2d/2t

f(t)=wd(t)cos0t

/2

/2

2 1( ) [ ( )] sin sin sin

2

dj t

d dd

dW j w t e dt fd d c fd

f

F

]cos)([)( 0ttwjF d F0

0

0

0 )(2

sin)(2

sin

dd

32

Example:

d/2d/2

1

t

wd(t)

d/2d/2t

f(t)=wd(t)cos0t

/2

/2

2( ) [ ( )] sin sin

2

dj t

d dd

dW j w t e dt d c fd

F

]cos)([)( 0ttwjF d F0

0

0

0 )(2

sin)(2

sin

dd

-60 -40 -20 0 20 40 60-0.5

0

0.5

1

1.5

F(j

)

d=2

0=5

33

Example:

t

attf

sin)( ?)( jF

Sol)

d/2d/2

1

t

wd(t)

2sin

2)(

djWd

)(22

sin2

)]([

dd w

td

tjtW FF

)(sin

)]([ 2

aw

t

attf FF

||1

||0

a

a

34

About writing

35

2[ ( )] sin 2 ( )

2d d

tdW jt w

t

F F

2

2

,2 2

sin[ ( )] ( )

( ) ( )[ ( )] sin ( ) Re Re

2 /

1 / 2( )

Re ( )

0

a

a

atf t w

t

a ff t cat w ct c

a a

if xx

be carreful ct x

otherwise

F F

F F

Fourier Transform of f’(t)

( ) and if lim ( ) 0t

f t F j f t

F

Pf)dtetftf tj

)(')]('[F

dtetfjetf tjtj )()(

)()(' jFjtf F

)( jFj36

Fourier Transform of f (n)(t)

( ) and if lim ( ) 0t

f t F j f t

F

)()()()( jFjtf nn F

37

Fourier Transform of Integral

( ) and if ( ) 0 0f t F j f t dt F

F

jFj

dxxft 1

)(F

Let dxxftt

)()( 0)(lim tt)()()]([)]('[ jjjFtft FF

)(1

)(

jFj

j

38

(Suite)

39

1( ) (0) ( ) ( )

t

f d F F jj

General case

By convolution with heaviside distribution

The Derivative of Fourier Transform

d

jdFtjtf FF )]([

Pf)

dtetfjF tj

)()(

dtetfd

d

d

jdF tj

)(

)(dtetf tj

)(

dtetjtf tj

)]([ )]([ tjtfF40

Continuous-Time Fourier Transform

Convolution

41

Basic Concept

Linear Systemfi(t) fo(t)=L[fi(t)]

fi(t)=a1 fi1(t) + a2 fi2(t) fo(t)=L[a1 fi1(t) + a2 fi2(t)]

A linear system satisfies fo(t) = a1L[fi1(t)] + a2L[fi2(t)]

= a1fo1(t) + a2fo2(t)42

Basic Concept

Time Invariant

System

fi(t) fo(t)

fi(t +t0) fo(t + t0)

fi(t t0) fo(t t0)

tfi(t)

tfo(t)

tfi(t+t0)

tfo(t+t0)

tfi(tt0)

tfo(tt0)

43

Basic Concept

Causal

System

fi(t) fo(t)

A causal system satisfies

fi(t) = 0 for t < t0 fo(t) = 0 for t < t0

44

Basic Concept

Causal

System

fi(t) fo(t)

tfi(t)

tfo(t)

t0t0

t0t

fi(t)t

fo(t)

t0fi(t)

t tfo(t)

t0t0

Which of the following

systems are causal?

45

Unit Impulse Response

LTI

System

(t) h(t)=L[(t)]

f(t) L[f(t)]=?

