signal & systems - kocwelearning.kocw.net/kocw/document/2015/chungnam/kimjin... · 2016. 9. 9. ·...
TRANSCRIPT
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Kim, J. Y.
IC & DSP
Research
Group
Signal & Systems
Chonnam National University
Dept. of Electronics Engineering
Kim, Jinyoung
http://www.chonnam.ac.kr/
-
Kim, J. Y.
IC & DSP
Research
Group
6. Representation of Signals Using Continuous-Time Complex Exponentials :
Laplace Transform
http://www.chonnam.ac.kr/
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Kim, J. Y.
IC & DSP
Research
Group
Laplace, Pierre Simon
1749~1827 프랑스의 수학자, 물리학자, 천문학자, 노르망디의 빈농출신. 어릴 때부터 비상한 재능이 있었고, 1784년에는 에콜 노르말의 교수가 되었다. 나폴레옹 1세 하에서 내상(1799)과 백작이 되었고, 또 왕정복고 시기에 후작의 지휘를 받았다. (1817). 정치적으로는 입장이 불명확했다. 해석학에 뛰어나, 이것을 천체역학이나 확률론에 응용하여 많은 성과를 얻었다. 명저[천체역학](Mecanique celeste, 5권 1799~1825)는 뉴튼 이래의 천체역학을 집대성하여, 태양의 천체세게 관련된 많은 현상을 해명했다. 특히 그 섭동론은 천왕성 운행의 이론적 계산치의차이를 이용하여 해왕성의 크기와 위치를 예언하고, 그 발견에 기여한 르베리에(Urbain J.J. Le Verrier, 1811~77, 프랑스의 천문학자. 해왕성은 J.Galle에 의해 1846년 9월에 발견됨)등의 사업의 기초를 마련했다. 또 태양계의 기원에 대해 칸트-라플라스설인 성운성을 완성시켰다. 그 외에도, 수학, 물리학 연구에 뛰어난 업적을 남겼다.
http://www.chonnam.ac.kr/
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Kim, J. Y.
IC & DSP
Research
Group
Shortage of Fourier Transform
1) Damped Signal
2) Non-stationary
t j te e
http://www.chonnam.ac.kr/http://kr.youtube.com/watch?v=0dPS-EHl-FE
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Kim, J. Y.
IC & DSP
Research
Group
6.1 Introduction
Laplace Transform
- A representation in terms of complex
exponential signals
- Generalization of FT
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Kim, J. Y.
IC & DSP
Research
Group
6.2 The Laplace Transform
LTI system response to complex
exponential function
Definition of Laplace transform
S-plane
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Kim, J. Y.
IC & DSP
Research
Group
Complex exponential : est
cos( ) sin( )st t te e t je t
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Kim, J. Y.
IC & DSP
Research
Group
6.2.1 Eigenfunction Property of est
H x(t)=est y(t)
( )
( ) { }
( )* ( )
( ) ( )
( )
( )
st
s t
st s
y t H e
h t x t
h x t d
h e d
e h e d
{ } ( )
, where ( ) ( )
st st
s
H e H s e
H s h e d
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Kim, J. Y.
IC & DSP
Research
Group
Express in polar form
LTI System Response to Complex Exponential Function
( )( ) ( ) j s sty t H s e e
Substitute s j
{ ( )}( ) ( )
( ) cos( ( ))
( ) sin( ( ))
t j t j
t
t
y t H j e e
H j e t j
j H j e t j
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Kim, J. Y.
IC & DSP
Research
Group
6.2.2 Laplace Transform Representation : Derivation of Laplace Transform
A representation for arbitrary signals as a
weighted superposition of eigenfunction est.
( )( ) ( ) ( )
( )
1( ) ( )
2
j t
t j t
t j t
H s H j h e d
h e e d
h t e H j e d
( )
1( ) ( )
2
1( )
2
t j t
j t
h t e H j e d
H j e d
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Kim, J. Y.
IC & DSP
Research
Group
Derivation of Laplace Transform
2
Substituting and js jdsd /
Laplace Transform
1( ) ( )
2
j
st
j
h t H s e dsj
( ) ( )
1( ) ( )
2
st
j
st
j
X s x t e dt
x t X s e dsj
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Kim, J. Y.
IC & DSP
Research
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6.2.3 Convergence
A necessary condition for convergence of
the Laplace transform is absolute
integrability of x(t)e-t.
