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Continuous-Time Fourier Transform

Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

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Page 1: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Continuous-Time Fourier Transform

Page 2: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Content

Introduction

Fourier Integral

Fourier Transform

Properties of Fourier Transform

Convolution

Parseval’s Theorem

Page 3: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Continuous-Time Fourier Transform

Introduction

Page 4: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

The Topic

Fourier

Series

Discrete

Fourier

Transform

Continuous

Fourier

Transform

Fourier

Transform

Continuous

Time

Discrete

Time

Per

iodic

Aper

iod

ic

Page 5: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Review of Fourier Series

Deal with continuous-time periodic signals.

Discrete frequency spectra.

A Periodic Signal

T 2T 3T

t

f(t)

Page 6: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Two Forms for Fourier Series

T

ntb

T

nta

atf

n

n

n

n

2sin

2cos

2)(

11

0Sinusoidal

Form

Complex

Form:

n

tjn

nectf 0)( dtetfT

cT

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

Page 7: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

How to Deal with Aperiodic Signal?

A Periodic Signal

T

t

f(t)

If T, what happens?

Page 8: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Continuous-Time Fourier Transform

Fourier Integral

Page 9: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

tjn

n

nT ectf 0)(

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 edef 00

0

2/

2/)(

2

1

LetT

20

0 dT

n

tjnT

T

jn

T edef 002/

2/)(

2

1

dedef tjj

T )(2

1

Page 10: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

dedeftf tjj)(2

1)(

Fourier Integral

F(j)

dtetfjF tj

)()(

dejFtf tj)(2

1)( Synthesis

Analysis

Page 11: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Fourier Series vs. Fourier Integral

n

tjn

nectf 0)(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 Spectra

Page 12: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Continuous-Time Fourier Transform

Fourier Transform

Page 13: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Fourier Transform Pair

dtetfjF tj

)()(

dejFtf tj)(2

1)( Synthesis

Analysis

Fourier Transform:

Inverse Fourier Transform:

Page 14: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Existence of the Fourier Transform

dttf |)(|

Sufficient Condition:

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

Page 15: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

dtetfjF tj

)()(

Continuous Spectra

)()()( jjFjFjF IR

)(|)(| jejF FR(j)

FI(j)

()

MagnitudePhase

Page 16: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Example

1-1

1

t

f(t)

dtetfjF tj

)()( dte tj

1

1

1

1

1

tje

j

)(

jj eej 2sin

2sin 2c f

Page 17: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

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(

)]

Page 18: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Example

dtetfjF tj

)()( dtee tjt

0

t

f(t)

et

dte tj

0

)(

j

1

Page 19: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

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

Page 20: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Continuous-Time Fourier Transform

Properties of

Fourier Transform

Page 21: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Notation

)()( jFtf F

)()]([ jFtfF

)()]([1 tfjF- F

Page 22: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Linearity

)()()()( 22112211 jFajFatfatfa F

Page 23: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Time Scaling

ajF

aatf

||

1)( F

Page 24: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

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

Page 25: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Time Shifting

0)( 0

tjejFttf

F

Pf)dtettfttf tj

)()]([ 00F dtettft

t

tj

)( 0

)()( 0

)(0

0

0 ttdetftt

tt

ttj

dtetfet

t

tjtj

)(0

dtetfe tjtj

)(0

tjejF 0)(

Page 26: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Frequency Shifting (Modulation)

)()( 00

jFetf

j F

Pf)dteetfetf tjtjtj

00 )(])([F

dtetftj

)( 0)(

)( 0 jF

Page 27: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Symmetry Property

)(2)]([ fjtFF

Pf)

dejFtf tj)()(2

dejFtf tj)()(2

dtejtFf tj

)()(2

Interchange symbols and t

)]([ jtFF

Page 28: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

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

)()(* )( jF

Page 29: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

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.

Page 30: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

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

Page 31: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Example:

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

Sol)

))((2

1cos)( 00

0

tjtjeetfttf

])([2

1])([

2

1]cos)([ 00

0

tjtjetfetfttf

FFF

)]([2

1)]([

2

100 jFjF

Page 32: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

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

Page 33: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Example:

d/2d/2

1

t

wd(t)

d/2d/2t

f(t)=wd(t)cos0t

2sin

2)]([)(

2/

2/

ddtetwjW

d

d

tj

dd 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

Page 34: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

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

Page 35: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Fourier Transform of f’(t)

0)(lim and )(

tfjFtft

F

Pf)dtetftf tj

)(')]('[F

dtetfjetf tjtj )()(

)()(' jFjtf F

)( jFj

Page 36: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Fourier Transform of f (n)(t)

0)(lim and )(

tfjFtft

F

)()()()( jFjtf nn F

Page 37: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Fourier Transform of f (n)(t)

