9.0 laplace transform 9.1 general principles of laplace transform linear time-invariant laplace...
TRANSCRIPT
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9.0 Laplace Transform 9.1 General Principles of Laplace
Transform
linear time-invariant
Laplace TransformEigenfunction Property
y(t) = H(s)esth(t)x(t) = est
dehsH s
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Chapters 3, 4, 5, 9, 10
jω
Chap 3 Chap 5Chap 4
Chap 10
Chap 9
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Laplace TransformEigenfunction Property
– applies for all complex variables s
dtethjH
jstj
Fourier Transform
Laplace Transform
dtethjH
jstj
eigenfunction of all linear time-invariant systems with unit impulse response h(t)
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Laplace Transform
A Generalization of Fourier Transform
dtetx
dtetxjX
tj
tj
e
t-
jsjs to fromjω
X( + jω)
tetx Fourier transform of
sXtx
jsdtetxsXL
st
,
Laplace Transform
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Laplace Transform
𝜎 2+ 𝑗𝜔1
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Laplace TransformA Generalization of Fourier Transform
– X(s) may not be well defined (or converged) for all s
– X(s) may converge at some region of s-plane, while x(t) doesn’t have Fourier Transform
– covering broader class of signals, performing more analysis for signals/systems
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Laplace TransformRational Expressions and Poles/Zeros
sD
sNsX
roots zeros
roots poles
– Pole-Zero Plots
– specifying X(s) except for a scale factor
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Laplace TransformRational Expressions and Poles/Zeros– Geometric evaluation of Fourier/Laplace transform
from pole-zero plots
jj
ii
s
sMsX
each term (s-βi) or (s-αj) represented by a vector with magnitude/phase
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Poles & Zeros
x x
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Region of Convergence (ROC)Property 1 : ROC of X(s) consists of strips parallel
to the jω -axis in the s-plane
– For the Fourier Transform of x(t)e-σt to converge
dtetx t
Property 2 : ROC of X(s) doesn’t include any poles
depending on σ only
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Property 1, 3
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Region of Convergence (ROC)Property 3 : If x(t) is of finite duration and absolutely
integrable, the ROC is the entire s-plane
dttxedtetx
dttxedtetx
TT
dttx
T
T
TT
T
t
T
T
TT
T
t
T
T
,0for
,0for
duration finite the: ,
2
1
22
1
2
1
12
1
2
1
21
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Region of Convergence (ROC)Property 4 : If x(t) is right-sided (x(t)=0, t < T1),
and {s | Re[s] = σ0}ROC, then
{s | Re[s] > σ0}ROC, i.e., ROC
includes a right-half plane
dtetxedtetx
dtetx
t
T
T
T
t
t
T
0
1
101
1
1
0
1
for
01
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Property 4
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Region of Convergence (ROC)Property 5 : If x(t) is left-sided (x(t)=0, t > T2),
and {s | Re[s] = σ0}ROC, then
{s | Re[s] < σ0}ROC, i.e., ROC
includes a left-half plane
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Property 5
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Region of Convergence (ROC)Property 6 : If x(t) is two-sided, and {s | Re[s] =
σ0}ROC, then ROC consists of a strip
in s-plane including {s | Re[s] = σ0}
txtxtx
-tx-tx
txtxtx
LR
LR
LR
ROCROCROC
sidedleft : sided,right :
See Fig. 9.9, 9.10, p.667 of text
*note: ROC[x(t)] may not exist
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Region of Convergence (ROC)A signal or an impulse response either doesn’t have a
Laplace Transform, or falls into the 4 categories of
Properties 3-6. Thus the ROC can be , s-plane, left-
half plane, right-half plane, or a signal strip
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Region of Convergence (ROC)Property 7 : If X(s) is rational, then its ROC is
bounded by poles or extends to infinity– examples:
assas
tue
assas
tue
Lat
Lat
Re ROC ,1
Re ROC ,1
– partial-fraction expansion
See Fig. 9.1, p.658 of text
i i
i
assX
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Region of Convergence (ROC)Property 8 : If x(s) is rational, then if x(t) is right-
sided, its ROC is the right-half plane to
the right of the rightmost pole. If x(t) is
left-sided, its ROC is the left-half plane
to the left of the leftmost pole.
