1
Chapter 9
The Laplace Transform
2
Chapter 9 The Laplace Transform
§ 9.1 The Laplace Transform
dtethsH st
ste stesH
Defining dtetxsX st
—— Laplace Transform
1,The relationship
tX s x t eF
3
Chapter 9 The Laplace Transform
2,Region of Convergence（收敛域）
Dirichlet Condition 1,
dtetx
t?
ROC：对给定的,使其拉氏变换存在的 σ对应的
S平面上的区域。
tx
sjX j X s
0 R O C
4
Chapter 9 The Laplace Transform
Example 9.1
atx t e u t
1 R eX s s asa
a? 0?
j
pole-zero plot
零极点图
Example 9.2
atx t e u t
1 R eX s s asa
a?
j
pole-zero plot
5
Chapter 9 The Laplace Transform
asastue at Re 1
asastue at Re 1
R O C ; sXtx
Particularly,
0Re 1 sstu 0
j
The Fourier transform of does not exist.tu
jtu F 1
6
Chapter 9 The Laplace Transform
Example 9.3
23 2 tuetuetx tt
2Re 233 2 sstue t
1Re 122 sstue t
23 2 tuetue tt
21
1
ss
s
1Res
j
1?2? 1
7
Chapter 9 The Laplace Transform
Example 9.3 3co s2 tutetuetx tt
2121 332 tueetueetuetx tjttjtt
2Re 212 sstue t
j-s
/tue tj
31
21
2
1 31
1ReRe as
js
/tue tj
31
21
2
1 31
1ReRe as
1022 1252 22 sss sssX
1Res
j
j31
2?
j31
a
8
Chapter 9 The Laplace Transform
Example 9.4
3134 2 tuetuettx tt
dtett st 1? entire S plane
sRe
1Re 1 3/434 sstue t
2Re 23/131 2 sstue t
21 1
2
ss
ssX 2Re?s
2Re0 s? does not exist.txF
j
1? 21
pole-zero plot
9
Chapter 9 The Laplace Transform
sD sNsX?
0?sDPoles:
0?sNZeros:
1,The direction of signals
2,The position of poles ROC of X(s)
§ 9.2 The Properties of ROC
Property 1,The ROC of X(s) consists of strips parallel to
the jω-axis in the s-plane
dtetx t? —— Depends only on σ
10
Chapter 9 The Laplace Transform
Property 2,For rational Laplace transforms,the ROC does
not contain any poles.
Property 3,If is of finite duration and is absolutely integrable,
then the ROC is the entire s-plane.
tx
2T t1T
tx
21 0 T,tT; ttx
dttxTT 2
1
dtetx tTT 2
1
① when 0
R O C0
dtetx t?
11
Chapter 9 The Laplace Transform
0②
12 21 TTee TT
dtetx tTT 2
1
R O C0
0③
12 21 TTee TT
dtetx tTT 2
1
R O C0
dttxe TTT 2
1
1?
dttxe TTT 2
1
2?
21 0 Tt,T; ttx sRe
12
Chapter 9 The Laplace Transform
ExampleTtututx
0Re 1 sstu
dteTtu stT
s
ee
s
sT
T
ts
1 0Re?s
1 seTtutu sTsRe
pole,0?s
zero,01 sTe?kjsT ee 2
zeros,?10,2,k
T
kjs
k
j
pole-zero plot
T
j?2
T
j?2?
零极点抵消
13
Chapter 9 The Laplace Transform
Property 4:
If is right sided,tx
R O C0
R O C0 dtetx t0?
1,0 Tttx
010 01 t ee tt
0 1?T②
T 0 1?①
dtetx tT 0
1
dtetx tT 1
1
dtetxdtetx ttT 11
1 0
0
finite
dtetx t00?
R O CRe 0s
tT etx 0
1
14
Chapter 9 The Laplace Transform
Property 5:
If is left sided,tx
R O C0
1,0 Tttx
R O CRe 0s
Property 6:
If is two sided,tx
R O C0
t-tx,
021 Re s
0T
t
tx
0T t
txL
0T t
txR
1Res 2Res?ROC:
15
Chapter 9 The Laplace Transform
stX s x t e d t
tX s F x t e d t
0 R O CsjX j X s
1,The direction of signals
2,The position of poles ROC of X(s)
16
Chapter 9 The Laplace Transform
Example 9.7
tbetx
tuetuetx tbtb
bsbstue bt Re 1
bsbstue bt Re 1
If b>0,
2 22 bs be tb
If b≤0,tx has no Laplace transform.
b?
j
b
bsb Re
17
Chapter 9 The Laplace Transform
txsX
m a xRes
tx
tx
mi nRes
Property 7:
If the Laplace transform of is rational,
① is right sided,
② is left sided,
Example 9.8
21
1
sssX
j
1?2
j
2? 1?
2Res 1Re2 s
j
1?2?
1Res
left sided two sided right sided
18
Chapter 9 The Laplace Transform
Basic Laplace Pairs
txsX Poles ROC
t? 1 nonesRe
s
1tu 0Re?s
tu 0Re?s
s
1
tue at asRe
tue at asRe
as?
1
as?
1
0?s
0?s
as
as
19
Chapter 9 The Laplace Transform
§ 9.3 The Inverse Laplace Transform
R O C
dsesXjtx stj
j?
2
1defining
a? 0?
j j?
j?
Example 9.9
21 1 sssX
Determine the inverse Laplace transform for all possible ROC.
20
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
j
MjX
i?
j
i?
ij ij
Pole vector,ijii eAj
Zero vector,ij
ii eBj
iA iB
i? i?
