c Ft Special

7/28/2019 c Ft Special

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Fourier Transforms of Special Functions

主講者：虞台文

http://www.google.com/search?hl=en&sa=X&oi=spell&resnum=0&ct=result&cd=1&q=unit+step+fourier+transform&spell=1



Content Introduction

More on Impulse Function Fourier Transform Related to Impulse Function

Fourier Transform of Some Special Functions

Fourier Transform vs. Fourier Series



Introduction Sufficient condition for the existence of a

Fourier transform

dt t f |)(|

That is, f (t ) is absolutely integrable. However, the above condition is not the

necessary one.



Some Unabsolutely Integrable Functions

Sinusoidal Functions: cos t , sin t ,…

Unit Step Function: u(t ).

Generalized Functions:

– Impulse Function (t ); and

– Impulse Train.




More on

Impulse Function



Dirac Delta Function

0

00

)( t

t

t and 1)(

dt t

0t

Also called unit impulse function.



Generalized Function The value of delta function can also be defined

in the sense of generalized function:

)0()()(

dt t t (t): Test Function

We shall never talk about the value of (t ).

Instead, we talk about the values of integrals

involving (t ).



Properties of Unit Impulse Function

)()()( 00 t dt t t t

Pf)

dt t t t )()( 0

Write t as t + t 0

dt t t t )()( 0

)( 0t




)0(

||

1)()(

a

dt t at

Pf)

dt t at )()(

Write t as t /a

Consider a>0

dt a

t t

a)(

1

)0(||

1

a

dt t at )()(

Consider a<0

dt a

t t

a)(

1

)0(||

1

a




)()0()()( t f t t f

Pf)

dt t t t f )()]()([

dt t t f t )]()()[(

)0()0( f

dt t t f )()()0(

dt t t f )()]()0([




)()0()()( t f t t f

Pf)

dt t at )()(

)(||

1

)( t aat

)0(||

1

a

dt t t

a)()(

||

1

dt t t

a)()(

||

1




)()0()()( t f t t f

)(||

1

)( t aat

0)( t t )()( t t



Generalized Derivatives

The derivative f ’ (t ) of an arbitrary

generalized function f (t ) is defined by:

dt t t f dt t t f )(')()()('

Show that this definition is consistent to the ordinarydefinition for the first derivative of a continuous function.

dt t t f )()(' dt t t f t t f

)(')()()(

=0



Derivatives of the -Function

)0(')(')()()('

dt t t dt t t

0

)()0(' ,

)()('

t dt

t d

dt

t d t

)0()1()()( )()( nnn dt t t

0

)()( )()0( ,

)()(

t n

nn

n

nn

dt

t d

dt

t d t



Product Rule)(')()()(')]'()([ t t f t t f t t f

dt t t t f )(')]()([

Pf) dt t t t f )(')]()([

dt t t f t )](')()[(

dt t t f t t f t )}()(')]'()(){[(

dt t t f t dt t t f t )]()'()[()]'()()[(

dt t t f t dt t t f t )]()'()[()]()()[('

dt t t f t t f t )()](')()()('[



Product Rule)()0(')(')0()(')( t f t f t t f

)()'()]'()([)(')( t t f t t f t t f

Pf)

)]'()0([ t f )(')0( t f

)()0(' t f



Unit Step Function u(t)

Define

0

)()()( dt t dt t t u

0t

u(t )

0001)(

t t t u



Derivative of the Unit Step Function

Show that )()(' t t u

dt t t u )()('

0)(' dt t

)]0()([ )0(

dt t t u )(')(

dt t t )()(



Derivative of the Unit Step Function

0t

u(t )

Derivative

0t

(t )




Fourier Transform

Related to

Impulse Function



Fourier Transform for (t)

1)(

F t

dt et t t j)()]([F 10

t

t je

0 t

(t )

0

1

F ( j)

F




Show that

d et t j

2

1)(

]1[)( 1 F t

d e t j1

2

1

d e t j

2

1

d e t j

2

1The integration converges to

in the sense of generalized function.

)(t




Show that

0

cos1

)( td t

d et t j

2

1)(

d t jt )sin(cos

2

1

td

jtd sin

2cos

2

1

0

cos1

td Converges to (t ) in the sense of

generalized function.



Two Identities for (t)

dxe y

jxy

2

1

)(

0

cos1

)( xydx y

These two ordinary integrations themselves are meaningless.

They converge to (t ) in the sense of generalized function.



