Leo Lam © 2010-2012

Signals and SystemsEE235

Leo Lam © 2010-2011

x squared equals 9x squared plus 1 equals yFind value of y

Leo Lam © 2010-2011

Today’s menu

• Fourier Transform Properties (cont’)• Duality• Loads of examples

Leo Lam © 2010-2011

Fourier Transform:

4

• Fourier Transform

• Inverse Fourier Transform:

Leo Lam © 2010-2012

FT Properties Example:

5

• Find FT for:

• We know the pair:

• So:

-8 0 8

G()

Leo Lam © 2010-2012

More Transform Pairs:

6

• More pairs:

time domain Fourier transform

21

aj 1

Leo Lam © 2010-2012

Periodic signals: Transform from Series

7

• Integral does not converge for periodic fns:

• We can get it from Fourier Series:• How? Find x(t) if• Using Inverse Fourier:

• So

)(2)( 0 X

tjtj edetx 0)(22

1)( 0

tje 0)(2 0

Leo Lam © 2010-2012

Periodic signals: Transform from Series

8

• We see this pair:

• More generally, if X(w) has equally spaced impulses:

• Then:

tje 0)(2 0

k

k kdX )(2)( 0

k

tjkk

tj eddeXtx 0)(2

1)(

Fourier Series!!!

Leo Lam © 2010-2012

Periodic signals: Transform from Series

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• If we know Series, we know Transform

• Then:

• Example: • We know:

• We can write:

k

k kdX )(2)( 0

k

tjkkedtx 0)(

)8cos()( ttf 8,5.0,5.0 011 dd

)8()8()( F

Leo Lam © 2010-2012

Summary

• Fourier Transform Pairs• FT Properties

Leo Lam © 2010-2011

Duality of Fourier Transform

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• Duality (very neat):

• Duality of the Fourier transform: If time domain signal f(t) has Fourier transform F(), then F(t) has Fourier transform 2 f(-)

• i.e. if:

• Then:

)(2

)(

0

0

0

0

tj

tj

e

ett

Changed sign

)(2)(

)()(

ftF

Ftf

Leo Lam © 2010-2011

Duality of Fourier Transform (Example)

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• Using this pair:

• Find the FT of– Where T=5

2

TsincT

T

trect

)(5

5)( tF2

tsinctg

52

52)(

rectrectG

)(2)( fG

Leo Lam © 2010-2011

Duality of Fourier Transform (Example)

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• Using this pair:

• Find the FT of

)(2)(

)(2)( )(

ueF

ueFa

a

jaatue at

10),(

jtatf

1)(

)(2)( fG

Leo Lam © 2010-2011

Convolution/Multiplication Example

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• Given f(t)=cos(t)e–tu(t) what is F()

)()(2

1)()( 2121

FFtftf

)1()1()()cos()( 11 Fttf

jFtuetf t

1

1)()()( 22

j

FFF

1

1)1()1(

2

1)()(

2

1)( 21

11

1

11

1

2

1)(

jjF

Leo Lam © 2010-2011

More Fourier Transform Properties

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Duality

Time-scaling

Multiplication

Differentiation

Integration

Conjugation

time domain Fourier transform

Dual of convolution

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( )n

n

d f t

dt ( )

nj F

( )t

f d

1( ) (0) ( )F F

j

*( )f t *( )F

Leo Lam © 2010-2011

Fourier Transform Pairs (Recap)

16 16

0cos( )t 0 0( ) ( )

0j te 02 ( )

0( )t t 0j te

1 2 ( )

1

j a

• Review:

Leo Lam © 2010-2011

Fourier Transform and LTI System

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• Back to the Convolution Duality:

• And remember:

• And in frequency domain

)()()()( 2121 FFtftf

Convolution in time

h(t) x(t)*h(t)x(t)Time domain

Multiplication in frequency

H(w) X(w)H(w)X(w)

Frequency domain

input signal’sFourier transform

output signal’sFourier transform

Leo Lam © 2010-2011

Fourier Transform and LTI (Example)

• Delay:

LTIh(t)

Time domain:

( )* ( 3) ( 3)x t t x t Frequency domain (FT):

3

3

( ) ( 3)

( ) ( ) ( ) ( )

j t j

j

H t e dt e

Y X H X e

Shift in time Add linear phase in frequency

( )

( ) 3

[ ( ) 3 ]

( ) ( )

( ) ( )

( )

j X

j X j

j X

X X e

Y X e e

X e

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Leo Lam © 2010-2011

Fourier Transform and LTI (Example)

• Delay:

• Exponential response

LTIh(t)

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Delay 3)3()3( tjtj ete

)3(3)( tjjtjtj eeeHe

Using Convolution Properties

Using FT Duality

Leo Lam © 2010-2011

Fourier Transform and LTI (Example)

• Delay:

• Exponential response

• Responding to Fourier Series

LTIh(t)

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Delay 3)3()( tjtj eHe

Delay 3jtjt ee

ttz

5.05.0

)cos()(

)3cos(

5.05.0 33

t

eeee jjtjjt

Leo Lam © 2010-2011

Another LTI (Example)

• Given Exponential response

• What does this system do? What is h(t)?

• And y(t) if

• Echo with amplification

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LTI532)(

)(

j

tj

eH

He

)5(3)(2)( ttth jtjt eettz 5.05.0)cos()(

)5cos(3)cos(2

)5.0(3)5.0(2)( )5()5(

tt

eeeety tjtjjtjt

Leo Lam © 2010-2011

Another angle of LTI (Example)

• Given graphical H(w), find h(t)

• What does this system do? What is h(t)?

• Linear phase constant delay

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)5()( tth

magnitude

w

w

phase

0

0

1

Slope=-5

5)( jeH

Leo Lam © 2010-2011

Another angle of LTI (Example)

• Given graphical H(w), find h(t)

• What does this system do (qualitatively

• Low-pass filter. No delay.

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magnitude

w

w

phase

0

0

1

Leo Lam © 2010-2011

Another angle of LTI (Example)

• Given graphical H(w), find h(t)

• What does this system do qualitatively?

• Bandpass filter. Slight delay.

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magnitude

w

w

phase

0

1

Leo Lam © 2010-2011

Another angle of LTI (Example)

• Given graphical H(w), find h(t)

• What does this system do qualitatively?

• Bandpass filter. Slight delay.

25

magnitude

w

w

phase

0

1

Leo Lam © 2010-2011

Summary

• Fourier Transforms and examples