21
Chapter 9 The Laplace Transform
21Re 2/11 sssX
Example 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
22
Chapter 9 The Laplace Transform
§ 9.4.2 Second-Order System
0,1,1 21
21
sssH
21,m a xRes
1
21
jjjH
21 2 1,2 22
nn ss
sH
2 122Hs ss n 2,1 / 2
23
Chapter 9 The Laplace Transform
§ 9.4.3 All-Pass Systems （全通系统）
C o n s t a n tjH
First-Order System
j
1?
1j
1A
1?
1j
1B
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?
24
Chapter 9 The Laplace Transform
§ 9.5 Properties of the Laplace Transform
§ 9.5.1 Linearity of the Laplace Transform
sbXsaXtbxtax L 2121
sXtx L 11 1RR o c?
sXtx L 22 2RR o c?
21 RRR o c
25
Chapter 9 The Laplace Transform
Example 9.13
1Re 21 12 ssssX
1Re 111 sssX
j
1?2?
j
1?
2121 ssXsXsX
2Res
2 tx t e u t
j
2?
26
Chapter 9 The Laplace Transform
§ 9.5.2 Time Shifting
0L0 stesXttx
sXtx L RRoc?
RRoc?
Example
kTttx
k
0
R e 0s?
11 sTXs e
j
pole-zero plot
T
j?2
T
j?2?
27
sXtx L
Chapter 9 The Laplace Transform
§ 9.5.3 Shifting in s-Domain
0L0 ssXetx ts
RRoc?
0Re sRR o c
ROC的边界平移
j
2r
21 Re rsr
1r
j
0201 ReReRe srssr
01 Re sr02 Re sr?
28
Chapter 9 The Laplace Transform
2
0
2
L
0c o s s
stut 0Re?s
2
0
2
0L
0s in?
stut
0Re?s
tute at 0co s
2
0
2
L
as
as asRe
tute at 0s i n
2
0
2
0L
as
asRe
29
Chapter 9 The Laplace Transform
§ 9.5.4 Time Scaling
sXtx L RRoc?
asXaatx /1L aRR oc?
sXtx L
RR oc
When 1a
30
Chapter 9 The Laplace Transform
122 se t
1Re1 s
1?
j
1?
442 2 se t
2Re2 s 2?
j
2?
4/112 2
1
se
t
21Re21 s 2
1?
j
2
1
31
Chapter 9 The Laplace Transform
§ 9.5.5 Conjugation
sXtx L RRoc?
sXtx L RRoc?
txtx sXsX
32
Chapter 9 The Laplace Transform
§ 9.5.6 Convolution Property
sXtx L 11 1RR o c?
sXtx L 22 2RR o c?
sXsXtxtx L 2121 21 RRR o c
2Re 211 ssssX
1Re 122 ssssX
121?sXsXsRettxtx 21
33
Chapter 9 The Laplace Transform
Example
213221 txtxtuetxtuetx tt
不存在傅立叶变换
5151 3221 tuetuetxtx tt
34
Chapter 9 The Laplace Transform
§ 9.5.7 Differentiation in the Time Domain
sXtx L RRoc?
RRocssX
dt
tdx L
1
0 t
tx
2 4 6 8
Example
DeterminesX
2
12 22
12 R e 0
1
ss
s
eeX s X s X s s
se
35
§ 9.5.8 Differentiation in the s-Domain
Chapter 9 The Laplace Transform
sXtx L RRoc?
RRoc
ds
sdXttx L
21 astute Lat asRe
32 121 astuet Lat asRe
36
Chapter 9 The Laplace Transform
more generally,
11!1 nLatn astuetn asRe
11!1 nLatn astuetn asRe
37
Chapter 9 The Laplace Transform
Example
1Re 21 12 sss esX s
Determinetx
Solution:
1111 1211
2
tuetuetuet
tuetuetutetx
ttt
ttt
38
Chapter 9 The Laplace Transform
Example
11 tuettx at
DeterminesX
s assX ln
as,0m axRe
39
Chapter 9 The Laplace Transform
§ 9.5.9 Integration in the Time Domain
sXtx L RRoc?
sXsdxt 1L 0Re sRR o c?
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 11 sssX
的积分不存在拉氏变换tx
0?
j
1?
40
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 ssdxt 1Re21 1L
0?
j
1?2?
s s-L 2Re21
s s-s 2Re02
1L
41
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,
ssXx s l i m0 为真分式sX
321 122 sss sssX
ssXx s l i m0 1
321
12l im 2?
sss
sss
s
42
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
ssXtxx st 0limlim
asassX Re 1
0?a① 0limlim
0 ssXtx st
0?a② 11limlim 0 sstx st
0?a③ 终值不存在。
43
11 R e ln1 sTX s s aa e 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 k T?
Example 1
DeterminesX
j
aT ln1
Example 2
1 R e 01 sX s sse
Determinetx
k0
1 kx t u t - k
44
Chapter 9 The Laplace Transform
Example 3
2
2
2 R e 0
1
X s s
s
Determinetx
00 22
0
s i n R e 0Lt u t ss
s i n c o sx t t t t u t
45
Chapter 9 The Laplace Transform
§ 9.7 Analysis and Characterization of LTI Systems
Using the Laplace Transform
tyth
sHsY
tx
sX
thtxty
sHsXsY?