0)(lim and )(

tfjFtft

F

)()()()( jFjtf nn F

Page 38: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Fourier Transform of Integral

00)( and )(

FdttfjFtf F

jFj

dxxft 1

)(F

Let dxxftt

)()( 0)(lim

t

t

)()()]([)]('[ jjjFtft FF

)(1

)(

jFj

j

Page 39: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

The Derivative of Fourier Transform

d

jdFtjtf FF )]([

Pf)

dtetfjF tj

)()(

dtetfd

d

d

jdF tj

)(

)(dtetf tj

)(

dtetjtf tj

)]([ )]([ tjtfF

Page 40: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Continuous-Time Fourier Transform

Convolution

Page 41: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

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)

Page 42: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

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)

Page 43: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Basic Concept

Causal

System

fi(t) fo(t)

A causal system satisfies

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

Page 44: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Basic Concept

Causal

System

fi(t) fo(t)

tfi(t)

tfo(t)

t0t0

t0

tfi(t)

tfo(t)

t0fi(t)

t tfo(t)

t0t0

Which of the following

systems are causal?

Page 45: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Unit Impulse Response

LTI

System

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

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

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

dtfLtfL )()()]([

dtLf )]([)(

dthf )()( Convolution

Page 46: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Unit Impulse Response

LTI

System

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

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

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

dtfLtfL )()()]([

dtLf )]([)(

dthf )()( Convolution

)(*)()]([ thtftfL

Page 47: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Unit Impulse Response

Impulse Response

LTI System

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

Page 48: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Convolution Definition

dtfftf )()()( 21

The convolution of two functions f1(t) and

f2(t) is defined as:

)(*)( 21 tftf

Page 49: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

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

dtfftftf )()()(*)( 2121

dtff )()( 21

)(])([)( 21

tdttftf

t

t

dftf )()( 21

dftf )()( 21

)(*)( 12 tftf

Page 50: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

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)

Page 51: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

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

Page 52: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

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

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

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

The following two

systems are identical

Page 53: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

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

dtfttf )()()(*)(

dtf )()(

)(tf

Page 54: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

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

)()(*)( TtfTttf

dTtfTttf )()()(*)(

dTtf )()(

)( Ttf

Page 55: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

f(t) f(t T)

)()(*)( TtfTttf

0 T

(tT)

tf (t)

0t

f (t)

0 T

Page 56: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

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.

Page 57: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

)()()(*)( 2121 jFjFtftf F

dtedtfftftfF tj

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

ddtetff tj)()( 21

dejFf j)()( 21

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

Page 58: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

)()()(*)( 2121 jFjFtftf F

dtedtfftftfF tj

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

ddtetff tj)()( 21

dejFf j)()( 21

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

Time Domain Frequency Domain

convolution multiplication

Page 59: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

)()()(*)( 2121 jFjFtftf F

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)

Page 60: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

)()()(*)( 2121 jFjFtftf F

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)

Page 61: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

)()()(*)( 2121 jFjFtftf F

An Ideal Low-Pass Filter

0

Fi(j)

0

Fo(j)

0

H(j)

pp

1

Page 62: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

)()()(*)( 2121 jFjFtftf F

An Ideal High-Pass Filter

0

Fi(j)

0

H(j)

pp

1

0

Fo(j)

Page 63: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

)(*)(2

1)()( 2121

jFjFtftf F

djFjFtftf )]([)(

2

1)()( 2121

F

Page 64: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

)(*)(2

1)()( 2121

jFjFtftf F

djFjFtftf )]([)(

2

1)()( 2121

F

Time Domain Frequency Domain

multiplication convolution

Page 65: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Continuous-Time Fourier Transform

Parseval’s Theorem

Page 66: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

djFjFdttftf ][)(

2

1)]()([ 2121

djFjFtftf )]([)(

2

1)()( 2121

F

djFjFdtetftf tj )]([)(

2

1)]()([ 2121

djFjFdttftf )]([)(

2

1)]()([ 2121=0

Page 67: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Properties of Convolution

djFjFdttftf ][)(

2

1)]()([ 2121

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

djFjFdttftf ][)(

2

1)]()([ *

2121

f2(t) real ][][ *

22 jFjF

Page 68: Continuous-Time Fourier Transform©orie du signal/CFT-R.Ceschi.pdf · The Topic Fourier Series Discrete Fourier Transform Continuous Fourier Transform Fourier Transform Continuous

Parseval’s Theorem:Energy Preserving

djFdttf 22 |)(|

2

1|)(|

dtetftf tj

)(*)](*[F

djFjF )]([*)(

2

1

dttftfdttf

)(*)(|)(| 2

*

)(

dtetf tj )(* jF

djF 2|)(|

2

1