Shifted Impulse Function

0)(0

t jet t

F

0)()]([ 0

t je j F t t f

F

0

1

|F ( j)|

F

Use the fact

0 t

(t t 0)

t 0




Fourier Transform of a

Some Special Functions



Fourier Transform of a Constant

)(2)()( A j F At f F

d Ae A j F t j][)( F

dt e At j )(

2

1

2

)(2 A



Fourier Transform of a Constant

)(2)()( A j F At f F

F

0t

A A2()

0

F ( j)



Fourier Transform of Exponential Wave

)(2)()( 00

j F et f t j F

)(2]1[ F

)]([])([ 00

j F et f t j

F

)(2][ 00 t j

eF



Fourier Transforms of Sinusoidal Functions

)()(cos 000 F t

)()(sin 000 j jt F

F

(+0)

0

F ( j)

(0)

0

0

t

f (t )=cos0t



Fourier Transform of Unit Step Function

)()]([ j F t uF Let )()]([ j F t uF

)0for (except1)()( t t ut u

]1[)]()([ F F t ut u

)(2)]([)]([ t ut u F F

)(2)()( j F j F

F ( j)=?

Can you guess it?




)(2)()( j F j F

Guess )()()( Bk j F

)()()()()()( B Bk k j F j F

)()()(2 B Bk

k

0 B() must be odd




Guess )()()( Bk j F k

)()(' t t u

)()]([ j F t uF

1)]([)]('[ t t u F F

)()]('[ j F jt uF

)]()([ B j

)()( B j j

0

j B

1

)(




Guess )()()( Bk j F k

j B

1

)(

j

t u1

)()( F




j

t u1

)()( F

F ()

0

| F ( j)|

0t

1

f (t )




Fourier Transform vs.

Fourier Series



Find the FT of a Periodic Function

Sufficient condition --- existence of FT

dt t f |)(|

Any periodic function does not satisfy this

condition.

How to find its FT (in the sense of general

function)?




We can express a periodic function f (t ) as:

T ect f

n

t jnn

2 ,)( 00

n

t jn

nect f j F 0

)]([)(F F

n

t jn

n ec ][0

F

n

n nc )(2 0

n

n nc )(2 0




We can express a periodic function f (t ) as:

T ect f

n

t jnn

2 ,)( 00

nn nc j F )(2)( 0

The FT of a periodic function consists of a sequence of

equidistant impulses located at the harmonic frequencies

of the function.



Example:

Impulse Train

0t

T 2T 3T T 2T 3T

n

T nT t t )()( Find the FT of the

impulse train.



Example:

Impulse Train

0t

T 2T 3T T 2T 3T

n


impulse train.

n

t jn

T eT

t 01

)(

cn

2



Example:

Impulse Train

0t

T 2T 3T T 2T 3T

n


impulse train.

n

t jn

T eT

t 01

)(

cn

n

T nT

t )(2

)]([ 0F

0

2



Example:

Impulse Train

0t

T 2T 3T T 2T 3T

n

T nT

t )(2

)]([ 0F

0

0 0 20 30020 30

2/T

F



Find Fourier Series Using

Fourier Transform

n

t jn

nect f 0)(

2/

2/

0)(1 T

T

t jn

n et f T

c

T /2 T /2

f (t )

t

T /2 T /2

f o(t )

t

t j

oo et f j F )()(

2/

2/)(

T

T

t jet f

)(

10 jn F T c on



Find Fourier Series Using

Fourier Transform

n

t jn

nect f 0)(

2/

2/

0)(1 T

T

t jn

n et f T

c

T /2 T /2

f (t )

t

T /2 T /2

f o(t )

t

t j

oo et f j F )()(

2/

2/)(

T

T

t jet f

)(

10 jn F T c on

Sampling the Fourier Transform of f o(t ) with period

2/T , we can find the Fourier Series of f (t ).



Example:

The Fourier Series of a Rectangular Wave

0

f (t )

d

1

t 0

t

f o(t )1

dt e j F d

d

t j

o

2/

2/)(

2sin

2 d

n

t jn

nect f 0)(

)(1

0 jn F T

c on

2

sin2 0

0

d n

Tn

2

sin1 0d n

n



Example:

The Fourier Transform of a Rectangular Wave

0

f (t )

d

1

t

n

t jn

nect f 0)(

)(1

0 jn F T

c on

sin2 0d n

sin1 0d n

F [ f (t )]=?

n

n nc j F )(2)( 0

)(2

sin2

)( 00

nd n

n j F

n

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