Facts: )()()()()( tfdtfdtf

dtfLtfL )()()]([

dtLf )]([)(

dthf )()( Convolution

46

Unit Impulse Response

Impulse Response

LTI System

h(t)f(t) f(t)*h(t)

47

)(*)()]([ thtftfL

Convolution Definition

dtfftf )()()( 21

The convolution of two functions f1(t) and

f2(t) is defined as:

)(*)( 21 tftf

48

Properties of Convolution

)(*)()(*)( 1221 tftftftf

dtfftftf )()()(*)( 2121

dtff )()( 21

)(])([)( 21

tdttftf

t

t

dftf )()( 21

dftf )()( 21 )(*)( 12 tftf

49

Impulse Response

LTI System

h(t)

f(t) f(t)*h(t)

Properties of Convolution

)(*)()(*)( 1221 tftftftf

Impulse Response

LTI System

f(t)

h(t) h(t)*f(t)

50

Properties of Convolution

)](*)([*)()(*)](*)([ 321321 tftftftftftf

51

Properties of Convolution

)](*)([*)()(*)](*)([ 321321 tftftftftftf

h1(t) h2(t) h3(t)

h2(t) h3(t) h1(t)

The following two

systems are identical

52

Properties of Convolution

)()(*)( tfttf (t)f(t) f(t)

dtfttf )()()(*)(

dtf )()(

)(tf

53

Properties of Convolution

)()(*)( tfttf (t)f(t) f(t)

)()(*)( TtfTttf

dTtfTttf )()()(*)(

dTtf )()(

)( Ttf 54

Properties of Convolution

f(t) f(t T)

)()(*)( TtfTttf

0 T

(tT)

tf (t)

0t

f (t)

0 T

55

Properties of Convolution

f(t) f(t T)

)()(*)( TtfTttf

0 T

(tT)

tf (t)

0t

f (t)

0 T

System function (tT) serves as an

ideal delay or a copier.

56

Properties of Convolution

)()()(*)( 2121 jFjFtftfF

dtedtfftftfF tj

)()()](*)([ 2121

ddtetff tj)()( 21

dejFf j)()( 21

defjF j)()( 12 )()( 21 jFjF57

Properties of Convolution

)()()(*)( 2121 jFjFtftfF

dtedtfftftfF tj

)()()](*)([ 2121

ddtetff tj)()( 21

dejFf j)()( 21

defjF j)()( 12 )()( 21 jFjF

Time Domain Frequency Domain

convolution multiplication

58

Properties of Convolution

)()()(*)( 2121 jFjFtftfF

Time Domain Frequency Domain

convolution multiplication

Impulse Response

LTI System

h(t)f(t) f(t)*h(t)

Impulse Response

LTI System

H(j)F(j) F(j)H(j)

59

Properties of Convolution

)()()(*)( 2121 jFjFtftfF

Time Domain Frequency Domain

convolution multiplication

F(j)

F(j)H1(j)

H2(j)H1(j) H3(j)

F(j)H1(j)H2(j)

F(j)H1(j)H2(j)H3(j)

60

Properties of Convolution

)()()(*)( 2121 jFjFtftfF

An Ideal Low-Pass Filter

0

Fi(j)

0

Fo(j)

0

H(j)

pp

1

61

Properties of Convolution

)()()(*)( 2121 jFjFtftfF

An Ideal High-Pass Filter

0

Fi(j)

0

H(j)

pp

1

0

Fo(j)

62

Properties of Convolution

)(*)(2

1)()( 2121

jFjFtftf F

djFjFtftf )]([)(

2

1)()( 2121

F

63

Properties of Convolution

)(*)(2

1)()( 2121

jFjFtftf F

djFjFtftf )]([)(

2

1)()( 2121

F

Time Domain Frequency Domain

multiplication convolution

64

Continuous-Time Fourier Transform

Parseval’s Theorem

65

Properties of Convolution

djFjFdttftf ][)(

2

1)]()([ 2121

djFjFtftf )]([)(

2

1)()( 2121

F

djFjFdtetftf tj )]([)(

2

1)]()([ 2121

djFjFdttftf )]([)(

2

1)]()([ 2121=0

66

Properties of Convolution

djFjFdttftf ][)(

2

1)]()([ 2121

If f1(t) and f2(t) are real functions,

djFjFdttftf ][)(

2

1)]()([ *2121

f2(t) real ][][*

22 jFjF67

Parseval’s Theorem:Energy Preserving

djFdttf 22 |)(|

2

1|)(|

dtetftf tj

)(*)](*[F

djFjF )]([*)(

2

1

2| ( ) | ( ) *( ) ( )f t dt f t f t dt Energy of f t

*

)(

dtetf tj )(* jF

djF 2|)(|

2

1

68

Isometry

69

1 2 1 2 1 21

( ), ( ) ( ), ( ) ,2

f t f t F F F F

2 2 22 2 21

2L L Lf t F F