ROC(region of convergence) : the range
of for which the Laplace transform
converges.
( ) tx t e dt
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Kim, J. Y.
IC & DSP
Research
Group
The Laplace transform applies to more general
signals than the Fourier transform does. (a) Signal for
which the Fourier transform does not exist.
(b) Attenuating factor associated with Laplace
transform. (c) The modified signal x(t)e-t is absolutely
integrable for > 1.
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Kim, J. Y.
IC & DSP
Research
Group
Right half plane
Left half plane
6.2.4 S-plane
6.2.5 Pole-Zero
Represent the complex frequency s in
terms of a complex plane termed the s-
plane
j
X(j)=X(s)|=0
Zero:X(s)=0
Pole:X(s)=
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Kim, J. Y.
IC & DSP
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General Form of Laplace
Transform
A ratio of two polynomials in s
ck : zeros, dk :poles
1
1 1 0
1
1 1 0
1
1
( )
( )( )
( )
M M
M M
N N
N
M
M kk
N
kk
b S b S b s bX s
S a S a s a
b s cX s
s d
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Kim, J. Y.
IC & DSP
Research
Group
Examples 1
Example6.1 Determine the Laplace
transform of x(t)=eatu(t)
j
a
( )
0
( )
0
( ) ( )
1
1, Re( )
at st
s a t
s a t
X s e u t e dt
e dt
es a
s as a
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Kim, J. Y.
IC & DSP
Research
Group
Examples 2
Example6.2 Determine the Laplace
transform of y(t)=-eatu(-t)
a
j
0
( )
0
( )
( ) ( )
1
1,Re( )
at st
s a t
s a t
Y s e u t e dt
e dt
es a
s as a
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Kim, J. Y.
IC & DSP
Research
Group
Examples 3
Comment
In the previous examples the Laplace
transforms X(s) and Y(s) are equal even
though the signals x(t) and y(t) are
different.
The ROC must be specified for the
Laplace transform to be unique.
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Kim, J. Y.
IC & DSP
Research
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6.3 The Unilateral Laplace
Transform
Definition
Properties
Examples
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Kim, J. Y.
IC & DSP
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Definition
Definition : unilateral Laplace transform
Example
0
( ) ( )
( ) ( )U
st
L
X s x t e dt
x t X s
1( )
1( ) with ROC { }
ULat
Lat
e u ts a
e u t re s as a
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Kim, J. Y.
IC & DSP
Research
Group
6.4 Properties of the Unilateral
Laplace Transform
Linearity
Scaling
Time Shift
( ) ( ) ( ) ( )UL
ax t by t aX s bY s
1( )
| |
UL sx at X
a a
( ) ( )
for all such that ( ) ( ) ( ) ( )
ULsx t e X s
τ x t u t x t u t
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Kim, J. Y.
IC & DSP
Research
Group
Time shifts for which the unilateral Laplace
transform time-shift property does not apply. (a) A
nonzero portion of x(t) that occurs at times t 0 is
shifted to times t < 0. (b) A nonzero portion x(t) that
occurs at times t < 0 is shifted to times t 0.
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Kim, J. Y.
IC & DSP
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Properties 2
s-Domain Shift
Convolution
Differentiation in the s-domain
0
0( ) ( )UL
s te x t X s s
( )* ( ) ( ) ( )UL
x t y t X s Y s
( ) ( )UL d
tx t X sds
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Kim, J. Y.
IC & DSP
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Properties 3
Differentiation in the time domain
General form for the differentiation
property
00 0
( ) ( ) ( ) ( )
( ) ( ) (0 )
U
U
Lst st st
L
d dx t x t e dt x t e s x t e dt
dt dt
dx t sX s x
dt
1 2
1 20 0
2 1
0
( ) ( ) ( ) | ( ) |
... ( ) | (0 )
Un n nLn
n n nt t
n n
t
d d dx t s X s x t s x t
dt dt dt
ds x t s x
dt
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Kim, J. Y.
IC & DSP
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Properties 4
Initial and final value theorems
Integration property
0
lim ( ) (0 )
lim ( ) ( )
s
s
sX s x
sX s x
( 1)
0
( 1)
(0 ) ( )( ) ,
where (0 ) ( )
Ut L x X s
x ds s
x x d
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Kim, J. Y.
IC & DSP
Research
Group
http://www.chonnam.ac.kr/
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Kim, J. Y.
