bab 18 sistem linier

8/20/2019 Bab 18 Sistem Linier

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Chapter 18Chapter 18Fourier TransformFourier Transform

Copyright © The McGraw-Hill Companies, Inc. Permission require !or repro uction or isplay.

"le#an er-$a i%u"le#an er-$a i%u

FuFu ndamentals ofndamentals ofElectric CircuitsElectric Circuits


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&

'ourier Trans!orm'ourier Trans!orm

Chapter 1(Chapter 1(

1(.1 )e!inition o! the 'ourier Trans!orm

1(.& Properties o! the 'ourier Trans!orm 1(.* Circuit "pplications


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*

1(.1 )e!inition o! 'ourier Trans!orm +11(.1 )e!inition o! 'ourier Trans!orm +1

It is an integral trans!ormation o! f(t) !romthe time omain to the !requency omain F( ω )

F( ω ) is a comple# !unction its magnitu e is

calle the amplitude spectrum , while its phaseis calle the phase spectrum .

Gi/en a !unction f(t) , its !ourier trans!ormenote 0y F( ω ), is e!ine 0y

∫ ∞

∞−−= )()( dt et f F t jω ω


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1(.1 )e!inition o! 'ourier Trans!orm +&1(.1 )e!inition o! 'ourier Trans!orm +&

2#ample 1

)etermine the 'ourier trans!orm o! a singlerectangular pulse o! wi e τ an height ", asshown 0elow.

A rectangular pulse


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1(.1 )e!inition o! 'ourier Trans!orm +*1(.1 )e!inition o! 'ourier Trans!orm +*

2sin

22

2/

2/)(

2/2/

2/

2/

ωτ τ

ω

τ

τ

ω

ω

ωτ ωτ

ω

τ

τ

ω

c A

jee A

e j A

dt Ae F

j j

t j

t j

=

−=

−−==

−

−

−∫

$olution4

Amplitude spectrum ofthe rectangular pulse


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1(.1 )e!inition o! 'ourier Trans!orm +1(.1 )e!inition o! 'ourier Trans!orm +

2#ample &4

60tain the 'ourier trans!orm o! the 7switche -

on8 e#ponential !unction as shown 0elow.


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1(.1 )e!inition o! 'ourier Trans!orm +31(.1 )e!inition o! 'ourier Trans!orm +3

$olution4

ω

ω

ω

ω ω

ja

dt e

dt eedt et f F

et uet f

t ja

t j jat t j

at at

+=

=

==

<>

==

∫ ∫ ∫

∞∞−

+−

∞

∞−−−∞

∞−−

−−

1

)()(

Hence,

0 t,0

0 t,)()(

)(


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(

1(.& Properties o! 'ourier Trans!orm +11(.& Properties o! 'ourier Trans!orm +1

[ ] )()()()( 22112211 ω ω F a F at f at f a F +=+

:inearity4

I! F 1( ω ) an F 2( ω ) are, respecti/ely, the 'ourier

Trans!orms o! f 1(t) an f 2(t)

2#ample *4

[ ] ( ) ( )[ ] [ ])()(21

)sin( 000 00 ω ω δ ω ω δ π ω ω ω

−−+=−= − je F e F

jt F t jt j


9/19


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1?

1(.& Properties o! 'ourier Trans!orm +*1(.& Properties o! 'ourier Trans!orm +*

[ ] )()( 00 ω ω

F et t f F t j−=−

Time $hi!ting4

I! F ( ω ) is the 'ourier Trans!orms o! f (t), then

2#ample 4

[ ]ω

ω

je

t ue F j

t

+=−−

−−

1)2(

2)2(


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11

1(.& Properties o! 'ourier Trans!orm +1(.& Properties o! 'ourier Trans!orm +

[ ] )()( 00 ω ω ω −= F et f F t j

'requency $hi!ting +"mplitu e Mo ulation 4

I! F ( ω ) is the 'ourier Trans!orms o! f (t), then

2#ample 34

[ ] )(21

)(21

)cos()( 000 ω ω ω ω ω ++−= F F t t f F


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1&


)()( s F jt udt df

F ω =

Time )i!!erentiation4

I! F ( ω ) is the 'ourier Trans!orms o! f (t), then the

'ourier Trans!orm o! its eri/ati/e is

2#ample 54

( )ω ja

t uedt d

F at +=

− 1)(


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1*


)()0()(

)( ω δ π ω

ω F

j F

dt t f F t =∫ ∞−

Time Integration4

I! F ( ω ) is the 'ourier Trans!orms o! f (t), then the

'ourier Trans!orm o! its integral is

2#ample 94

[ ] )(1)( ω πδ ω

+= j

t u F


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1


[ ] )(*)()( ω ω F F t f L =−=−

@e/ersal4

I! F( ω ) is the 'ourier Trans!orms o! f (t), then

re/ersing f(t) a0out the time a#is re/erses F( ω ) a0out !requency.

2#ample (4

[ ] [ ] )(2)()(1 ω πδ =−+= t ut u F F


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1(.& Properties o! 'ourier Trans!orm +(1(.& Properties o! 'ourier Trans!orm +(

[ ] [ ] )(2)( )()( ω π ω −=⇒= f t F F F t f F

)uality4

I! F( ω ) is the 'ourier Trans!orms o! f (t), then the

'ourier trans!orm o! F(t) is 2 π f(- ω ).

12

)(

then,)(If

2 +=

= −

ω ω F

et f t 2#ample ;4

[ ] ω π ω π −=+=

e f F

t F(t)

2)(2

then1

2 If 2Duality

property


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I! X( ω ) , H( ω ) an Y( ω ) are the 'ourier trans!orms

o! x(t) , h(t) , an y(t) , respecti/ely, then

It is e!ine as

1(.& Properties o! 'ourier Trans!orm +;1(.& Properties o! 'ourier Trans!orm +;

[ ] )(*)(21

)()()( ω ω π

ω X H t xt h F Y ==

In the /iew o! uality property o! 'ouriertrans!orms, we e#pect

[ ] )()()(*)()( ω ω ω X H t xt h F Y ==

∫ ∞∞− =−= )(*)()(or)()()( t ht xt yd t h xt y λ λ λ


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'ourier trans!orms can 0e applie to circuits with non-sinusoi al e#citation in e#actly the same way as phasortechniques 0eing applie to circuits with sinusoi ale#citations.

Ay trans!orming the !unctions !or the circuit elements intothe !requency omain an ta%e the 'ourier trans!orms o!the e#citations , con/entional circuit analysis techniquescoul 0e applie to etermine un%nown response in!requency omain.

'inally, apply the in/erse 'ourier trans!orm to o0tain theresponse in the time omain.

1(.* Circuit "pplications +11(.* Circuit "pplications +1

Y( ω ) = H( ω )X( ω )


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1(

2#ample 1?4 'in v 0(t) in the circuit shown 0elow !or

v i (t)=2e - t u(t)

1(.* Circuit "pplications +&1(.* Circuit "pplications +&


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1;

$olution4

1(.* Circuit "pplications +*1(.* Circuit "pplications +*

)()(4.0)( givesansformFourier tr inversetheTaking

).0)(!(1

)("

Hence,

211

)(")("

)( iscircuittheof functiontransferThe

!

2)(" issigna#in$uttheof ansformFourier tr The

!.00

0

i

0

i

t ueet v

j j

j H

j

t t −− −=

++=

+==

+=

ω ω ω

ω ω

ω ω

ω ω

bab 18 sistem linier

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