sH —— System Function or Transfer Function
46
Chapter 9 The Laplace Transform
For a system with a rational system function,
causal
m a xRe sR O C
§ 9.7.2 Stability (稳定性）
stable a x i sjωR O C
§ 9.7.1 Causality
Causal
m a xRe sR O C
47
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 cs
anticausal,unstable system
（反因果）
48
系统因果、稳定
Chapter 9 The Laplace Transform
sH 的极点均在 轴左侧，
且
j
m a xRes
如果 为有理函数sH
Stability of Causal System
Consider the following causal systems
11 a ssH
—— Stable
21 1 b sssH —— unstable
49
Chapter 9 The Laplace Transform
0 2 c 22 2 n
nn
n
sssH
21
2
cscssH
n
121 nnc
122 nnc
0,1 21 cc?① Stable system
10② 0Re,Re 21 cc Stable system
01 δ-③ 0Re,Re 21 cc Unstable system
1δ④ 0,21 cc Unstable system
50
Chapter 9 The Laplace Transform
§ 9.7.3 LTI Systems Characterized by Linear Constant-Coefficient
Differential Equations
txty
dt
tdy 3
k
kM
k
kk
kN
k
k dt
txdb
dt
tyda
00
ROC
k
k
N
k
k
k
M
k
sa
sb
0
0
sX sYsH?
51
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
yt
32y t y t y t x t
Determine the unit step response of the system.
21122 tts t e e u t
52
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
53
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 R e 112sH s s - ss
3 2 3y t y t y t x t x t
54
Chapter 9 The Laplace Transform
Example Consider a causal LTI system,
tbutueth
dt
tdh t 42,2
t-etyt-etx tt 61,1 22
b—— unknown constant
Determine the system function and b.sH
2 R e 04H s sss
55
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
40hsH
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
40hsH
42 4 ss ssH 2Res
56
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
57
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。
58
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
tfty
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？
tfty
tf
59
Chapter 9 The Laplace Transform
§ 9.8 System Function Algebra and Block Diagram Representations
（方框图）
tyth
sHsY
tx
sX
thtxty
sHsXsY?
§ 9.8.1 System Functions for Interconnections of LTI Systems
1,Series interconnection
2,Parallel interconnection
3,Feedback interconnection
60
Chapter 9 The Laplace Transform
Example 9.28
Consider the causal LTI system
31 ssH
sYsX +
-
s/1
3
sYsX +
s/1
3?
+
Example 9.29
Consider the causal LTI system
32 sssH
231 sssH
61
Chapter 9 The Laplace Transform
Example 9.30
Consider the causal LTI system
2312 sssH
Example 9.31
Consider the causal LTI system
23 642 2 2 ss sssH
(a) direct form
(b) cascade form
(c) parallel form
62
Chapter 9 The Laplace Transform
§ 9.8.2 LTI 系统的信号流图表示一 定义节点,表示信号和变量。
支路,表示节点间信号的传输路径和方向。
sH
sYsX
sYsXsH
支路增益,转移函数 。?sH
源点,只有输出支路的节点。
阱点,只有输入支路的节点。
前向通路,从源点到阱点沿箭头方向连通的路径
（每个节点只通过一次）。
环路,起点即终点的通路（每个节点只通过一次）。
63
Chapter 9 The Laplace Transform
通路增益,通路中所有支路增益的乘积。
sH1
sH2
sYsX +
-
系统的方框图表示
sYsX 11
sH2?
sH1
系统的信号流图表示环路增益,环路中所有支路增益的乘积。
源点 阱点前向通路环路
64
Chapter 9 The Laplace Transform
二 信号流图的基本性质
1,信号只能沿箭头方向流动。
2,节点信号为连接到接点的所有输入支路信号的叠加，
并传输到每一输出支路。
3,对于给定系统，信号流图表示不唯一。
4,如果信号流图倒置，系统的转移函数不变。
1H
2H
4H
5H
1x
2x
3x
4x
5x
22113 xHxHx
344 xHx?
355 xHx?
65
Chapter 9 The Laplace Transform
三 Mason 增益公式（源点到阱点的增益）
kk
k
gH ov1
△ —— 信号流图的特征行列式
△ =1 -（所有不同的环路增益之和）
+（每两个互不接触的环路增益的乘积之和）
-（每三个互不接触的环路增益的乘积之和）
+（每四个互不接触的环路增益的乘积之和）
66
Chapter 9 The Laplace Transform
k—— 前向通路的序号
kk
k
gH ov1
gk—— 第 k条前向通路的增益
△ k—— 去掉第 k条前向通路后,余下的子流图的特征行列式
G2
sXsY
G1 G3
-H
例 1,试求系统的转移函数
1L
321 GHGL
11 L 3211 GGGg? 11
32
321
11 1
1
GHG
GGGgH
ov
67
Chapter 9 The Laplace Transform
例 2,g
sX
sY
a b
c
d e
f
1L 2L
bcL?1 efL?2
211 LL 21LL?
gedbag 1? 11 b c e fefbc
a b d g egH
ov 1
1
11
1L 2L
bcL?1 efL?2
211 LL
gebag 1? 1
1 efbc
a b e ggH
ov 1
1
11
sXsYa b
c
e
f
g例 3：
68
Chapter 9 The Laplace Transform
例 4：
sXsY
a
b
c
d
e
f
h
i
j
k
g l
1L 2L
3L 4L
bcL?1 deL?2
hiL?3 jkL?4
43211 LLLL 31( LL? 41LL? 32LL? )42LL?
a b d fg?1431 1 LL
g h jlg?2212 1 LL
22111 ggH ov
69
Chapter 9 The Laplace Transform
例 5：
sXsY
a b
c
d
e f h
i
g
1L
2L
bcL?1
fgL?2
211 LL 21LL?
a b dg?1 21 1 L
efhg?2 12 1 L
a ihg?3 13
3322111 gggH ov
70
Example 9.31
222 4 632ssHs ss
sX
s/1 s/1
2?3?