IC & DSP
Research
Group
Examples 1
Example6.3 Find the unilateral Laplace
transform of x(t)=(-e3tu(t))*(tu(t))
(sol)
Apply the s-domain differentiation property
3 1( )3
1( )
U
U
Lt
L
e u ts
u ts
2
1( )
UL
tu ts
3
2
1( ) ( ( ))*( ( ))
( 3)
ULtx t e u t tu t
s s
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Kim, J. Y.
IC & DSP
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Example 2
Example 6.4 RC filter output x(t)=te2tu(t)
/( )
2
2
1 5( ) ( )
5
1( )
( 2)
5( )
( 2) ( 5)
uLt RCh t e u t
RC s
X ss
Y ss s
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Kim, J. Y.
IC & DSP
Research
Group
Examples 3
Example6.5 Verify the differentiation
property for the signal x(t)=eatu(t)
(sol)
, 0
( ) ( )U
at at
Lat
de ae t
dt
d ax t ae u t
dt s a
1( ) ( ) (0 ) 1
ULdx t sX s x s
dt s a
a
s a
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Kim, J. Y.
IC & DSP
Research
Group
Examples 4
Example6.5 Determine the initial and
final values of a signal x(t) whose
unilateral LT is
(sol)
7 10( )
( 2)
sX s
s s
0
7 10(0 ) lim 7
( 2)
7 10( ) lim 5
( 2)
s
s
sx s
s s
sx s
s s
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Kim, J. Y.
IC & DSP
Research
Group
6.5 Inversion of the Laplace
Transform
Direct inversion of Laplace transform
given by
requires an understanding of contour
integration .
Determine inverse Laplace transform
using the one-to-one relationships
between a signal and its unilateral
Laplace transform.
1( ) ( )
2
j
st
j
x t X s e dsj
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Kim, J. Y.
IC & DSP
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Inverse Lapace Transform based on Several Basic Transform Pairs 1
Laplace transforms that are a ratio of
polynomials in s.
1
1 1 0
1
1 1 0
1
1 1 0
1
1
( )
( )
( )
( )
M M
M M
N N
N
M M
M M
N
k
k
Nk
k k
b S b S b s bX s
S a S a s a
b S b S b s bX s
s d
AX s
s d
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Kim, J. Y.
IC & DSP
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Inverse Lapace Transform based on Several Basic Transform Pairs 2
Basic Laplace transform pair
(CF) If a pole di is repeated r times, then
there are r terms in thr partial fraction
expansion associated with this pole.
( )U
k
Ld t k
k
k
AA e u t
s d
1 2
2
1
, , ,
( )( 1)!
r
U
k
i i i
r
i i i
n Ld t
n
k
A A A
s d s d s d
At Ae u t
n s d
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Kim, J. Y.
IC & DSP
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Example
Example6.7
(sol)
2
3 4( )
( 1)( 2)
sX s
s s
31 2
2
2
( )1 2 2
1 1 2( )
1 2 2
AA AX s
s s s
X ss s s
2 2( ) ( ) ( ) 2 ( )t t tx t e u t e u t te u t
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Kim, J. Y.
IC & DSP
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A pair of Complex-Conjugate
Poles 1
+j0 and -j0 are a pair of complex-
conjugate poles.
1 21 2
0 0
1 2
0 0
2 01 2 1
2 2 2 2 2 2
0 0 0
1 21 1 2 1 2 0 2 2
0
1 2
( * )
( )( )
( )
( ) ( ) ( )
( , ( ) /( )
( , : )
A AA A
s j s j
B s B
s j s j
CB s B C s
s s s
B s BC B C B B
s
B B real
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Kim, J. Y.
IC & DSP
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A pair of Complex-Conjugate
Poles 2
Basic Laplace transform pairs
11 0 2 2
0
2 01 0 2 2
0
( )cos( ) ( )
( )
sin( ) ( )( )
U
U
Lt
Lt
C sC e t u t
s
CC e t u t
s
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Kim, J. Y.
IC & DSP
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Example
Example6.9
(sol)
2
3 2
4 6( )
2
sX s
s s
1 2
2
2
2 2
( )1 ( 1) 1
2 2 2
1 ( 1) 1
2 1 12 4
1 ( 1) 1 ( 1) 1
B s BAX s
s s
s
s s
s
s s s
( ) 2 ( ) 2 cos( ) ( ) 4 sin( ) ( )t t tx t e u t e t u t e t u t
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Kim, J. Y.
