4. cft

A presentation on Continuous-Time Fourier Transform

By:-Sunil Kumar Yadav (2013UEE1176)Pawan Kumar Jangid (2013UEE1166)Manish Kumar Bagara (2013UEE1180)Chandan Kumar (2013UEE1203)

Subject:- Network, Signal & Systems

ContentIntroductionFourier IntegralFourier TransformProperties of Fourier TransformConvolutionParseval’s Theorem

Continuous-Time Fourier Transform

Introduction

The Topic

FourierSeries

FourierSeries

DiscreteFourier

Transform

DiscreteFourier

Transform

ContinuousFourier

Transform

ContinuousFourier

Transform

FourierTransform

FourierTransform

ContinuousTime

DiscreteTime

Per

iodi

cA

peri

odic

Review of Fourier Series

Deal with continuous-time periodic signals. Discrete frequency spectra.

A Periodic SignalA Periodic Signal

T 2T 3T

t

f(t)

Two Forms for Fourier Series

T

ntb

T

nta

atf

nn

nn

2sin

2cos

2)(

11

0

T

ntb

T

nta

atf

nn

nn

2sin

2cos

2)(

11

0SinusoidalForm

ComplexForm:

n

tjnnectf 0)(

n

tjnnectf 0)( dtetf

Tc

T

T

tjnn

2/

2/

0)(1

dttfT

aT

T2/

2/0 )(2

tdtntfT

aT

Tn 0

2/

2/cos)(

2

tdtntfT

bT

Tn 0

2/

2/sin)(

2

How to Deal with Aperiodic Signal?

A Periodic SignalA Periodic Signal

T

t

f(t)

If T, what happens?


Fourier Integral

tjn

nnT ectf 0)(

Fourier Integral

n

tjnT

T

jnT edef

T00

2/

2/)(

1

dtetfT

cT

T

tjnTn

2/

2/

0)(1

T

20

2

1 0

T

n

tjnT

T

jnT edef 00

0

2/

2/)(

2

1

LetT

20

0 dT

n

tjnT

T

jnT edef 00

2/

2/)(

2

1

dedef tjjT )(

2

1

dedeftf tjj)(2

1)(

Fourier Integral

F(j)

dtetfjF tj

)()(

dejFtf tj)(2

1)( Synthesis

Analysis

Fourier Series vs. Fourier Integral

n

tjnnectf 0)(

n

tjnnectf 0)(

FourierSeries:

FourierIntegral:

dtetfT

cT

T

tjnTn

2/

2/

0)(1 dtetf

Tc

T

T

tjnTn

2/

2/

0)(1

dtetfjF tj

)()( dtetfjF tj

)()(

dejFtf tj)(2

1)(

dejFtf tj)(2

1)(

Period Function

Discrete Spectra

Non-PeriodFunction

Continuous Spectra


Fourier Transform

Fourier Transform Pair

dtetfjF tj

)()( dtetfjF tj

)()(

dejFtf tj)(2

1)(

dejFtf tj)(2

1)( Synthesis

Analysis

Fourier Transform:

Inverse Fourier Transform:

Existence of the Fourier Transform

dttf |)(|

dttf |)(|

Sufficient Condition:

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

dtetfjF tj

)()(

Continuous Spectra

)()()( jjFjFjF IR

)(|)(| jejF FR(j)

FI(j)

|F(j)|

()

MagnitudePhase

Example

1-1

1

t

f(t)

dtetfjF tj

)()( dte tj

1

1

1

1

1

tje

j

)(

jj eej

sin2

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

4ar

g[F

()]

-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

4ar

g[F

()]

Example

dtetfjF tj

)()( dtee tjt

0

t

f(t)

et

dte tj

0

)(

j

1

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

-10 -5 0 5 100

0.5

1

|F(j

)|

-10 -5 0 5 10-2

0

2

arg[

F(j

)]

=2


Properties of

Fourier Transform

Notation

)()( jFtf F )()( jFtf F

)()]([ jFtfF )()]([ jFtfF

)()]([1 tfjF- F )()]([1 tfjF- F

Linearity

)()()()( 22112211 jFajFatfatfa F )()()()( 22112211 jFajFatfatfa F

Time Scaling

a

jFa

atf||

1)( F

a

jFa

atf||

1)( F

Time Reversal

jFtf F)( jFtf F)(

Pf) dtetftf tj

)()]([F dtetft

t

tj

)(

)()( tdetft

t

tj

)()( tdetft

t

tj

dtetft

t

tj

)( dtetft

t

tj

)(

dtetf tj

)( )( jF

Time Shifting

0)( 0tjejFttf F 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)(

Frequency Shifting (Modulation)

)()( 00 jFetf j F )()( 0

0 jFetf j F

Pf)dteetfetf tjtjtj

00 )(])([F

dtetf tj

)( 0)(

)( 0 jF

Symmetry Property

)(2)]([ fjtFF )(2)]([ fjtFF

Pf)

dejFtf tj)()(2

dejFtf tj)()(2

dtejtFf tj

)()(2

Interchange symbols and t

)]([ jtFF

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

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.

F R( j) = F R(j) F I( j) = F I(j)

Magnitude spectrum |F(j)| is even, and phase spectrum () is odd.

