Leo Lam © 2010-2013

Signals and SystemsEE235

Leo Lam © 2010-2013

Fourier Transform:

2

• Fourier Formulas:• Inverse Fourier Transform:

• Fourier Transform:

Time domain toFrequency domain

Leo Lam © 2010-2013

Fourier Transform (delta function):

3

• Fourier Transform of

• Standard Fourier Transform pair notation

dtetF tj

)()(

( ) ( )f t t

1)()(

dttF

1)( t

Leo Lam © 2010-2013

Fourier Transform (rect function):

4

• Fourier Transform of

• Plot for T=1?

t-T/2 0 T/2

1

Define

-10 -8 -6 -4 -2 0 2 4 6 8 10-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Leo Lam © 2010-2013

Fourier Transform (rect function):

5

• Fourier Transform of

• Observation:– Wider pulse (in t) <-> taller narrower spectrum– Extreme case:

• <->

-10 -8 -6 -4 -2 0 2 4 6 8 10-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

)2

()(T

sincTF

Peak=pulse width (example: width=1)

T

n2Zero-Crossings:

)(DC

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Fourier Transform - Inverse relationship

6

• Inverse relationship between time/frequency

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Fourier Transform - Inverse

7

• Inverse Fourier Transform (Synthesis)

• Example:

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Fourier Transform - Inverse

8

• Inverse Fourier Transform (Synthesis)

• Example:

• Single frequency spike in w: exponential time signal with that frequency in t

A single spike in frequencyComplex exponential in time

Leo Lam © 2010-2013

Fourier Transform Properties

9

• A Fourier Transform “Pair”: f(t) F()• Re-usable!

Scaling

Additivity

Convolution

Time shift

time domain Fourier transform

Leo Lam © 2010-2013

How to do Fourier Transform

10

• Three ways (or use a combination) to do it:– Solve integral– Use FT Properties (“Spiky signals”)– Use Fourier Transform table (for known

signals)

Leo Lam © 2010-2013

FT Properties Example:

11

• Find FT for:

• We know the pair:

• So:

-8 0 8

G()

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More Transform Pairs:

12

• More pairs:

time domain Fourier transform

21

aj 1

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Periodic signals: Transform from Series

13

• 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 © 2010-2013

Periodic signals: Transform from Series

14

• 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 © 2010-2013

Periodic signals: Transform from Series

15

• 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 © 2010-2013

Summary

• Fourier Transform Pairs• FT Properties

Leo Lam © 2010-2013

Duality of Fourier Transform

17

• 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 © 2010-2013

Duality of Fourier Transform (Example)

18

• 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 © 2010-2013

Duality of Fourier Transform (Example)

19

• Using this pair:

• Find the FT of

)(2)(

)(2)( )(

ueF

ueFa

a

jaatue at

10),(

jtatf

1)(

)(2)( fG