IC & DSP
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6.6 Solving Differential Equations with Initial Conditions
Primary application of the unilateral
Laplace transform in systems : solving
differential equations with nonzero initial
conditions.
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Kim, J. Y.
IC & DSP
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Example 6.10 : RC circuit analysis
2( ) 5 ( ) ( ), ( ) 3 ( ), (0 ) 2td
y t y t x t x t e u t ydt
2 5
( ) (0 ) 5 ( ) ( )
1( ) ( ) (0 )
5
3( )
2
3 2( )
( 2)( 5) 5
1 3
2 2
( ) ( ) 3 ( )t
sY s y Y s X s
Y s X s ys
X ss
Y ss s s
s s
y t e tu t e u t
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Kim, J. Y.
IC & DSP
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General Description of Differential Equation in Laplace Transform Domain 1
1
1 1 01
1
1 1 01
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
N N
N NN N
M M
M NM M
d d da y t a y t a y t a y t
dt dt dt
d d db y t b x t b x t b x t
dt dt dt
1
1 1 0
1
1 1 0
11
1 0 0
11
1 0 0
( ) ( ) ( ) ( ) ( ) ( ),
( )
( )
( ) ( )
( ) ( )
N N
N N
M M
M N
lN kk l
k lk l t
lM kk l
k lk l t
A s Y s C s B s X s D s where
A s a s a s a s a
B s b s b s b s b
dC s a s y t
dt
dD s b s x t
dt
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Kim, J. Y.
IC & DSP
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General Description of Differential Equation in Laplace Transform Domain 2
Clear separation between
the natural response of the system to initial conditions and
the forced response of the system associated with the input
( ) ( )
( )
( )
( ) ( ) ( ) ( )( )
( ) ( )
( ) ( )
( ) ( ) ( )( )
( )
( )( )
( )
f n
f
n
B s X s D s C sY s
A s A s
Y s Y s
B s X s D sY s
A s
C sY s
A s
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Kim, J. Y.
IC & DSP
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Example 6.11
2
2
0
( ) 5 ( ) 6 ( ) 2 ( ) ( )
where, ( ) ( ), (0 ) 1, ( ) | 2t
d d dy t y t y t x t x t
dt dt dt
dx t u t y y t
dt
2
0
2
0
2
( 5 6) ( ) (0 ) ( ) | 5 (0 ) (2 1) ( ) 2 (0 )
(2 1) ( ) 2 (0 )( )
5 6
(0 ) ( ) | 5 (0 )
5 6
t
t
ds s Y s sy y t y s X s x
dt
s X s xY s
s s
dsy y t y
dt
s s
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Kim, J. Y.
IC & DSP
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Example 6.11 : continued
( )
( )
( ) 2 3
( ) 2 3
1 1/ 6 1/ 2 1/ 3( )
( 2)( 3) 2 3
7 5 4( )
( 2)( 3) 2 3
1 1 1( ) ( ) ( ) ( )
6 2 3
( ) 5 ( ) 4 ( )
f
n
f t t
n t t
Y ss s s s s s
sY s
s s s s
y t u t e u t e u t
y t e u t e u t
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Kim, J. Y.
IC & DSP
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6.7 Laplace Transform Method
in Circuit Analysis - 생략
Basic electrical circuit elements
- Resistance
- Inductance
- Capacitance
( ) ( )
( ) ( )
R R
R R
v t Ri t
V s RI s
( )
( ) ( ) (0 )
L L
L L L
dv L i t
dt
V s sLI s Li
0
1( ) ( ) (0 )
(0 )1( ) ( )
t
C C C
CC C
v t i d vC
vV s I s
sC sC
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Kim, J. Y.
IC & DSP
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Laplace Transform Circuit
Model 2
Laplace transform circuits models for use
with Kirchhoff’s voltage law
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Kim, J. Y.
IC & DSP
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Laplace Transform Circuit
Model 3
Laplace transform circuits models for use
with Kirchhoff’s current law
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Kim, J. Y.
IC & DSP
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Laplace Transform Circuit
Model 4
Example6.13 : use Laplace transform circuit
modes to determine the voltage y(t) in the
circuit of the following figure.
x(t)=3e-10tu(t) and vC(0-)=5
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Kim, J. Y.
