บทที่ 5 การแปลงลา ... - DSS · PDF fileSignal-B 1/2555 บทที่ 2 สัญญาณและระบบเวลาต่อเนื่อง

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Signal-B 1/2555 2 1
5
Laplace Transform
( j ) ( s j ) (Ordinary Linear Differential Equation System) (R) (C) (L)
5.1 : (Laplace Transform: Definition)
x t( )
stL x t X s x t e dt[ ( )] ( ) ( )
(5.1)
s (Complex frequency) 1/ s j t () st (Complex Plane) s (5.1) ( ) (Region of Convergence: ROC)
s j 0 (5.1)

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2 Signal & System .. B-Elect.20120626
j t
s jwX s x t e dt x t( ) ( ) [ ( )]
(5.2)
j tst t tX s x t e dt x t e e dt x t e( ) ( ) ( ) ( )
(5.3)
(5.3) x t( ) tx t e( )
2 (two-side transform bilateral transform) (Forward Transform) (Invert Transform)
X s L x t( ) [ ( )] x t L X s1( ) [ ( )] x t X s( ) ( )
t
t e for tx t e u t
for t
0( ) ( )
0 0
( 0 ) te u t( ) (Causal real exponential signal)
(5.1)
t st
j tt
X s e u t e dt
e e dt( )
0
( ) ( )
-2 -1.5 -1 -0.5 0 0.5 1 1.5 20
0.2
0.4
0.6
0.8
1
t
exp(-4*t)
x(t)

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Signal-B 1/2555 2 3
t t
tj t
t
e e dt
e
j
s
( )
0
{( ) }
0( )
1
s j s
te u t s
s
1( ) ; Re( )
(5.3)
s
sRe( )
sIm( )
ROC s: Re( )
sRe( )
sIm( )
ROC s: Re( )
sRe( )
sIm( )
ROC s: Re( ) ( 0 ) tx t e u t( ) ( ) t 0 1 t 0
te t 0
X ss
1( )
s
te u t s
s
1( ) ; Re( )
(5.4)
X ss
1( )
sRe( )
sIm( )
ROC s: Re( )
te u t( )
t
x t( )

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X ss
1( )
sRe( )
sIm( )
ROC s: Re( ) t
x t( )
te u t( )
tx t e u t( ) ( ) ( 0 ) 3 ( Anti-causal)
s tt st t stX s e u t e dt e e dt e dts
0 0
( ) 1( ) ( )
t 0 s
te u t s
s
1( ) ; Re( )
(5.5)
X ss
1( )
sRe( )
sIm( )
ROC s: Re( )
x t( )
te u t( )
t
X ss
1( )
t
x t( )
te u t( )
sRe( )
sIm( )
ROC s: Re( )
(5.3) (5.5) x t( ) ROC X s( ) ROC x t( ) tx t e u t( ) ( ) 0 ( x t( ) t )

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Signal-B 1/2555 2 5
X ss
2( )
2
x t( )
tx t e u t2( ) 2 ( ) sRe( ) 2 tx t e u t2( ) 2 ( ) sRe( ) 2
tx t e x t1( ) ( ) X s( )
s tt stX s e x t e dt x t e dt X s
( )1 1 1( ) ( ) ( ) ( )
dx tx tdt
( )( ) X s( )
() (integration by parts)
st
st st st
X s x t e dt
dx t e x t e s x t e dt
x sX s
0
00 0
( ) ( )
( ) ( ) ( )
(0 ) ( )
Ldi t
v t Ldt
( )( ) C
dv ti t C
dt
( )( )
tx t t e u t t u t t u t21
3( ) ( ) ( ) 2 cos(2 ) ( ) 2 sin(3 ) ( )
x t( )
) t( ) 1 s
) te u t ss
21
3
1( ) ; Re( ) 2
3 6
()

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) x t t u t3 ( ) 2 cos(2 ) ( ) 1 j t j te e
t
2 2
cos(2 )2
j t j tx t t u t e e2 23 ( ) 2 cos(2 ) ( )
sX s ss j s j s
3 2
1 1 2( ) ; Re( ) 0
2 2 4
( s 0 )
) x t t u t4 ( ) sin(2 ) ( ) 2 anti-causal
j t j t
j j t j te et e e
j
2 22 2
2sin(2 )
2
j t j t
j t j te ex t t u t j e e
j
2 22 2
4 ( ) 2 sin(2 ) ( ) 22
jX s j j ss j s j s s
4 2 2
41 1 4( ) Re( ) 0
2 2 4 4
t
t
X s L t e u t t u t t u t
L t L e u t L t u t L t u t
21
3
21
3
( ) ( ) ( ) 2 cos(2 ) ( ) 2 sin(3 ) ( )
( ) ( ) 2 cos(2 ) ( ) 2 sin(3 ) ( )
s
s s s
s s s s s s s
s s s
s s s
s s s
2 2
3 2 2 2
3 2
3 2
3 2
1 2 41
3 6 4 4
3 6 12 24 4 6 12 12 24
3 6 12 24
3 13 36 52
3 6 12 24
ROC ROC x t t u t3 ( ) 2 cos(2 ) ( ) x t t u t4 ( ) sin(2 ) ( )
5.2 x t( ) tx t e( )
j ttx t e X j X j e d
1 1( ) ( ) ( )
2
s j ds jd
1 st
s0 2 2
0
cos
2 t
s0 2 2
0
sin

