1. bieán ñoåi fourier vaø aûnh fourier. 2. caùc tính chaát cuûa … · 2009. 10. 28. ·...
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7.3 Phöông phaùp bieán ñoåi Fourier
1. Bieán ñoåi Fourier vaø aûnh Fourier.
2. Caùc tính chaát cuûa bieán ñoåi Fourier.
3. Bieán ñoåi Fourier cuûa caùc haøm thoâng duïng.
4. Aùp duïng bieán ñoåi Fourier.
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1. Bieán ñoåi Fourier vaø aûnh FourierBieán ñoåi Fourier cho tín hieäu khoâng tuaàn hoaøn f(t) : laømoät coâng cuï toaùn coù phaïm vi aùp duïng raát lôùn trong caùc baøi toaùn kyõ thuaät , noù ñöôïc ñònh nghóa laø moät caëp bieán ñoåi thuaän – ngöôïc nhö sau :
vaø :
Ñeå coù bieán ñoåi Fourier, tín hieäu f(t) cuõng phaûi thoûa maõn ñieàu kieän Dirichlets.
( ) ( ). j tF f t e dt
1( ) ( ).
2
j tf t F e d
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Ñaëc ñieåm cuûa haøm F( )Phoå taàn soá :
Phoå bieân ñoä: bieåu dieãn |F(j )| theo . Phoå pha : ( ) theo .Phoå bieân ñoä vaø phoå pha cuûa tín hieäu khoâng tuaàn hoaøn laøcaùc haøm lieân tuïc theo .
( )( ) ( ) jF F e
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2. Caùc tính chaát cuûa bieán ñoåi Fourier
Vôùi F( ) = P( ) + jQ( ) thì: P( ) laø haøm chaün theo taàn soá vaø Q( ) laø haøm leû theo taàn soá .Tuyeán tính (Linearity) :
Neùn tín hieäu (Time scaling):
Treã tín hieäu (Time shifting) :
Ñieàu cheá (Modulation):
Ñaïo haøm trong mieàn thôøi gian
1 2 1 2. ( ) . ( ) . ( ) . ( )a f t b f t a F bF
1( ) .f at F
a a
0
0( ) ( ).j tf t t F e
0
0( ) ( )j te f t F
( )( ). ( )
df t j Fdt
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Caùc tính chaát cuûa bieán ñoåi Fourier (tieáp theo)
Tích phaân trong mieàn thôøi gian (Integration in the time domain):
Tích chaäp trong mieàn thôøi gian (Convolution in the time domain):
Ñònh lyù Parseval (Parseval’s Theorem):cho ta moät söï lieân heägiöõa naêng löôïng ôû mieàn thôøi gian vaø naêng löôïng trong mieàn taàn soá.
1( ) . ( ) . (0). ( )
t
f d F Fj
; (0) ( )F f t dt
1 2 1 2 1 2( )* ( ) ( ). ( ) ( ). ( )f t f t f f t d F F
22 1( ) ( )
2f t dt F d
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3. Bieán ñoåi Fourier cuûa caùc haøm thoâng duïng
Sign(t)
e-at.1(t)
2 ( )1 (nguoàn DC)
1(t)
1(t)Aûnh FourierHaøm goác
1( )
j
1
a j2
j
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Bieán ñoåi Fourier cuûa caùc haøm thoâng duïng (tieáp theo)
Haøm muõ hai phía
Haøm quaù ñoä AC :sin( 0t).1(t)
Haøm quaù ñoä AC : cos( 0t).1(t)
Haøm AC : sin( 0t)
Haøm AC : cos( 0t)Aûnh FourierHaøm goác
0 0( ) ( )
0 0( ) ( )j
0 0 2 2
0
( ) ( )2
j
00 0 2 2
0
( ) ( )2j
te 2 2
2
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4. ÖÙng duïng bieán ñoåi FourierTruyeàn tín hieäu qua maïch tuyeán tính:Xaùc ñònh bieán ñoåi Fourier cuûa taùc ñoäng x(t) vaø haøm truyeàn theo taàn soá K(j ) cuûa maïch . Sau ñoù xaùc ñònh :
Y( ) = K(j ).X( )Bieán ñoåi ngöôïc Y( ) tìm y(t).Löu yù : khoâng coù khaùi nieäm ñieàu kieän ñaàu nhö Ch 6 !
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Bieán ñoåi Fourier : Ví duï 1Tìm ñaùp öùng xaùc laäp u(t) khi e(t) = 10cos(2t) V
GiaûiHaøm truyeàn maïch ôû mieàn taàn soá :
Aûnh Fourier cuûa taùc ñoäng :
Tín hieäu ra mieàn taàn soá :
2
2( )
3 4 4K j
j( ) 10 ( 2) ( 2)E
2
2
10 ( 2) ( 2)( )
3 4 4U
j
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Ví duï 1 (tieáp theo)Tìm haøm goác :
Löu yù laø :
1( ) ( )
2
j tu t U e d
0
0( )j tj te d e
2 22 2
2 2
5(2 ) 5( 2 )( )
3(2 ) 8 4 3( 2 ) 8 4
j t j tu t e ej j
2 220 20( )
8(1 ) 8(1 )
j t j tu t e ej j
5( ) (1 )(cos 2 sin 2 ) (1 )(cos 2 sin 2 )
4u t j t j t j t j t
5( ) cos 2 sin 2 3, 53 cos(2 45 )
2
ou t t t t
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Bieán ñoåi Fourier : Ví duï 2Tìm ñaùp öùng quaù ñoä u(t) khi e(t) = 5e-2t.1(t) V
GiaûiHaøm truyeàn maïch ôû mieàn taàn soá :
Aûnh Fourier cuûa taùc ñoäng :Tín hieäu ra mieàn taàn soá :
Vaäy :
+
_ e(t) u(t)
+
-
1 H
10
10( )
10
RK jR j L j 5
( )2
Ej
50 1 1( ) ( ). ( )
8 2 10U K j E
j j2 10( ) 6,25 .1( )t tu t e e t V
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7.4 Bieåu dieãn ñoà thò cuûa haøm truyeàn1 Ñaëc tuyeán taàn soá cuûa maïch .