IC & DSP
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Laplace Transform Circuit
Model 6
Example6.11 : (sol) continued
1 2
14
2
20
( ) 1000( ( ) ( ))
1 5( ) ( ) ( )
10
( ) ( ) 1000 ( )
10 5( ) ( )
20 20
2 1( ( ) 3 )
20 10
( ) 2 ( )t
Y s I s I s
X s Y s I ss s
X s Y s I s
sY s X s
s s
X ss s
y t e u t
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Kim, J. Y.
IC & DSP
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6.8 The Bilateral Laplace
Transform
The bilateral Laplace Transform
( ) ( ) ( )L
stx t X s x t e dt
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Kim, J. Y.
IC & DSP
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Properties of the Bilateral Laplace
Transform 1
The linearity, scaling, s-domain, convolution, and differentiation is the s-domain properties are identical for the bilateral and unilateral Laplace transform, although operations associated with these properties may change the ROC.
The bilateral Laplace transform properties involving time shift and differentiation in time domain, the time integration differ slightly from their unilateral counterparts.
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Kim, J. Y.
IC & DSP
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Properties of the Bilateral Laplace
Transform 2
Time shift
Differentiation in the time domain
Time Integration
Integration introduces a pole at s=0
( ) ( )L
sx t e X s
( ) ( ) with ROC at least L
x
dx t sX s R
dt
( )( ) with ROC Re( ) 0
s
t L
x
X sx d R s
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Kim, J. Y.
IC & DSP
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Example 6.15 Find the Laplace
transform
x(t)
(sol)
2
3( 2)
2( ) ( 2)t
dx t e u t
dt
3
3( 2) 2
2 23( 2) 2
2
1( ) , with ROC Re( ) 3
3
1( 2) , with ROC Re( ) 3
3
( ) ( 2) , with ROC Re( ) 33
Lt
Lt s
Lt s
e u t ss
e u t e ss
d sx t e u t e s
dt s
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Kim, J. Y.
IC & DSP
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6.9 Properties of the Region of
Convergence
Convergence
- ROC cannot contain any poles
- Convergence of the Laplace transform for a signal x(t) implies that
-The quantity is the real part of s, so the ROC depends only on the real part of s. ->ROC consists of strips parallel to the j-axis in s-plane
( ) | ( ) | tI x t e dt
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Kim, J. Y.
IC & DSP
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Properties of the Region of
Convergence 2
ROC for x(t) that is finite duration signal.
: ROC for a finite-duration signal
includes the entire s-plane
( ) 0 for and x t t a t b
( ) | ( ) |
( / ) , 0
( ), 0
b
t
a
bt
a
I Ae dt x t A
A e
A b a
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Kim, J. Y.
IC & DSP
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Properties of the Region of
Convergence 3
ROC for x(t) that is infinite duration
signal.
0
0
( ) | ( ) |
( ) ( )
, ( ) | ( ) |
( ) | ( ) |
t
t
t
I x t e dt
I I
where I x t e dt
I x t e dt
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-
Kim, J. Y.
IC & DSP
Research
Group
Properties of the Region of
Convergence 4
ROC for x(t) that is infinite duration
signal. : continued
np
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-
Kim, J. Y.
IC & DSP
Research
Group
Properties of the Region of
Convergence 5
ROC for x(t) that is infinite duration
signal. : continued
| ( ) | , 0| ( ) | , 0
p
n
t
t
x t Ae t
x t Ae t
00
( ) ( )
( ) ( )
0 0
( )
( )
n n
p p
t t
n
t t
p
AI A e dt e
AI A e dt e
and n p
p n
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-
Kim, J. Y.
IC & DSP
Research
Group
Properties of the Region of
Convergence 6
Change in ROC : example
( ) ( ) with ROC
( ) ( ) with ROC
( ) ( ) ( ) ( )
with ROC at least
L
x
L
y
L
x y
x t X s R
y t Y s R
ax t by t aX s bY s
R R
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-
Kim, J. Y.
IC & DSP
Research
Group
Properties of the Region of
Convergence 7
Relationship between the time extent of a signal
and the ROC
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-
Kim, J. Y.
IC & DSP
Research
Group
Example 6.16
Identify the ROC associated with the
bilateral Laplace transform
2
1
2
2
( ) ( ) ( )
( ) ( ) ( )
t t
t t
x t e u t e u t
x t e u t e u t
( ) ROC Re(s)
( ) ROC Re(s)
at
at
e u t a
e u t a
2
1
2
2
1( ) ( ) ( ) , 2 Re( ) 1
( 1)( 2)
( ) ( ) ( ) X
Lt t
Lt t
x t e u t e u t ROC ss s
x t e u t e u t
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Kim, J. Y.
