Post on 02-Jan-2016
1
Signals and SystemsLecture 25
•The Laplace Transform
•ROC of Laplace Transform
•Inverse Laplace Transform
2
Appendix Partial Fraction Expansion
Consider a fraction polynomial:
)(
)(
)()(
012
21
1
012
21
1
mnwhere
asasasas
bsbsbsbsb
sD
sNsX
nn
nn
n
mm
mm
mm
Discuss two cases of D(s)=0, for distinct root
and same root.
Chapter 9 The Laplace Transform
3
(1) Distinct root:
)())((
)(
21
012
21
1
n
nn
nn
n
sss
asasasassD
n
i i
i
n
n
n
mm
mm
mm
s
A
s
A
s
A
s
A
sss
bsbsbsbsbsX
1
2
2
1
1
21
012
21
1
)())(()(
thus
Chapter 9 The Laplace Transform
4
Calculate A1 :
Multiply two sides by (s-1):
n
n
s
sA
s
sAAsXs
)()(
)()( 1
2
1211
1|)()( 11 ssXsALet s=1, so
isi sXsAi |)()(Generally
Chapter 9 The Laplace Transform
5
(2) Same root:
)())(()(
)(
211
012
21
1
nrrr
nn
nn
n
ssss
asasasassD
n
n
r
rrrr
nrrr
mm
mm
mm
s
A
s
A
s
A
s
A
s
A
ssss
bsbsbsbsbsX
1
1
1
11
1
12
1
11
211
012
21
1
)()()(
)())(()()(
thus
),,2,1(
|)()(
nrri
sXsAiisi
For first order poles:
Chapter 9 The Laplace Transform
6
r
n
n
r
r
rr
r
ss
A
s
A
sAsAAsXs
)]([
)()()()(
11
1
111112111
Multiply two sides by (s-1)r :
For r-order poles:
1|)()( 111 s
r sXsASo
1|)]'()[( 112 s
r sXsA
1|)]()[(
)!1(
1 111
srr
r sXsr
A
Chapter 9 The Laplace Transform
7
9.3 The Inverse Laplace Transform
dejXtx
or
dejX
jXFetx
etxF
jsdteetxjX
dtetxsX
tj
tj
t
t
tjt
st
)(
1
)(2
1)(
)(2
1
)]([)(
])([
)()()(
)()(
So
j
j
stdsesXj
tx
)(
2
1)(
Chapter 9 The Laplace Transform
8
The calculation for inverse Laplace transform:
(1) Integration of complex function by equation.
(2) Compute by Fraction expansion.
General form of X(s):
n
i i
i
n
n
s
A
s
A
s
A
s
AsX
1
2
2
1
1)(
Important transform pair:
polerighttue
polelefttue
s t
t
ii
i
),(
),(1
Example 9.9 9.10 9.11
Chapter 9 The Laplace Transform
9
Chapter 9 The Laplace Transform
§9.3 The Inverse Laplace Transform
ROC
dsesXj
tx stj
j
21
defining
a 0
j j
j
Example 9.9
21
1
sssX
Determine the inverse Laplace transform for all possible ROC.
10
Chapter 9 The Laplace Transform
§9.4 Geometric evaluation of the Fourier transform
几何求值 from the Pole-Zero plot
1
1
i
n
i
i
m
i
αj
jMjX
i
j
i
ij ij
Pole vector: ijii eAj
Zero vector: ijii eBj
iAiB
i i
11
Chapter 9 The Laplace Transform
2
1Re
2/1
1
s
ssXExample 9.12
§9.4.1 First-Order System
txtyty
tueth t
/1
τ——time constant (时间常数)
controls the speed of response of first-order systems
12
Chapter 9 The Laplace Transform
§9.4.2 Second-Order System
0, 1
.1 2121
ss
sH
21,maxRe s
1
21
jjjH
2
1
2
1 .2
22
nnsssH
2
1
2 2H s
s s
n 2 , 1/ 2
13
Chapter 9 The Laplace Transform
§9.4.3 All-Pass Systems (全通系统)
Constant jH
First-Order System
j
1
1 j1A
1
1 j1B
1
1
j
jjH
零极点相对于 jω轴对称