2
4
sY
6?
Chapter 9 The Laplace Transform
1L 2L
71
Chapter 9 The Laplace Transform
§ 9.8.3 系统的模拟
1,加法器
2,标量乘法器
3,积分器
sX1
sX2
sXsX 21?
一 基本的模拟单元
1
1
sX1
sX2
sXsX 21?
sX
a
saXasXsaX
1/ssXsX
s
1sX
s
1
sXs1
72
Chapter 9 The Laplace Transform
二 方框图模拟
Example 9.29
Consider the causal LTI system
32 sssH
txtxtyty 23
dydxtxty tt 32
tw
ty
s/1
3?
s/1
2
tw
tx
S1 S2
73
Chapter 9 The Laplace Transform
交换 S1和 S2的连接顺序
ty
s/1
3?
s/1
2
tx
S1S2
输入相同输出相同系统等价为：
ty
3?
s/1
2
tx
74
Chapter 9 The Laplace Transform
三 信号流图模拟两个基本约定：
1,假定所有的环路均相互接触；
i
i
L 1
2,假定每一前向通路与所有的环路相互接触；
1k
i
i
k
k
kk
k L
g
gH ov
11
75
Example 9.29
Consider the causal LTI system
32 sssH
Chapter 9 The Laplace Transform
sssH /31 /21
1L
sX
1 1/s
-3
sY
2
1
公共点
ty
3?
s/1
2
tx
76
Chapter 9 The Laplace Transform
Example 9.31
Consider the causal LTI system
23 642 2 2 ss sssH
22/2/31 /6/42 ss sssH
sY
-61/s1/s
sX
1
公共点
-3 -2
4
2
77
(a) direct form
(b) cascade form
(c) parallel form
Example Consider the causal LTI system
531 42 sss sssH
Chapter 9 The Laplace Transform
15239 8623 2 sss sssH
78
Chapter 9 The Laplace Transform
§ 9.9 The Unilateral Laplace Transform
（单边拉氏变换）
Defining
dtetxs st 0 IX
tutxs L?IX
m a xRes
0 u n k o w n
0 X
2
1
I
t
tdses
jtx
stj
j
If is causal,tx
ssX IX?
79
Chapter 9 The Laplace Transform
Example 9.33 tuetx at
Example 9.33 tuetuttx t
12?
Example 1?tx
Example tetx t,2
80
Chapter 9 The Laplace Transform
X I 232 sss
Example 9.36 Consider the unilateral transform
2Res
X I 212 sss 2Res
tuettutx t21 2 0tf o r
§ 9.9.2 Properties of the Unilateral Laplace Transform
ssX IX?
Causal Signals:
81
Chapter 9 The Laplace Transform
1,Differentiation in the time-domain
RR O C X I stx IL
RR O C 0X
I
xss
dt
tdx IL
000 121 nnnn
n
n
xsxsssdt txd x?IXIL
Example Consider the signal
determine the unilateral Laplace Transform of
tuetutx at
dt
tdx
82
Chapter 9 The Laplace Transform
0xss
dt
tdx
IX
IL
① The Initial-Value Theorem
ssXx Is lim0
② The Final-Value Theorem
ssXtxx Ist 0l iml im
2,Integration in the time-domain
RR O C X I stx IL
0sReRR O C 01X1 1I t-xsssdx IL
83
Chapter 9 The Laplace Transform
§ 9.9.3 Solving Differential Equations
Using the Unilateral Laplace Transform
时域解经典解法零输入、零状态解法频域解 jHjXjYjYty -
f 1F
tytyty fx
tytyty pc
复频域解双边拉氏变换初始状态为零
sHsXsY?
sYty -f 1L
单边拉氏变换初始状态不为零tytyty fx,,?
微分方程的求解
84
Chapter 9 The Laplace Transform
Example 9.38
Suppose a causal LTI system
with initial conditions:
Let the input to this system be,
Determine the full response of the system,
txtytyty 23
0,0 yy
tutx
23 /23 0300 22 ss sss yysysY I?
ty xILty fIL
5030,2 γy,βy -- L e t?
0t tt eety 231Full response
85
Homework:
9.2 9.5 9.7 9.8 9.9
9.13 9.21(a,b,i,j)
9.22(a,b,c,d) 9.28
9.31 9.32 9.33 9.35 9.45
86
1 2 1 2 1 2 La x t b x t a X s b X s R o c R R
0L0 stx t t X s e
Lx t X s R oc R?
R oc R?
L x t X s R o c R
0 L 00 R estx t e X s s R o c R s
Chapter 9 The Laplace Transform
87
Chapter 9 The Laplace Transform
L 1 / x a t X s a R o c a Ra
L x t X s R o c R
1 2 1 2 1 2 Lx t x t X s X s R o c R R
L d x t s X s R o c R
dt
L d X st x t R o c Rds
L 1 R e 0t x d X s R o c R ss
88
Chapter 9 The Laplace Transform
Causal
m a xRe sR O C
For a system with a rational system function,
causal
m a xRe sR O C
stable a x i sjωR O C
89
Chapter 9 The Laplace Transform
stX s x t e d t
tX s F x t e d t
0 R O CsjX j X s
1,The direction of signals
2,The position of poles ROC of X(s)
90
Chapter 9 The Laplace Transform
stu L 1 0?sRe
1 LtsRe1.
stu L 1 0?sRe2.
astue Lat 1 asRe
astue Lat 1 asRe3.
