meiling chensignals & systems1 lecture #06 laplace transform
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meiling chen signals & systems 1
Lecture #06
Laplace Transform
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Eigenfunction
xAx
Ax x
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LTI system)()( ttx )()( thty
h(t) is the impulse response of the LTI systemAccording to the convolution:
dtxhtxthtxHty
)()()()()}({)(
stetxlet )(
dehe
dehtxHty
sst
ts
)(
)()}({)( )(
We define that
dehsH s
)()(
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stst esHeHtxHty )(}{)}({)(
We identify as an eigenfunction of the LTI system and H(s) as the corresponding eigenvalue.
ste
js In which s is a complex frequency
deeh
dehdehjH
j
jj
])([
)()()( )()(
)( jH Is the Fourier transform of eh )(
dejHeh tj
)(2
1)(
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dejH
dejHeth
dejHeth
tj
tj
tj
)()(2
1
)(2
1)(
)(2
1)(
dsesHj
thj
j
st
)(
2
1)(j
dsdjslet
j
j
st
st
dsesXj
tx
dtetxsX
)(
2
1)(
)()(Laplace transform
Inverse Laplace transform
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j
j
st
st
dsesXj
sXLtx
dtetxtxLsX
)(
2
1)}({)(
)()}({)(
1
0
Unilateral Laplace transform for causal system
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Laplace transform properties
)()()]()([ sGsFtgtfL
)()]([ asFtfeL at
)()]()([ sfetutfL s
)0()()]([ fssFtfL
)0()0()0()()]([ 121)( nnnnn ffsfssFstfL
s
sFdfL
t )(])([
0
ds
sdFttfL
)()]([
n
nnn
ds
sFdtftL
)()1()]([
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Time convolution )()()}()({ 2121 sFsFtftfL
)()(
])([)(
])([)(
])([)()}()({
,
])([)(
])()([)}()({
)()()()(
21
0
1
0
2
0
1
0
2
)(
0
1
0
221
0
1
0
2
0
21
0
21
0
2121
sFsF
defdef
ddeeff
ddefftftfL
ddtt
ddtetff
dtedftftftfL
dftftftf
ss
ss
s
st
st
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Initial Value TheoremInitial-Value Theorem
If is continuous at and may different and if is not impulse function or derivative of impulse function, then
)(tf 0t )0( f )0( f
)(tf
)(lim)0()(lim0
ssFftfst
Example 1ttf
s
ssF
cos)()(
22
1lim)(limcoslim)(lim22
2
00
s
sssFttf
sstt
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Final Value Theorem
Final-Value Theorem
)(lim)(lim0
ssFtfst
If and are Laplace transformable, if exists and if is analytic on the imaginary axis and in right half of the s-plane, then
)(tf )(tf )(lim tft
)(ssF
1. No any pole on the imaginary axis or in right half of s-plane.2. System is stable.
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Example 2
Example 3
)2(
5)(
2
ssssF
2
5)(lim)(lim
0
ssFtf
st
ttfs
sF
sin)()(
22
ttfttt
sinlim)(lim
not exist
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Remark 1 )0()(lim
fssF
s
Remark 2 If include impulse function at . )(tf 0t
)0()(lim
fssF
s
1)(lim0)0(
1)()()(
ssFbutf
ssFtutf
s
Example 4
Example 5
)(lim0)0(
1)()()(
ssFbutf
sFttf
s
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Inverse Laplace transformF(s) is a strictly proper rational function
)(deg)(deg)(
)()( sDsN
sD
sNsF
Degree of denominator
Case I simple root as
as
AsF )( atAetf )(
assFasA )]()[(
assD
sNA
)(
)(where
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Example 6
36189
)1(18)(
23
s
C
s
B
s
A
sss
ssF
1)3)(6(
)1(180
sss
sA 1
18183
)1(1802
sss
sAor
5)3(
)1(186
sss
sB 5
18183
)1(1862
sss
sBor
4)6(
)1(183
sss
sC 4
18183
)1(1832
sss
sCor
tt eetf 36 451)(
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Inverse Laplace transform
Case II complex root ))(( *asas
*)(
as
A
as
AsF
assD
sNA
)(
)(
jaja *,
**
)(
)(A
sD
sNA
as
AjAj eAAeAA *,let
taat eAAetf**)(
tjAjtjAj eeAeeAtf )()()(
)()( )()( AtjAtjt eeeAtf
)cos(2)( AteAtf t
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Example 7
js
B
js
B
ssss
ssF
112
2/1
)22)(2(
)3()(
*
2
4
3
4
1
)1)(2(
31 j
jss
sB js
3tan,4
10 1 BB
)3tancos(2
10
2
1)( 12 teetf tt
)cos(2)( BteBtf t
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Inverse Laplace transform
Case III repeated root nas )(
)()()()(
)()( 1
11
as
A
as
A
as
A
sD
sNsF
nn
nn
nksFasds
d
knA
sFasds
dA
sFasds
dA
sFasA
asn
kn
kn
k
asn
n
asn
n
asn
n
,,2,1,)()()!(
1
)()(!2
1
)()(
)()(
2
2
2
1
))!2()!1(
()( 12
2
1
1
AtAn
tA
n
tAetf
n
n
n
nat
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Example 8
13)4()4()4)(3)(1(
)2(48)(
22
s
D
s
C
s
B
s
A
sss
ssF
32)3)(1(
)2(484
sss
sA
3
80]
3)3)(1(
)2(48[ 4
sss
s
ds
dB
3
8,24 DC
tttt eeetetf 3
824
3
8032)( 344