2

3

How Do We Perform Inverse Laplace Transform?

• In 6.003, we will only deal with Laplace transform that are:

1) Rational, i.e. X(s) = N(s)/D(s) ;

And/or:

2) exponential, i.e. X(s) = e-sT .

•For case 2), use shift property

•For case 1), use PFE.

•For case 1) & 2), i.e.

x(t!T)" # $ e!sT X(s) (Similar to the FT property

x(t!T)" # $ e!j

%

TX(j

%

)

X(s)=e!sT N(s) / D(s)

( )

X1(s)"x1(t)

1 2 4 4 3 4 4

Then b

x(t)=x1(t)t!t"T=x1(t"T)

4

Inverse Laplace Transforms Via Partial Fraction

Expansion and Properties

Example:

Three possible ROC’s — corresponding to three different signals

Recall

X(s)=s+3

s+1

( )

s!2

( )

=A

s+1+B

s!2

A=!2

3,B=5

3

1

s+a, Re(s)<!a" # $ !e!at u(!t) left - sided

and 1

s+a, Re(s)>!a" # $ e!at u(t) right - sided

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3

5

Example (cont.)

ROC I: — Left-sided signal.

x(t) =

ROC II: — Two-sided signal, has Fourier Transform.

x(t) =

ROC III: — Right-sided signal.

x(t) =

!Ae!tu(!t)!Be 2tu(!t)

!2

3

e!tu(t)+5

3

e2tu(!t)

"

#

$ %

&

' (0 as t(±)

Ae!tu(t)+Be 2tu(t)

=2

3

e!t!5

3

e2t"

#

$ %

&

' u(!t) Diverges as t( ! )

=!2

3

e!t+5

3

e2t"

#

$ %

&

' u(t) Diverges as t(+)

6

Properties of Laplace Transforms

•Many parallel properties of the CTFT, but for Laplace

transforms we need to determine implications for the ROC

•For example:

Linearity

ROC at least the intersection of ROC’s of X1(s) and X2(s)

ROC can be bigger (due to pole-zero cancellation)

E.g. x1(t) = x2(t) and a = -b

Then a x1(t) + b x2(t) = 0 → X(s) = 0

⇒ ROC entire s-plane

ax1(t)+bx2(t)! " # aX1(s)+bX2(s)

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##### Document information

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pakhi

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Address:
Computer science

University:
Birla Institute of Technology and Science

Subject:
Signals and Systems

Upload date:
18/07/2012