Chapter 9
The Laplace Transform
2
Chapter 9 The Laplace Transform
§ 9.1 The Laplace Transform
dtethsH st
ste stesH
Defining dtetxsX st
—— Laplace Transform
1,The relationship
tX s x t eF
3
Chapter 9 The Laplace Transform
2,Region of Convergence（收敛域）
Dirichlet Condition 1,
dtetx
t?
ROC：对给定的,使其拉氏变换存在的 σ对应的
S平面上的区域。
tx
sjX j X s
0 R O C
4
Chapter 9 The Laplace Transform
Example 9.1
atx t e u t
1 R eX s s asa
a? 0?
j
pole-zero plot
零极点图
Example 9.2
atx t e u t
1 R eX s s asa
a?
j
pole-zero plot
5
Chapter 9 The Laplace Transform
asastue at Re 1
asastue at Re 1
R O C ; sXtx
Particularly,
0Re 1 sstu 0
j
The Fourier transform of does not exist.tu
jtu F 1
6
Chapter 9 The Laplace Transform
Example 9.3
23 2 tuetuetx tt
2Re 233 2 sstue t
1Re 122 sstue t
23 2 tuetue tt
21
1
ss
s
1Res
j
1?2? 1
7
Chapter 9 The Laplace Transform
Example 9.3 3co s2 tutetuetx tt
2121 332 tueetueetuetx tjttjtt
2Re 212 sstue t
j-s
/tue tj
31
21
2
1 31
1ReRe as
js
/tue tj
31
21
2
1 31
1ReRe as
1022 1252 22 sss sssX
1Res
j
j31
2?
j31
a
8
Chapter 9 The Laplace Transform
Example 9.4
3134 2 tuetuettx tt
dtett st 1? entire S plane
sRe
1Re 1 3/434 sstue t
2Re 23/131 2 sstue t
21 1
2
ss
ssX 2Re?s
2Re0 s? does not exist.txF
j
1? 21
pole-zero plot
9
Chapter 9 The Laplace Transform
sD sNsX?
0?sDPoles:
0?sNZeros:
1,The direction of signals
2,The position of poles ROC of X(s)
§ 9.2 The Properties of ROC
Property 1,The ROC of X(s) consists of strips parallel to
the jω-axis in the s-plane
dtetx t? —— Depends only on σ
10
Chapter 9 The Laplace Transform
Property 2,For rational Laplace transforms,the ROC does
not contain any poles.
Property 3,If is of finite duration and is absolutely integrable,
then the ROC is the entire s-plane.
tx
2T t1T
tx
21 0 T,tT; ttx
dttxTT 2
1
dtetx tTT 2
1
① when 0
R O C0
dtetx t?
11
Chapter 9 The Laplace Transform
0②
12 21 TTee TT
dtetx tTT 2
1
R O C0
0③
12 21 TTee TT
dtetx tTT 2
1
R O C0
dttxe TTT 2
1
1?
dttxe TTT 2
1
2?
21 0 Tt,T; ttx sRe
12
Chapter 9 The Laplace Transform
ExampleTtututx
0Re 1 sstu
dteTtu stT
s
ee
s
sT
T
ts
1 0Re?s
1 seTtutu sTsRe
pole,0?s
zero,01 sTe?kjsT ee 2
zeros,?10,2,k
T
kjs
k
j
pole-zero plot
T
j?2
T
j?2?
零极点抵消
13
Chapter 9 The Laplace Transform
Property 4:
If is right sided,tx
R O C0
R O C0 dtetx t0?
1,0 Tttx
010 01 t ee tt
0 1?T②
T 0 1?①
dtetx tT 0
1
dtetx tT 1
1
dtetxdtetx ttT 11
1 0
0
finite
dtetx t00?
R O CRe 0s
tT etx 0
1
14
Chapter 9 The Laplace Transform
Property 5:
If is left sided,tx
R O C0
1,0 Tttx
R O CRe 0s
Property 6:
If is two sided,tx
R O C0
t-tx,
021 Re s
0T
t
tx
0T t
txL
0T t
txR
1Res 2Res?ROC:
15
Chapter 9 The Laplace Transform
stX s x t e d t
tX s F x t e d t
0 R O CsjX j X s
1,The direction of signals
2,The position of poles ROC of X(s)
16
Chapter 9 The Laplace Transform
Example 9.7
tbetx
tuetuetx tbtb
bsbstue bt Re 1
bsbstue bt Re 1
If b>0,
2 22 bs be tb
If b≤0,tx has no Laplace transform.
b?
j
b
bsb Re
17
Chapter 9 The Laplace Transform
txsX
m a xRes
tx
tx
mi nRes
Property 7:
If the Laplace transform of is rational,
① is right sided,
② is left sided,
Example 9.8
21
1
sssX
j
1?2
j
2? 1?
2Res 1Re2 s
j
1?2?
1Res
left sided two sided right sided
18
Chapter 9 The Laplace Transform
Basic Laplace Pairs
txsX Poles ROC
t? 1 nonesRe
s
1tu 0Re?s
tu 0Re?s
s
1
tue at asRe
tue at asRe
as?
1
as?
1
0?s
0?s
as
as
19
Chapter 9 The Laplace Transform
§ 9.3 The Inverse Laplace Transform
R O C
dsesXjtx stj
j?
2
1defining
a? 0?
j j?
j?
Example 9.9
21 1 sssX
Determine the inverse Laplace transform for all possible ROC.
20
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
j
MjX
i?
j
i?
ij ij
Pole vector,ijii eAj
Zero vector,ij
ii eBj
iA iB
i? i?