2. Ñaëc tuyeán logarithm – Taàn soá logarithm.
3. Giaûn ñoà Bode.
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1. Ñaëc tuyeán taàn soá cuûa maïchTrong haøm truyeàn toaùn töû , khi ta thay s = j , ta coù haøm truyeàn cuûa maïch trong mieàn taàn soá :
Caùc ñaëc tuyeán :|K(j )| : Ñaëc tuyeán bieân taàn.
( ) : Ñaëc tuyeán pha taàn.P( ) : Ñaëc tuyeán phoå thöïc .Q( ) : Ñaëc tuyeán phoå aûo .
( )( ) ( )
( ) ( ) ( )
jK j K j eK j P jQ
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Caùc ñaëc tuyeán taàn soá cuûa maïch RC
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2. Ñaëc tuyeán bieân ñoä logarithm vaøtaàn soá logarithm
Taàn soá tuyeán tính (LIN) : giaù trò treân truïc taàn soá Vi = k i+a.Töùc laø : i - i-1 = const .Taàn soá logarithm 2 (OCT) : giaù trò treân truïc taàn soá Vi =log2( i) . Töùc laø : i =2 i-1 .Taàn soá logarithm 10 (DEC hay LOG) : giaù trò treân truïc taàn soá Vi = log10( i) . Töùc laø : i =10 i-1 .Khi bieåu dieãn caùc ñaëc tuyeán taàn soá , ngöôøi ta ít duøng haøm |K(j )| maø thöôøng duøng bieåu dieãn haøm 20log10 |K(j )| , ñôn vò dB , theo log10( ) ñöôïc goïi laø ñaëc tuyeán bieân ñoä logarithm. Öu ñieåm cuûa caùch bieåu dieãn naøy laø coù theå moâ taû haøm truyeàn trong moät khoaûng raát roäng cuûa taàn soá.
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Minh hoïa Ñaëc tuyeán bieân ñoälogarithm
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Giaûn ñoà Bode Neáu ta coù bieåu dieãn haøm truyeàn döôùi daïng :
Zi laø caùc ñieåm khoâng cuûa haøm truyeàn . Zp laø caùc ñieåm cöïc cuûa haøm truyeàn .
2
1
2
1
1 1 2 ...
( )
1 1 2 ...
NN N N
M M M
jj j jz
K j Kj j jp
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Giaûn ñoà Bode (tieáp theo) Bieán ñoåi ñeå coù ñaëc tuyeán bieân ñoä logarithm:
Vaø ñaëc tuyeán pha :
10 10 10 10
1
2
10 10
1
20log [| ( ) |] 20log 20 log ( ) 20log 1
20log 1 2 ( ) ( ) ... 20log 1 ...N N N
jK j K N jz
jj jp
1
2 2
1
0 0( ) .90
180 0
2 ( ) 2 ( )... ...
1 ( ) 1 ( )
oo
o
N N M M
N M
KK j N arctg
zK
arctg arctg arctgp
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Thaønh phaàn haèng soá
Giaù trò ñaëc tuyeán bieân ñoä logarithm vaø pha :
log10
( )
20log10
(|K(j )|)
20log10
(K)
0
log10
( )
)
180o
0
1020log
0 0
0180
o
o
a K
Kb
K
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Ñieåm cöïc vaø khoâng baèng khoâng
Giaù trò ñaëc tuyeán bieân ñoä logarithm vaø pha :
10
10
20 log ( )
20 log ( )
90
90
o
o
zeroa
pole
zerob
pole
log10
( )
20log10
(|K(j )|)
20 dB
0
log10
( )
)
90o
0
1
log10
( )
20log10
(|K(j )|)
-20 dB
0
log10
( )
)
-90o
0
1
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Ñieåm cöïc vaø khoâng khaùc khoâng
Giaù trò ñaëc tuyeán bieân ñoä logarithm vaø pha :
10
10
20log ( )
20log ( )
45
45
o
o
zerozapole
p
zerob
pole
log10
( )
20log10
(|K(j )|)
20 dB
0
log10
( )
)
45o
0
1
log10
( )
20log10
(|K(j )|)
-20 dB
0
log10
( )
)
-45o
0
1 2
2
= z) = 10z)
1 2
1 290o
-90o
= p) = 10p)
= z)
= p)
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Ñieåm cöïc vaø ñieåm khoâng phöùc
Giaù trò ñaëc tuyeán bieân ñoä logarithm vaø pha :
10
10
40log ( )
40log ( )
90
90
o
o
zerozapole
p
zerob
pole
log10
( )
20log10
(|K(j )|)
40 dB
0
log10
( )
)
90o
0
1
log10
( )
20log10
(|K(j )|)
-40 dB
0
log10
( )
)
-90o
0
1 2
2
= z) = 10z)
1 2
1 2180o
-180o
= p) = 10p)
= z)
= p)
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Ví duï veõ giaûn ñoà Bode Veõ giaûn ñoà Bode cho :
10 110
( )
1 150
j
K jjj
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