Signals and Systems

Chapter 4 the Laplace Transform

4.4 The Inverse Laplace Transform

Review

desXsX tj)(

2

1)(1-F

Multiplying both sides by te

desXtx ts)(

2

1)(

s =σ+ jω ， ds =jdω

j

j

ts dsesXj

tx

)(

2

1)(

tetx )(

ttjt etxFdteetxsX

)()()(

The Inverse Laplace Transform:

This equation states that x(t) can be represented as a weighted integral of complex exponentials.The contour of integration is the straight line in the s-plane corresponding to all points s satisfying Re(s)=

j

j

ts dsesXj

tx

)(

2

1)(

The formal evaluation of the integral for a general X(s) requires the use of contour integration (围线积分 ) in the complex plane,a topic that we will not consider here.The inverse Laplace transform can be determined by partial-fraction expansion.

For example: )2)(1(

1

sssX determine x(t).

Firstly, perform a partial-fraction expansion to obtain:

.21)2)(1(

1

s

B

s

A

sssX

Secondly, evaluate the coefficients:

.1

,1

B

A

thus ,the partial-fraction expansion for X(s) is :

.2

1

1

1

sssX

2Re2

1)(2

s

stue Lt

1Re1

1)(

s

stue Lt

So , )()()( 2 tuetuetx tt

Partial-fraction ExpansionThe procedure consists of expanding the rational algebraic expression into a linear combination of lower order terms.

011

1

011

1

)(

)()(

bsbsbsb

asasasa

sB

sAsX n

nn

n

mm

mm

rationalproper is , if sXnm

)())((

)())((

)(

)()(

21

21

nn

mm

pspspsb

zszszsa

sB

sAsX

sXpppp n ofpoles,,, 321

sXzzzz m ofzeros,,, 321

m

i i

i

as

AsX

1

)(

Assuming no multiple-order poles and that the order of the denominator polynomial is greater than the order of the numerator polynomial,we can expand X(s) in the form:

ROC σ> -ai (right sided signal) σ< -ai (left sided signal)

)()( tueAtx taii

i

)()( tueAtx taii

i

Adding the inverse transforms of the individual terms ,then yields the inverse transform of X(s).

Discussing 1 ： no multiple-order poles,the first order

,.....,, 321 npppp

n

n

ps

A

ps

A

ps

AsX

2

2

1

1)(

)()()()( 112

211 ps

ps

Aps

ps

AAsXps

n

n

1)()( 11 pssXpsA

iii pssXpsA )()(

different poles,real numbers or complex numbers

tpn

tptptp neAeAeAeAtx ....)( 321321

321

321

s

A

s

A

s

AsX

11 )()1(

s

sXsA 1)3)(2)(1(

332)1(

1

2

ssss

sss

22 )()2(

s

sXsA 5)3)(2)(1(

332)2(

2

2

ssss

sss

33 )()3(

s

sXsA 6)3)(2)(1(

332)3(

3

2

ssss

sss

For example: )3)(2)(1(

332 2

sss

sssX

3

6

2

5

1

1)(

sss

sX

)2j1)(2j1)(2(

32

sss

ssX

2j12j1221

s

B

s

B

s

A

5

7)2(

2

ssXsA

5

2j1

)2j1)(2j1)(2(

3)2j1(

2j1

2

1

s

sss

ssB

)52)(2(

3)( 2

2

sss

ssX 1

)( 2sin5

22cos

5

1e2 )(e

5

7 2 tutttutx tt

For example:

Homework

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