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

Example:

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

Sol)

))((2

1cos)( 00

0tjtj eetfttf

])([2

1])([

2

1]cos)([ 00

0tjtj etfetfttf FFF

)]([2

1)]([

2

100 jFjF

Example:

d/2d/2

1

t

wd(t)

d/2d/2t

f(t)=wd(t)cos0t

2sin

2)]([)(

2/

2/

ddtetwjW

d

d

tjdd F

]cos)([)( 0ttwjF d F0

0

0

0 )(2

sin)(2

sin

dd

Example:

d/2d/2

1

t

wd(t)

d/2d/2t

f(t)=wd(t)cos0t

2sin

2)]([)(

2/

2/

ddtetwjW

d

d

tjdd 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

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

0

0.5

1

1.5

F(j

)

d=2

0=5

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

Fourier Transform of f’(t)

0)(lim and )(

tfjFtft

F 0)(lim and )(

tfjFtft

F

Pf) dtetftf tj

)(')]('[F

dtetfjetf tjtj )()(

)()(' jFjtf F )()(' jFjtf F

)( jFj

Fourier Transform of f (n)(t)

0)(lim and )(

tfjFtft

F 0)(lim and )(

tfjFtft

F

)()()()( jFjtf nn F )()()()( jFjtf nn F

Fourier Transform of f (n)(t)

0)(lim and )(

tfjFtft

F 0)(lim and )(

tfjFtft

F

)()()()( jFjtf nn F )()()()( jFjtf nn F

Fourier Transform of Integral

00)( and )(

FdttfjFtf F 00)( and )(

FdttfjFtf F

jFj

dxxft 1

)(F

jFj

dxxft 1

)(F

Let dxxftt

)()( 0)(lim

t

t

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

)(1

)(

jFj

j

The Derivative of Fourier Transform

d

jdFtjtf FF )]([

d

jdFtjtf FF )]([

Pf)dtetfjF tj

)()(

dtetfd

d

d

jdF tj

)()(

dtetf tj

)(

dtetjtf tj

)]([ )]([ tjtfF


Convolution

Basic Concept

Linear SystemLinear 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)

Basic Concept

Time InvariantSystem

Time InvariantSystem

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)

Basic Concept

CausalSystemCausalSystem

fi(t) fo(t)

A causal system satisfies

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

Basic Concept

CausalSystemCausalSystem

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?

Which of the following systems are causal?

Unit Impulse Response

LTISystem

LTISystem

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

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

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

dtfLtfL )()()]([

dtLf )]([)(

dthf )()( Convolution


LTISystem

LTISystem

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

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

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

dtfLtfL )()()]([

dtLf )]([)(

dthf )()( Convolution

)(*)()]([ thtftfL )(*)()]([ thtftfL


Impulse ResponseLTI System

h(t)


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

Convolution Definition

dtfftf )()()( 21

The convolution of two functions f1(t) and f2(t) is defined as:

)(*)( 21 tftf

Properties of Convolution

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

dtfftftf )()()(*)( 2121

dtff )()( 21

)(])([)( 21

tdttftf

t

t

dftf )()( 21

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


h(t)


h(t)

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


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


f(t)


f(t)

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


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


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

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

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

The following two systems are identical

The following two systems are identical


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

dtfttf )()()(*)(

dtf )()(

)(tf


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

)()(*)( TtfTttf )()(*)( TtfTttf

dTtfTttf )()()(*)(

dTtf )()(

)( Ttf


f(t) f(t T)


0 T

(tT)

tf (t)

0t

f (t)

0 T


f(t) f(t T)


0 T

(tT)

tf (t)

0t

f (t)

0 T

System function (tT) serves as an ideal delay or a copier.System function (tT) serves as an ideal delay or a copier.


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

dtedtfftftfF tj

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

ddtetff tj)()( 21

dejFf j)()( 21

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



dtedtfftftfF tj

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

ddtetff tj)()( 21

dejFf j)()( 21

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

Time Domain Frequency Domain

convolution multiplication






h(t)


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


H(j)


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





F(j)

F(j)H1(j)

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

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

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



An Ideal Low-Pass Filter

0

Fi(j)

0

Fo(j)

0

H(j)

pp

1



An Ideal High-Pass Filter

0

Fi(j)

0

H(j)

pp

1

0

Fo(j)


)(*)(2

1)()( 2121

jFjFtftf F )(*)(

2

1)()( 2121

jFjFtftf F

djFjFtftf )]([)(

2

1)()( 2121

F

djFjFtftf )]([)(

2

1)()( 2121

F


)(*)(2

1)()( 2121

jFjFtftf F )(*)(

2

1)()( 2121

jFjFtftf F

djFjFtftf )]([)(

2

1)()( 2121

F

djFjFtftf )]([)(

2

1)()( 2121

F


multiplication convolution


Parseval’s Theorem


djFjFdttftf ][)(

2

1)]()([ 2121

djFjFdttftf ][)(

2

1)]()([ 2121

djFjFtftf )]([)(

2

1)()( 2121

F

djFjFtftf )]([)(

2

1)()( 2121

F

djFjFdtetftf tj )]([)(

2

1)]()([ 2121

djFjFdttftf )]([)(

2

1)]()([ 2121=0=0


djFjFdttftf ][)(

2

1)]()([ 2121

djFjFdttftf ][)(

2

1)]()([ 2121

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

djFjFdttftf ][)(

2

1)]()([ *

2121

djFjFdttftf ][)(

2

1)]()([ *

2121

f2(t) real ][][ *22 jFjF

Parseval’s Theorem:Energy Preserving

djFdttf 22 |)(|

2

1|)(|

djFdttf 22 |)(|

2

1|)(|

dtetftf tj

)(*)](*[F

djFjF )]([*)(

2

1

dttftfdttf

)(*)(|)(| 2

*

)(

dtetf tj )(* jF

djF 2|)(|

2

1

Thank You

Engineering

4. cft