1
1
j
jjH
1
12 1 tgjHjH
全通系统:零极点个数相同,且相对于 jω轴对称。
11 BA
14
Chapter 9 The Laplace Transform
§9.5 Properties of the Laplace Transform
§9.5.1 Linearity of the Laplace Transform
sbXsaXtbxtax L2121
sXtx L11 1RRoc
sXtx L22 2RRoc
21 RRRoc
15
Chapter 9 The Laplace Transform
Example 9.13
1Re 21
12
s
sssX
1Re 1
11
s
ssX
j
12
j
1
2
121
s
sXsXsX
2Re s
2tx t e u t
j
2
16
Chapter 9 The Laplace Transform
§9.5.2 Time Shifting
0L0
stesXttx
sXtx L RRoc
RRoc
Example kTttx
k
0
Re 0s
1
1 sTX s
e
j
pole-zero plot
Tj2
Tj2
17
sXtx L
Chapter 9 The Laplace Transform
§9.5.3 Shifting in s-Domain
0L0 ssXetx ts
RRoc
0Re sRRoc
ROC的边界平移
j
2r
21 Re rsr
1r
j
0201 ReReRe srssr
01 Re sr 02 Re sr
18
Chapter 9 The Laplace Transform
20
2L
0cos
s
stut 0Re s
20
20L
0sin
s
tut 0Re s
tute at0cos
20
2L
as
as as Re
tute at0sin
20
20L
as
as Re
19
Chapter 9 The Laplace Transform
§9.5.4 Time Scaling
sXtx L RRoc
asXa
atx /1L aRRoc
sXtx L
RRoc
When 1a
20
Chapter 9 The Laplace Transform
1
22
se t
1Re1 s 1
j
1
4
42
2
se t
2Re2 s 2
j
2
4/1
12
2
1
se
t
2
1Re
2
1 s 2
1
j
2
1
21
Chapter 9 The Laplace Transform
§9.5.5 Conjugation
sXtx L RRoc
sXtx L RRoc
txtx sXsX
22
Chapter 9 The Laplace Transform
§9.5.6 Convolution Property
sXtx L11 1RRoc
sXtx L22 2RRoc
sXsXtxtx L2121 21 RRRoc
2Re 2
11
ss
ssX
1Re 1
22
ss
ssX
121 sXsX sRe ttxtx 21
23
Chapter 9 The Laplace Transform
Example ? 213
22
1 txtxtuetxtuetx tt
不存在傅立叶变换
5
1
5
1 3221 tuetuetxtx tt
24
Chapter 9 The Laplace Transform
§9.5.7 Differentiation in the Time Domain
sXtx L RRoc
RRoc ssXdt
tdx L
1
0 t
tx
2 4 6 8
Example
Determine sX
2
1 2 2 2
1 2 Re 0
1
s s
s
e eX s X s X s s
s e
25
§9.5.8 Differentiation in the s-Domain
Chapter 9 The Laplace Transform
sXtx L RRoc
RRoc ds
sdXttx L
2
1
astute Lat
as Re
3
2 1
2
1
astuet Lat
as Re
26
Chapter 9 The Laplace Transform
more generally,
1
1
!
1
n
Latn
astuet
n as Re
1
1
!
1
n
Latn
astuet
n as Re
27
Chapter 9 The Laplace Transform
Example
1Re 21
12
sss
esX
s
Determine tx
Solution:
1111 1211
2
tuetuetuet
tuetuetutetxttt
ttt
28
Chapter 9 The Laplace Transform
Example 11
tuet
tx at
Determine sX
s
assX
ln
as ,0maxRe
29
Chapter 9 The Laplace Transform
§9.5.9 Integration in the Time Domain
sXtx L RRoc
sXs
dxt 1L
0Re sRRoc
ROC的变化:
① R 与 无公共部分,积分的拉氏变换不存在。 0Re s
tuetx t
1Re 1
1
s
ssX
的积分不存在拉氏变换 tx
0
j
1
30
Chapter 9 The Laplace Transform
② R 与 部分重叠。 0Re s
tuetx t 2
dxt
0
j
2
③ R 与 部分重叠。 0Re s
1Re 21
sss
ssX
s ss
dxt
1Re21
1L
0
j
12
s s-
L 2Re2
1
s s-s
2Re02
1L
31
Chapter 9 The Laplace Transform
§9.5.10 The Initial- and Final-Value Theorems
初值定理和终值定理1. The Initial-Value Theorem
0 , 0 ttx Contains no impulses or higher order singularities at the origin.