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Signal-B 1/2555 2 7
j t
st
x t X j e dsj
jX s e ds
( )1( ) ( )
2
( )2
(5.6)
(5.6)
x t x t x t x t X s X s X s X s1 2 3 1 2 3( ) ( ) ( ) ( ) ..... ( ) ( ) ( ) ( ) ..... (5.7)
X ss s
1( )
1 2
s sRe( ) 1 sRe( ) 2
()
A BX s
s s s s
1( ) ;
( 1)( 2 ) 1 2
A s B s( 2 ) ( 1) 1
s 1 A 1 s
AS
1
11
( 2 )
s 2 B 1 s
BS
2
11
( 1)
X ss s s s
1 1 1( )
( 1)( 2 ) 1 2
te u t s
s
1( ) ; Re{ } 1
1
te u t s
s
2 1( ) ; Re{ } 2
2
s sRe( ) 1 -1 -2
t tL x t e e u ts s
1 21( ) ( )
( 1)( 2 )

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te u t s
s
1( ) ; Re{ } 1
1
te u t s
s
2 1( ) ; Re{ } 2
2
s sRe( ) 2 -2 -1
t tL x t e e u ts s
1 21( ) ( )
( 1)( 2 )
s sX s
s s
2
2
2 5( )
3 5
()
s s A B CX s
s ss s s
2
2 2
2 5( )
3 53 5 5
s
s sA
s
2
2
3
2 52
5
s
s sC
s
2
5
2 510
3
s 0 B
B1 2 10
15 3 5 25 B 1
X s
s s s2
2 1 10( )
3 5 5
3
t t t t tx t e e te e t e
3 5 5 3 5( ) 2 10 2 (1 10 )
s sX ss s
2
2
3 15 14( )
3 2
( )
s s sX s
s ss s s s
2
2 2
3 15 14 6 8 2 4( ) 3 3
1 23 2 3 2
t tx t t e e
2( ) 3 ( ) 2 4
3 n
n
nt
s1
!
te x t X s( ) ( ) n t
n
nt e
s1
!
( )

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Signal-B 1/2555 2 9
5.3
stL x t X s x t e dt[ ( )] ( ) ( )
x t X s( ) ( )
(linearity)
ax t bx t aX s bX s OC R R1 2 1 2 1 2( ) ( ) ( ) ( ) R
(time shifting)
stx t t X s e OC R00( ) ( ) R
()
stjx t X s e ds( ) ( )
2
s t t
st st
jx t t X s e ds
je X s e ds
0
0
( )0( ) ( )
2
( )2
stx t t e X s00( ) ( )
(time scaling)
s
ax at X
a
1( ) ( ) x t X s( ) ( )
at a
dt d1

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st
s a
a
s a
a
s
aa
L x x e dt
x e d a
x e d a
X
( / )1
( / )1
1
[ ( )] ( )
( ) ; 0
( ) ; 0
( )
(conjugation)
x t X s( ) ( )
(time domain differentiation)
() dx sX s xdt
( ) (0 ) n
n nn n nx t s X s s x s x x
( ) ( 1 )1 2( ) ( ) (0 ) (0 ).... (0 )
n nt s( ) ( )
(s-domain differentiation)
nn
n
d X st x t
ds
( )( )
( )
( )( ) ( )
nat
n
te u t
n s a
( )( ) 1
( )( 1) ! ( )
s aRe{ }
t
x d X ss
1( ) ( )
z t x t y t x y t d( ) ( ) * ( ) ( ) ( )

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Signal-B 1/2555 2 11
st
st
L z t x y t d e dt
x d y t e dt
( ) ( ) ( )
( ) ( )
s
s
x d Y s e
x e d Y s X s Y s
( ) ( )
( ) ( ) ( ) ( )
x t y t X s Y s( ) ( ) ( ) ( )
2.7 (to be added)
//E

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1.
) te ) te0.4 ) te 2 ) te ) te 0.4 ) te 20.5
) te( 1) ) te(0.4 0.5 ) ) te( 2 2 ) ) te( 2 ) ) te(1 0.4 ) ) t
e( 2 1)
0.5
2. y t( ) LTI ()
.
2 1 1( )
0
for tx t
elsewhere
1.5 1 2( )
0
for th t
elsewhere
. 2
( ) ( )t
x t e u t
( ) ( )h t u t
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บทที่ 5 การแปลงลา ... - DSS · PDF fileSignal-B 1/2555 บทที่ 2 สัญญาณและระบบเวลาต่อเนื่อง