21
Chapter 9 The Laplace Transform
21Re 2/11 sssX
Example 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
22
Chapter 9 The Laplace Transform
§ 9.4.2 Second-Order System
0,1,1 21
21
sssH
21,m a xRes
1
21
jjjH
21 2 1,2 22
nn ss
sH
2 122Hs ss n 2,1 / 2
23
Chapter 9 The Laplace Transform
§ 9.4.3 All-Pass Systems （全通系统）
C o n s t a n tjH
First-Order System
j
1?
1j
1A
1?
1j
1B
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?
24
Chapter 9 The Laplace Transform
§ 9.5 Properties of the Laplace Transform
§ 9.5.1 Linearity of the Laplace Transform
sbXsaXtbxtax L 2121
sXtx L 11 1RR o c?
sXtx L 22 2RR o c?
21 RRR o c
25
Chapter 9 The Laplace Transform
Example 9.13
1Re 21 12 ssssX
1Re 111 sssX
j
1?2?
j
1?
2121 ssXsXsX
2Res
2 tx t e u t
j
2?
26
Chapter 9 The Laplace Transform
§ 9.5.2 Time Shifting
0L0 stesXttx
sXtx L RRoc?
RRoc?
Example
kTttx
k
0
R e 0s?
11 sTXs e
j
pole-zero plot
T
j?2
T
j?2?
27
sXtx L
Chapter 9 The Laplace Transform
§ 9.5.3 Shifting in s-Domain
0L0 ssXetx ts
RRoc?
0Re sRR o c
ROC的边界平移
j
2r
21 Re rsr
1r
j
0201 ReReRe srssr
01 Re sr02 Re sr?
28
Chapter 9 The Laplace Transform
2
0
2
L
0c o s s
stut 0Re?s
2
0
2
0L
0s in?
stut
0Re?s
tute at 0co s
2
0
2
L
as
as asRe
tute at 0s i n
2
0
2
0L
as
asRe
29
Chapter 9 The Laplace Transform
§ 9.5.4 Time Scaling
sXtx L RRoc?
asXaatx /1L aRR oc?
sXtx L
RR oc
When 1a
30
Chapter 9 The Laplace Transform
122 se t
1Re1 s
1?
j
1?
442 2 se t
2Re2 s 2?
j
2?
4/112 2
1
se
t
21Re21 s 2
1?
j
2
1
31
Chapter 9 The Laplace Transform
§ 9.5.5 Conjugation
sXtx L RRoc?
sXtx L RRoc?
txtx sXsX
32
Chapter 9 The Laplace Transform
§ 9.5.6 Convolution Property
sXtx L 11 1RR o c?
sXtx L 22 2RR o c?
sXsXtxtx L 2121 21 RRR o c
2Re 211 ssssX
1Re 122 ssssX
121?sXsXsRettxtx 21
33
Chapter 9 The Laplace Transform
Example
213221 txtxtuetxtuetx tt
不存在傅立叶变换
5151 3221 tuetuetxtx tt
34
Chapter 9 The Laplace Transform
§ 9.5.7 Differentiation in the Time Domain
sXtx L RRoc?
RRocssX
dt
tdx L
1
0 t
tx
2 4 6 8
Example
DeterminesX
2
12 22
12 R e 0
1
ss
s
eeX s X s X s s
se
35
§ 9.5.8 Differentiation in the s-Domain
Chapter 9 The Laplace Transform
sXtx L RRoc?
RRoc
ds
sdXttx L
21 astute Lat asRe
32 121 astuet Lat asRe
36
Chapter 9 The Laplace Transform
more generally,
11!1 nLatn astuetn asRe
11!1 nLatn astuetn asRe
37
Chapter 9 The Laplace Transform
Example
1Re 21 12 sss esX s
Determinetx
Solution:
1111 1211
2
tuetuetuet
tuetuetutetx
ttt
ttt
38
Chapter 9 The Laplace Transform
Example
11 tuettx at
DeterminesX
s assX ln
as,0m axRe
39
Chapter 9 The Laplace Transform
§ 9.5.9 Integration in the Time Domain
sXtx L RRoc?
sXsdxt 1L 0Re sRR o c?
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 11 sssX
的积分不存在拉氏变换tx
0?
j
1?
40
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 ssdxt 1Re21 1L
0?
j
1?2?
s s-L 2Re21
s s-s 2Re02
1L
41
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,
ssXx s l i m0 为真分式sX
321 122 sss sssX
ssXx s l i m0 1
321
12l im 2?
sss
sss
s
42
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
ssXtxx st 0limlim
asassX Re 1
0?a① 0limlim
0 ssXtx st
0?a② 11limlim 0 sstx st
0?a③ 终值不存在。
43
11 R e ln1 sTX s s aa e 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 k T?
Example 1
DeterminesX
j
aT ln1
Example 2
1 R e 01 sX s sse
Determinetx
k0
1 kx t u t - k
44
Chapter 9 The Laplace Transform
Example 3
2
2
2 R e 0
1
X s s
s
Determinetx
00 22
0
s i n R e 0Lt u t ss
s i n c o sx t t t t u t
45
Chapter 9 The Laplace Transform
§ 9.7 Analysis and Characterization of LTI Systems
Using the Laplace Transform
tyth
sHsY
tx
sX
thtxty
sHsXsY?