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Property 8
ROC
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Region of Convergence (ROC)An expression of X(s) may corresponds to different
signals with different ROC’s.– an example:
21
1
sssX
– ROC is a part of the specification of X(s)
See Fig. 9.13, p.670 of text
The ROC of X(s) can be constructed using these properties
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Inverse Laplace Transform
jddsdsesXj
tx
dejXtx
dejXjXFetx
stj
j
tj
tjt
21
21
21
1
1
1
1
1
111
– integration along a line {s | Re[s]=σ1}ROC for a fixed σ1
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Laplace Transform
( 合成 ?)
(basis?)X
X
( 分析 ?)X
[𝑒(𝜎1+ 𝑗 𝜔 ) 𝑡 ]∗=𝑒𝜎1 𝑡 ⋅𝑒− 𝑗 𝜔𝑡( 分析 )≠ �⃗�⋅𝑣
basis
( 合成 )
𝑥 (𝑡 )𝑒−𝜎 1𝑡= 12𝜋
− ∞
∞
𝑋 (𝜎 1+ 𝑗 𝜔)𝑒 𝑗𝜔𝑡 𝑑𝑡
( 分析 )
𝑋 (𝜎 1+ 𝑗 𝜔)=−∞
∞
[𝑥 (𝑡 )𝑒−𝜎1𝑡]⋅𝑒− 𝑗𝜔𝑡 𝑑𝑡
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Practically in many cases : partial-fraction expansion works
Inverse Laplace Transform
1
m
i i
i
as
AsX
– ROC to the right of the pole at s = -ai
tueA tai
i
– ROC to the left of the pole at s = -ai
tueA tai
i
Known pairs/properties practically helpful
each termfor i
i
as
A
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9.2 Properties of Laplace Transform
Linearity
222
111
ROC ,
ROC ,
ROC ,
RsXtx
RsXtx
RsXtx
L
L
L
212121 ROC , RRsbXsaXtbxtax L
Time Shift
RsXettx stL ROC ,00
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Time Shift
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Shift in s-plane
0
0
00
Reby shifted ROC
Re
ReROC ,0
s
Rsss
sRssXtxe Lts
See Fig. 9.23, p.685 of text
– for s0 = jω0
RjsXtxe Ltj ROC ,00
shift along the jω axis
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Shift in s-plane
𝑅𝑒 [𝑠0]
𝑠0= 𝑗𝜔0
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Time Scaling (error on text corrected in class)
RsasaRa
sX
aatx L
ROC ,
1
– Expansion (?) of ROC if a > 1
Compression (?) of ROC if 1 > a > 0
reversal of ROC about jw-axis if a < 0
(right-sided left-sided, etc.)
See Fig. 9.24, p.686 of text
RssRsXtx L ROC ,
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Conjugation
real if
ROC ,
txsXsX
RsXtx L
– if x(t) is real, and X(s) has a pole/zero at
then X(s) has a pole/zero at
0
0
ss
ss
Convolution
212121 ROC , RRsXsXtxtx L
– ROC may become larger if pole-zero cancellation occurs
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Differentiation
RssXdt
tdx L ROC ,
ROC may become larger if a pole at s = 0 cancelled
R
ds
sdXttx L ROC ,
(− 𝑗𝑡𝑥 (𝑡)𝐹↔
𝑑𝑑𝜔
𝑋 ( 𝑗𝜔 ))
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Integration in time Domain
0Re ROC ,1
,
0Re ROC ,1
sss
tu
tutxdx
ssRsXs
dx
L
t
tL
(𝑢 (𝑡 )𝐹↔
1𝑗𝜔
+𝜋𝛿(𝜔))(
− ∞
𝑡
𝑥 (𝜏 ) 𝑑𝜏=𝑥 (𝑡 )∗𝑢(𝑡)𝐹↔
𝑋 ( 𝑗𝜔 )𝑑𝜔
+𝜋 𝑋 ( 𝑗0)𝛿(𝜔))
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Initial/Final – Value Theorems
Theorem value-Final limlim
Theorem value-Initial lim
0ssXtx
ssXox
st
s
x(t) = 0, t < 0
x(t) has no impulses or higher order singularities at t = 0
Tables of Properties/Pairs
See Tables 9.1, 9.2, p.691, 692 of text
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9.3 System Characterization with Laplace Transform
system function
transfer function
y(t)=x(t) * h(t)h(t)
x(t)
X(s) Y(s)=X(s)H(s)H(s)
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Causality
– A causal system has an H(s) whose ROC is a right-half plane
h(t) is right-sided
– For a system with a rational H(s), causality is equivalent to its ROC being the right-half plane to the right of the rightmost pole
– Anticausality
a system is anticausal if h(t) = 0, t > 0
an anticausal system has an H(s) whose ROC is a left-half plane, etc.