ssXxs
lim0 为真分式 sX
321
122
sss
sssX
ssXxs
lim0
1321
12lim
2
sss
ssss
32
Chapter 9 The Laplace Transform
2. The Final-Value Theorem
0 , 0 ttx
的极点均在 jω轴左侧,允许在 s=0有一个一阶极点 sX
ssXtxxst 0limlim
asas
sX
Re 1
0 a① 0limlim0
ssXtxst
0 a② 11
limlim0
s
stxst
0 a③ 终值不存在。
33
1 1 Re ln
1 sTX s s aae T
Chapter 9 The Laplace Transform
§9.5.11 运用基本性质求解拉氏变换
0
k
k
x t a t kT
Example 1
Determine sX
j
aT
ln1
Example 2
1 Re 0
1 sX s s
s e
Determine tx
k 0
1 k
x t u t - k
34
Chapter 9 The Laplace Transform
Example 3
22
2 Re 0
1X s s
s
Determine tx
00 2 2
0
sin Re 0Ltu t ss
sin cosx t t t t u t
35
Chapter 9 The Laplace Transform
§9.7 Analysis and Characterization of LTI Systems
Using the Laplace Transform
ty th
sH sY
tx
sX
thtxty
sHsXsY
sH ——System Function or Transfer Function
36
Chapter 9 The Laplace Transform
For a system with a rational system function,
causal maxRe sROC
§9.7.2 Stability (稳定性)
stable axisjω ROC
§9.7.1 Causality
Causal maxRe sROC
37
Chapter 9 The Laplace Transform
Example 9.20 21
1
ss
ssH
j
21
j
21
j
21
2Re a s
Causal , unstable system
2Re1 b s-
noncausal , stable system
1Re c s
anticausal , unstable system(反因果)
38
系统因果、稳定
Chapter 9 The Laplace Transform
sH 的极点均在 轴左侧,
且
j
maxRe s
如果 为有理函数 sH
Stability of Causal System
Consider the following causal systems
1
1 a
ssH ——Stable
21
1 b
sssH ——unstable
39
Chapter 9 The Laplace Transform
Causal maxRe sROC
For a system with a rational system function,
causal maxRe sROC
stable axisjω ROC
40
Chapter 9 The Laplace Transform
§9.7.3 LTI Systems Characterized by Linear Constant-Coefficient
Differential Equations
txtydt
tdy3
k
kM
kkk
kN
kk dt
txdb
dt
tyda
00
ROCk
k
N
k
kk
M
k
sa
sb
0
0
sX
sYsH
41
Chapter 9 The Laplace Transform
Example Consider a causal LTI system whose input and
output related through an linear constant-coefficient
differential equation of the form
tx
y t
3 2y t y t y t x t
Determine the unit step response of the system.
21 1
2 2t ts t e e u t
42
Chapter 9 The Laplace Transform
Example 9.24
Consider a RLC
circuit in Figure 9.27+
R L
C
-
+ ty
tx
-
Figure 9.27
LCsLRs
LCsH
/1/
/12
43
Chapter 9 The Laplace Transform
Example 9.25
Consider an LTI system with input ,
Output .
(a) Determine the system function.
(b) Justify the properties of the system.
(c) Determine the differential equation of the system.
tuet x t3
tueet y tt 2
3 Re 1
1 2
sH s s -
s s
3 2 3y t y t y t x t x t
44
Chapter 9 The Laplace Transform
Example Consider a causal LTI system ,
tbutuethdt
tdh t 42 .2
t-etyt-etx tt 6
1 .1 22
b——unknown constant
Determine the system function and b. sH
2 Re 0
4H s s
s s
45
Chapter 9 The Laplace Transform
Example 9.26 An LTI system:
1. The system is causal.
2. is rational and has only two poles: s= - 2 and s=4.
3. 4. Determine 01 tytx
sH
40 h sH
Example 9.26 An LTI system:
1. The system is causal.
2. is rational and has only two poles: s=-2 and s=-4.
3. 4. Determine 01 tytx
sH
40 h sH
42
4
ss
ssH 4Re s
46
Chapter 9 The Laplace Transform
Example 9.27
已知一因果稳定系统, 为有理函数,有一极点
在 s=-2处,原点( s=0)处没有零点,其余零极点未知,
判断下列说法是否正确。
sH
1. 的傅立叶变换收敛。 teth 3
2. 0
dtth
3. 为一因果稳定系统的单位冲激响应。 tth
4. 至少有一个极点。
dt
tdh
5. 为有限长度信号。 th
47
Chapter 9 The Laplace Transform
6. sHsH
在 s=-2处有极点 在 s=+2处有极点
7. 2lim
sHs
无法判断正确与否。
48
Chapter 9 The Laplace Transform
例 设信号 是系统函数为
的因果全通系统的输出。
1. 求出至少有两种可能的输入 都能产生 。
tuety t2 1
1
s
ssH
tf ty
2. 若已知
问输入 是什么?
dttf
tf
3. 如果已知存在某个稳定(但不一定因果)的系统,
它若以 作输入,则输出为 ,问这个输入
是什么?系统的单位冲激响应是什么?
tf ty
tf
49
Problem Set
• P728 9.28
• P729 9.31