sH —— System Function or Transfer Function
46
Chapter 9 The Laplace Transform
For a system with a rational system function,
causal
m a xRe sR O C
§ 9.7.2 Stability (稳定性）
stable a x i sjωR O C
§ 9.7.1 Causality
Causal
m a xRe sR O C
47
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 cs
anticausal,unstable system
（反因果）
48
系统因果、稳定
Chapter 9 The Laplace Transform
sH 的极点均在 轴左侧，
且
j
m a xRes
如果 为有理函数sH
Stability of Causal System
Consider the following causal systems
11 a ssH
—— Stable
21 1 b sssH —— unstable
49
Chapter 9 The Laplace Transform
0 2 c 22 2 n
nn
n
sssH
21
2
cscssH
n
121 nnc
122 nnc
0,1 21 cc?① Stable system
10② 0Re,Re 21 cc Stable system
01 δ-③ 0Re,Re 21 cc Unstable system
1δ④ 0,21 cc Unstable system
50
Chapter 9 The Laplace Transform
§ 9.7.3 LTI Systems Characterized by Linear Constant-Coefficient
Differential Equations
txty
dt
tdy 3
k
kM
k
kk
kN
k
k dt
txdb
dt
tyda
00
ROC
k
k
N
k
k
k
M
k
sa
sb
0
0
sX sYsH?
51
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
yt
32y t y t y t x t
Determine the unit step response of the system.
21122 tts t e e u t
52
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
53
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 R e 112sH s s - ss
3 2 3y t y t y t x t x t
54
Chapter 9 The Laplace Transform
Example Consider a causal LTI system,
tbutueth
dt
tdh t 42,2
t-etyt-etx tt 61,1 22
b—— unknown constant
Determine the system function and b.sH
2 R e 04H s sss
55
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
40hsH
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
40hsH
42 4 ss ssH 2Res
56
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
57
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。
58
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
tfty
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？
tfty
tf
59
Chapter 9 The Laplace Transform
§ 9.8 System Function Algebra and Block Diagram Representations
（方框图）
tyth
sHsY
tx
sX
thtxty
sHsXsY?
§ 9.8.1 System Functions for Interconnections of LTI Systems
1,Series interconnection
2,Parallel interconnection
3,Feedback interconnection
60
Chapter 9 The Laplace Transform
Example 9.28
Consider the causal LTI system
31 ssH
sYsX +
-
s/1
3
sYsX +
s/1
3?
+
Example 9.29
Consider the causal LTI system
32 sssH
231 sssH
61
Chapter 9 The Laplace Transform
Example 9.30
Consider the causal LTI system
2312 sssH
Example 9.31
Consider the causal LTI system
23 642 2 2 ss sssH
(a) direct form
(b) cascade form
(c) parallel form
62
Chapter 9 The Laplace Transform
§ 9.8.2 LTI 系统的信号流图表示一 定义节点,表示信号和变量。
支路,表示节点间信号的传输路径和方向。
sH
sYsX
sYsXsH
支路增益,转移函数 。?sH
源点,只有输出支路的节点。
阱点,只有输入支路的节点。
前向通路,从源点到阱点沿箭头方向连通的路径
（每个节点只通过一次）。
环路,起点即终点的通路（每个节点只通过一次）。
63
Chapter 9 The Laplace Transform
通路增益,通路中所有支路增益的乘积。
sH1
sH2
sYsX +
-
系统的方框图表示
sYsX 11
sH2?
sH1
系统的信号流图表示环路增益,环路中所有支路增益的乘积。
源点 阱点前向通路环路
64
Chapter 9 The Laplace Transform
二 信号流图的基本性质
1,信号只能沿箭头方向流动。
2,节点信号为连接到接点的所有输入支路信号的叠加，
并传输到每一输出支路。
3,对于给定系统，信号流图表示不唯一。
4,如果信号流图倒置，系统的转移函数不变。
1H
2H
4H
5H
1x
2x
3x
4x
5x
22113 xHxHx
344 xHx?
355 xHx?
65
Chapter 9 The Laplace Transform
三 Mason 增益公式（源点到阱点的增益）
kk
k
gH ov1
△ —— 信号流图的特征行列式
△ =1 -（所有不同的环路增益之和）
+（每两个互不接触的环路增益的乘积之和）
-（每三个互不接触的环路增益的乘积之和）
+（每四个互不接触的环路增益的乘积之和）
66
Chapter 9 The Laplace Transform
k—— 前向通路的序号
kk
k
gH ov1
gk—— 第 k条前向通路的增益
△ k—— 去掉第 k条前向通路后,余下的子流图的特征行列式
G2
sXsY
G1 G3
-H
例 1,试求系统的转移函数
1L
321 GHGL
11 L 3211 GGGg? 11
32
321
11 1
1
GHG
GGGgH
ov
67
Chapter 9 The Laplace Transform
例 2,g
sX
sY
a b
c
d e
f
1L 2L
bcL?1 efL?2
211 LL 21LL?
gedbag 1? 11 b c e fefbc
a b d g egH
ov 1
1
11
1L 2L
bcL?1 efL?2
211 LL
gebag 1? 1
1 efbc
a b e ggH
ov 1
1
11
sXsYa b
c
e
f
g例 3：
68
Chapter 9 The Laplace Transform
例 4：
sXsY
a
b
c
d
e
f
h
i
j
k
g l
1L 2L
3L 4L
bcL?1 deL?2
hiL?3 jkL?4
43211 LLLL 31( LL? 41LL? 32LL? )42LL?
a b d fg?1431 1 LL
g h jlg?2212 1 LL
22111 ggH ov
69
Chapter 9 The Laplace Transform
例 5：
sXsY
a b
c
d
e f h
i
g
1L
2L
bcL?1
fgL?2
211 LL 21LL?
a b dg?1 21 1 L
efhg?2 12 1 L
a ihg?3 13
3322111 gggH ov
70
Example 9.31
222 4 632ssHs ss
sX
s/1 s/1
2?3?