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Causality
ROCis
right half Plane
right-sided CausalX
Causal
𝑋 (𝑠 )=∑𝑖
𝐴𝑖
𝑠+𝑎𝑖,
𝐴𝑖
𝑠+𝑎𝑖
→ 𝐴𝑖𝑒−𝑎𝑖 𝑡𝑢 (𝑡)
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Stability
– A system is stable if and only if ROC of H(s) includes the jω -axis
h(t) absolutely integrable, or Fourier transform converges
See Fig. 9.25, p.696 of text
– A causal system with a rational H(s) is stable if and only if all poles of H(s) lie in the left-half of s-plane
ROC is to the right of the rightmost pole
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Stability
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Systems Characterized by Differential Equations
N
kkk
M
k
kk
M
k
kk
N
k
kk
k
kM
kk
N
kk
k
k
sa
sb
sX
sYsH
sXsbsYsa
dt
txdb
dt
tyda
0
0
00
00
zeros
poles
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System Function Algebra
– Parallel
h(t) = h1(t) + h2(t)
H(s) = H1(s) + H2(s)
– Cascade
h(t) = h1(t) * h2(t)
H(s) = H1(s) ‧ H2(s)
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System Function Algebra
– Feedback
x(t) +
-+
h1(t)H1(s)
e(t)y(t)
h2(t)H2(s)
z(t)
sHsH
sH
sX
sYsH
sYsHsXsHsY
21
1
21
1
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9.4 Unilateral Laplace Transform
Transform Laplace bilateral
Transform Laplace unilateral 0
dtetxsX
dtetxsX
st
stu
impulses or higher order singularities at t = 0 included in the integration
uL sXtx
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– ROC for X(s)u is always a right-half plane
– a causal h(t) has H(s) = H(s)
– two signals differing for t < 0
but identical for t ≥ 0 have identical unilateral Laplace transforms
– similar properties and applications
9.4 Unilateral Laplace Transform
sHsH u
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Examples• Example 9.7, p.668 of text
0b Transform, Laplace No
0b ,sReb ,211
sRe ,1
sRe ,1
)(
22
bbsb
bsbse
bbs
tue
bbs
tue
tuetueetx
Ltb
Lbt
Lbt
btbttb
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Examples• Example 9.7, p.668 of text
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Examples• Example 9.9/9.10/9.11, p.671-673 of text
)()()( ,1sRe2
)( )( ,2sRe
)( )( ,1sRe
21
11
211)(
2
2
2
tuetuetx
tueetx
tueetx
sssssX
tt
tt
tt
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Examples• Example 9.9/9.10/9.11, p.671-673 of text
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Examples• Example 9.25, p.701 of text
stable is )( plane half-left in the poles Allcausal is )(
polerightmost theofright the tois )( of 2 ,1at poles ,
1sRe ,21
3)()(
)(
1sRe ,21
1)(
3sRe ,3
1)(
)( )()()( 23
sH
sHsHROC
-s-sROCROCROCss
ssXsY
sH
sssY
ssX
tueetytuetx
HXY
ttt
plane half-left in the poles All
left-half plane
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Problem 9.60, p.737 of text
poles no , all ,1
3
22
33
3
see
eesH
TtTtth
sTsT
sTsT
Echo in telephone communication
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Problem 9.60, p.737 of text
)22
(log1
,01
of zeros find To)2(22 2
Tm
Tj
Ts
ejee
sHmjsTsT
zeros
𝜋
2𝑇𝜋
2𝑇+
2𝜋𝑇
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Problem 9.60, p.737 of text
TtTt
sH
Ttee
jT
jT
s
tT
jTts
3by that cancels )(by generated Signal
eeigen valu ,0
2 when )1(1
)2
(log1for
3
22
log1
000
0
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Problem 9.44, p.733 of text
Tteee
jT
js
Tjmee
eee
nTte
TtT
jts
jmTs
Tsn
snTnT
n
nT
when
2-1for
2-1s ,1
poles find to
11sX
e tx
)21(
000
)2()1(
)1(0
T-
0
0