1 Signals and Systems Lecture 25 The Laplace Transform ROC of Laplace Transform Inverse Laplace Transform

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


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


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:


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


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)(


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


9


§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


§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


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


§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


§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


§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


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


§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


§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


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


§9.5.4 Time Scaling

sXtx L RRoc

asXa

atx /1L aRRoc

sXtx L

RRoc

When 1a

20


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


§9.5.5 Conjugation

sXtx L RRoc

sXtx L RRoc

txtx sXsX

22


§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


Example ? 213

22

1 txtxtuetxtuetx tt

不存在傅立叶变换

5

1

5

1 3221 tuetuetxtx tt

24


§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


sXtx L RRoc

RRoc ds

sdXttx L

2

1

astute Lat

as Re

3

2 1

2

1

astuet Lat

as Re

26


more generally,

1

1

!

1

n

Latn

astuet

n as Re

1

1

!

1

n

Latn

astuet

n as Re

27


Example

1Re 21

12

sss

esX

s

Determine tx

Solution:

1111 1211

2

tuetuetuet

tuetuetutetxttt

ttt

28


Example 11

tuet

tx at

Determine sX

s

assX

ln

as ,0maxRe

29


§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


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


§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


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


§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


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


§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


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


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


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


Causal maxRe sROC

For a system with a rational system function,

causal maxRe sROC

stable axisjω ROC

40


§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


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


Example 9.24

Consider a RLC

circuit in Figure 9.27+

R L

C

-

+ ty

tx

-

Figure 9.27

LCsLRs

LCsH

/1/

/12

43


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


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


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


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


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


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

Documents

1 Signals and Systems Lecture 25 The Laplace Transform ROC of Laplace Transform Inverse Laplace Transform