IC & DSP
Research
Group
ROCs for signals in Example 6.16. (a) The shaded regions denote
the ROCs of each individual term, e–2tu(t) and e–tu(–t). The doubly
shaded region is the intersection of the individual ROCs and
represents the ROC of the sum. (b) The shaded regions represent
the individual ROCs of
e–2tu(–t) and e–tu(t). In this case there is no intersection, and the
Laplace transform of the sum does not converge for any value of s.
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-
Kim, J. Y.
IC & DSP
Research
Group
Inversion of The Bilateral Laplace
Transform 1
Use the ROC to determine a unique
inverse transform in the bilateral case.
The ROC of a right-sided exponential
signal lies to the right of a pole, while the
ROC of a left-sided exponential signal lies
to the left of a pole
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-
Kim, J. Y.
IC & DSP
Research
Group
Inversion of The Bilateral Laplace
Transform 2
Example 6.17 Inverting a Laplace
transform
(sol)
5 7( ) with ROC 1 Re( ) 1
( 1)( 1)( 2)
sX s s
s s s
2
1 2 1( )
1 1 2
( ) ( ) 2 ( )
( )
t t
t
X ss s s
x t e u t e u t
e u t
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-
Kim, J. Y.
IC & DSP
Research
Group
Inversion of The Bilateral Laplace
Transform 3
Example 6.18 : Inverting an improper
rational Laplace transform
3 2
2
2 9 4 10( ) , with ROC Re( ) 1
s 3 4
s s sX s s
s
(1) 4
1 2( ) 2 3
1 4
( ) 2 ( ) 3 ( ) ( ) 2 ( )t t
X s ss s
x t t t e u t e u t
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-
Kim, J. Y.
IC & DSP
Research
Group
Inversion of The Bilateral Laplace
Transform 4
Partial fraction expansion
1
( )N
k
k k
AX s
s d
( ) with ROC Re( )
( ) with ROC Re( )
k
k
Ld t k
k k
k
Ld t k
k k
k
AA e u t s d
s d
AA e u t s d
s d
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-
Kim, J. Y.
IC & DSP
Research
Group
Inversion of The Bilateral Laplace
Transform 5
Some Laplace transform pairs
1
1
( ) ; right sided signal( 1)! ( )
( ) ; left sided signal( 1)! ( )
k
k
n Ld t
n
k
n Ld t
n
k
At Ae u t
n s d
At Ae u t
n s d
11 0 2 2
0
11 0 2 2
0
( )cos( ) ( )
( )
; right sided signal
( )cos( ) ( )
( )
; left sided signal
Lt
Lt
C sC e t u t
s
C sC e t u t
s
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-
Kim, J. Y.
IC & DSP
Research
Group
6.11 Transform Function
The transfer function of a system : the
Laplace transform of the impulse
response. ( ) ( )* ( )
( ) ( ) ( )
( ) ( ) / ( )
y t h t x t
Y s H s X s
H s Y s X s
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-
Kim, J. Y.
IC & DSP
Research
Group
The Transfer Function and
Differential Equations 1
Differential-equation description for a
LTI system
x(t)=est is an eigenfunction of the system
0 0
( ) ( )k kN M
k kk kk k
d da y t b x t
dt dt
0 0
0
0
,
( )
k k kN Mst st st k st
k kk k kk k
Mk
k
k
Nk
k
k
d d da e b e e s e
dt dt dt
b s
H s
a s
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-
Kim, J. Y.
IC & DSP
Research
Group
The Transfer Function and
Differential Equations 2
Example6.19
Pole-zero form
2
2( ) 3 ( ) 2 ( ) 2 ( ) 3 ( )
d d dy t y t y t x t x t
dt dt dt
2
2 3( )
3 2
sH s
s s
1
1
( / ) ( )( )
( )
M
M N kk
N
kk
b a s cH s
s d
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-
Kim, J. Y.
IC & DSP
Research
Group
The Transfer Function and
Differential Equations 3
Example6.18
1
2
Torque produced by the motor : ( ) ( )
Back electromortive fource : ( ) ( )
t K i t
dv t K y t
dt
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-
Kim, J. Y.
