bab 18 sistem linier
TRANSCRIPT
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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
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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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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&
1(.& Properties o! 'ourier Trans!orm +31(.& Properties o! 'ourier Trans!orm +3
)()( 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*
1(.& Properties o! 'ourier Trans!orm +51(.& Properties o! 'ourier Trans!orm +5
)()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(.& Properties o! 'ourier Trans!orm +91(.& Properties o! 'ourier Trans!orm +9
[ ] )(*)()( 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