2
4
sY
6?
Chapter 9 The Laplace Transform
1L 2L
71
Chapter 9 The Laplace Transform
§ 9.8.3 系统的模拟
1,加法器
2,标量乘法器
3,积分器
sX1
sX2
sXsX 21?
一 基本的模拟单元
1
1
sX1
sX2
sXsX 21?
sX
a
saXasXsaX
1/ssXsX
s
1sX
s
1
sXs1
72
Chapter 9 The Laplace Transform
二 方框图模拟
Example 9.29
Consider the causal LTI system
32 sssH
txtxtyty 23
dydxtxty tt 32
tw
ty
s/1
3?
s/1
2
tw
tx
S1 S2
73
Chapter 9 The Laplace Transform
交换 S1和 S2的连接顺序
ty
s/1
3?
s/1
2
tx
S1S2
输入相同输出相同系统等价为：
ty
3?
s/1
2
tx
74
Chapter 9 The Laplace Transform
三 信号流图模拟两个基本约定：
1,假定所有的环路均相互接触；
i
i
L 1
2,假定每一前向通路与所有的环路相互接触；
1k
i
i
k
k
kk
k L
g
gH ov
11
75
Example 9.29
Consider the causal LTI system
32 sssH
Chapter 9 The Laplace Transform
sssH /31 /21
1L
sX
1 1/s
-3
sY
2
1
公共点
ty
3?
s/1
2
tx
76
Chapter 9 The Laplace Transform
Example 9.31
Consider the causal LTI system
23 642 2 2 ss sssH
22/2/31 /6/42 ss sssH
sY
-61/s1/s
sX
1
公共点
-3 -2
4
2
77
(a) direct form
(b) cascade form
(c) parallel form
Example Consider the causal LTI system
531 42 sss sssH
Chapter 9 The Laplace Transform
15239 8623 2 sss sssH
78
Chapter 9 The Laplace Transform
§ 9.9 The Unilateral Laplace Transform
（单边拉氏变换）
Defining
dtetxs st 0 IX
tutxs L?IX
m a xRes
0 u n k o w n
0 X
2
1
I
t
tdses
jtx
stj
j
If is causal,tx
ssX IX?
79
Chapter 9 The Laplace Transform
Example 9.33 tuetx at
Example 9.33 tuetuttx t
12?
Example 1?tx
Example tetx t,2
80
Chapter 9 The Laplace Transform
X I 232 sss
Example 9.36 Consider the unilateral transform
2Res
X I 212 sss 2Res
tuettutx t21 2 0tf o r
§ 9.9.2 Properties of the Unilateral Laplace Transform
ssX IX?
Causal Signals:
81
Chapter 9 The Laplace Transform
1,Differentiation in the time-domain
RR O C X I stx IL
RR O C 0X
I
xss
dt
tdx IL
000 121 nnnn
n
n
xsxsssdt txd x?IXIL
Example Consider the signal
determine the unilateral Laplace Transform of
tuetutx at
dt
tdx
82
Chapter 9 The Laplace Transform
0xss
dt
tdx
IX
IL
① The Initial-Value Theorem
ssXx Is lim0
② The Final-Value Theorem
ssXtxx Ist 0l iml im
2,Integration in the time-domain
RR O C X I stx IL
0sReRR O C 01X1 1I t-xsssdx IL
83
Chapter 9 The Laplace Transform
§ 9.9.3 Solving Differential Equations
Using the Unilateral Laplace Transform
时域解经典解法零输入、零状态解法频域解 jHjXjYjYty -
f 1F
tytyty fx
tytyty pc
复频域解双边拉氏变换初始状态为零
sHsXsY?
sYty -f 1L
单边拉氏变换初始状态不为零tytyty fx,,?
微分方程的求解
84
Chapter 9 The Laplace Transform
Example 9.38
Suppose a causal LTI system
with initial conditions:
Let the input to this system be,
Determine the full response of the system,
txtytyty 23
0,0 yy
tutx
23 /23 0300 22 ss sss yysysY I?
ty xILty fIL
5030,2 γy,βy -- L e t?
0t tt eety 231Full response
85
Homework:
9.2 9.5 9.7 9.8 9.9
9.13 9.21(a,b,i,j)
9.22(a,b,c,d) 9.28
9.31 9.32 9.33 9.35 9.45
86
1 2 1 2 1 2 La x t b x t a X s b X s R o c R R
0L0 stx t t X s e
Lx t X s R oc R?
R oc R?
L x t X s R o c R
0 L 00 R estx t e X s s R o c R s
Chapter 9 The Laplace Transform
87
Chapter 9 The Laplace Transform
L 1 / x a t X s a R o c a Ra
L x t X s R o c R
1 2 1 2 1 2 Lx t x t X s X s R o c R R
L d x t s X s R o c R
dt
L d X st x t R o c Rds
L 1 R e 0t x d X s R o c R ss
88
Chapter 9 The Laplace Transform
Causal
m a xRe sR O C
For a system with a rational system function,
causal
m a xRe sR O C
stable a x i sjωR O C
89
Chapter 9 The Laplace Transform
stX s x t e d t
tX s F x t e d t
0 R O CsjX j X s
1,The direction of signals
2,The position of poles ROC of X(s)
90
Chapter 9 The Laplace Transform
stu L 1 0?sRe
1 LtsRe1.
stu L 1 0?sRe2.
astue Lat 1 asRe
astue Lat 1 asRe3.