IC & DSP
Research
Group
The Transfer Function and
Differential Equations 4
Example6.20:solution
2
12
2
1 122
2
1 2 1
2
1( ) ( ), ( ) [ ( ) ( )]
( ) [ ( ) ( )] [ ( ) ( )]
( ) ( ) ( )
dJ y t K i t i t x t v t
dt R
K Kd dJ y t x t v t x t K y t
dt R R dt
K K Kd dJ y t y t x t
dt R dt R
11
2 1 2 1 2
( )
KK
RJRH sK K K K
Js s s sR R
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-
Kim, J. Y.
IC & DSP
Research
Group
6.12 Causality and Stability
Causality : Impulse response of a causal
system is zero for t
-
Kim, J. Y.
IC & DSP
Research
Group
The relationship between the locations of poles and the impulse
response in a causal system. (a) A pole in the left half of the s-plane
corresponds to an exponentially decaying impulse response. (b) A
pole in the right half of the s-plane corresponds to an exponentially
increasing impulse response. The system is unstable in this case.
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-
Kim, J. Y.
IC & DSP
Research
Group
Stability
Stability : The impulse response is
absolutely integrableThe Fourier
transform exists and thus the ROC
includes the j-axis in the s-plane.
This knowledge is sufficient to uniquely
determine the inverse Laplace transform
of the transfer function.
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-
Kim, J. Y.
IC & DSP
Research
Group
The relationship between the locations of poles and the impulse
response in a stable system. (a) A pole in the left half of the s-
plane corresponds to a right-sided impulse response. (b) A pole
in the right half of the s-plane corresponds to an left-sided
impulse response. In this case, the system is noncausal.
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-
Kim, J. Y.
IC & DSP
Research
Group
A system that is both stable and causal must have a
transfer function with all of its poles in the left half of
the s-plane, as shown here.
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-
Kim, J. Y.
IC & DSP
Research
Group
Example 6.21
Transfer function
(a) stable (b) causal
-3 2 -3 2
2 1( )
3 2H s
s s
3 2( ) 2 ( ) ( )t th t u t e u t 3 2( ) 2 ( ) ( )t th t u t e u t
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-
Kim, J. Y.
IC & DSP
Research
Group
Inverse Systems 1
Inverse system
Minimum phase : H(s) has all of its poles and
zeros in the left half of the s-plane : A
nonminimum phase system cannot have a
stable and causal inverse system.
1
1 1
( )* ( ) ( )
1( ) ( ) 1 or ( )
( )
h t h t t
H s H s H sH s
1 1
1
( )( )
( / ) ( )
N
kk
M
M N kk
s dH s
b a s c
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-
Kim, J. Y.
IC & DSP
Research
Group
Inverse Systems 2
Example6.21 Find the inverse system
(sol)
:The inverse system cannot be both stable and causal
2
2( ) 3 ( ) ( ) ( ) 2 ( )
d d dy t y t x t x t x t
dt dt dt
2
2
1
2
( )( 3) ( )( 2)
( ) 2( )
( ) 3
1 3 3( )
( ) 2 ( 1)( 2)
Y s s X s s s
Y s s sH s
X s s
s sH s
H s s s s s
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-
Kim, J. Y.
IC & DSP
Research
Group
6.13 Determining the Frequency Response from Poles and Zeros
s j
at some fixed , 0
1
1
( / ) ( )( )
( )
M
M N kk
N
kk
b a j cH j
j d
010
01
/( )
M
M N kk
N
kk
b a j cH j
j d
0 0
1
0
1
arg ( ) arg / arg
arg
M
M N k
k
N
k
k
H j b a j c
j d
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-
Kim, J. Y.
IC & DSP
Research
Group
The function |j – g| corresponds to the lengths of
vectors from g to the j-axis in the s-plane. (a) Vectors
from g to j for several frequencies. (b) |j – g| as a
function of j.
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-
Kim, J. Y.
IC & DSP
Research
Group
Components of the magnitude response.
(a) Magnitude response associated with a zero.
(b) Magnitude response associated with a pole.
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-
Kim, J. Y.
IC & DSP
Research
Group
Example 6.23
Sketch the magnitude and phase response of
the system having the transfer function
(sol)
( 0.5)( )
( 0.1 5 )( 0.1 5 )
sH s
s j s j
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-
Kim, J. Y.
IC & DSP
Research
Group
하나의 꿈, 다양한 지식
Sagres castle
바스코 다 가마 항로 (1497)
조선 지도
천문학
항해
탐험
수학자 고문서 학자
신학자
http://www.chonnam.ac.kr/