1. (2) Region of Convergence ( ROC )
9 The Laplace Transform
9. The Laplace Transform
9.1 The Laplace Transform
(1) Definition
(2) Region of Convergence ( ROC )
ROC: Range of  for X(s) to converge
Representation:
A. Inequality
B. Region in S-plane

2. 9 The Laplace Transform
Example for ROC Re
S-plane Im -a

3. (3) Relationship between Fourier and Laplace transform
9 The Laplace Transform
(3) Relationship between Fourier and Laplace
transform
Example

4. 9.2 The Region of Convergence for Laplace Transform
9 The Laplace Transform
9.2 The Region of Convergence for Laplace Transform
Property1: The ROC of X(s) consists of strips parallel
to j-axis in the s-plane.
Property2: For rational Laplace transform, the ROC
does not contain any poles.
Property3: If x(t) is of finite duration and is absolutely
integrable, then the ROC is the entire s-
plane

5. Property4: If x(t) is right sided, and if the line Re{s}=0
9 The Laplace Transform
Property4: If x(t) is right sided, and if the line Re{s}=0
is in the ROC, then all values of s for which
Re{s}>0 will also in the ROC.

6. Property5: If x(t) is left sided, and if the line Re{s}=0
9 The Laplace Transform
Property5: If x(t) is left sided, and if the line Re{s}=0
is in the ROC, then all values of s for which
Re{s}<0 will also in the ROC.
x(t) T2 t
e-0t
e-1t

7. Property6: If x(t) is two sided, and if the line Re{s}=0
9 The Laplace Transform
Property6: If x(t) is two sided, and if the line Re{s}=0
is in the ROC, then the ROC will consist of
a strip in the s-plane that includes the line
Re{s}=0 .

8. 9 The Laplace Transform
S-plane Re Im R L

9. Property7: If the Laplace transform X(s) of x(t) is
rational, then its ROC is bounded by poles or extends to infinity. In addition, no
poles
of X(s) are contained in the ROC.
Property8: If the Laplace transform X(s) For rational Laplace transform, the ROC
does not contain any poles.
Property3: If x(t) is of finite duration and is absolutely
integrable, then the ROC is the entire s-
plane

10. Property7: If the Laplace transform X(s) of x(t) is
rational, then its ROC is bounded by
poles or extends to infinity. In addition,
no poles of X(s) are contained in the ROC.
Property8: If the Laplace transform X(s) of x(t) is
rational, then if x(t) is right sided, the ROC
is the region in the s-plane to the right of the
rightmost pole. If x(t) is left sided, the ROC
is the region in the s-plane to the left of the
leftmost pole.
Example

11. Consider a fraction polynomial:
9 The Laplace Transform
Appendix Partial Fraction Expansion
Consider a fraction polynomial:
Discuss two cases of D(s)=0, for distinct root
and same root.

12. 9 The Laplace Transform
(1) Distinct root:
thus

13. Multiply two sides by (s-1):
9 The Laplace Transform
Calculate A1 :
Multiply two sides by (s-1):
Let s=1, so
Generally

14. 9 The Laplace Transform
(2) Same root:
thus
For first order poles:

15. Multiply two sides by (s-1)r :
9 The Laplace Transform
For r-order poles:
Multiply two sides by (s-1)r : So

16. 9.3 The Inverse Laplace Transform
9 The Laplace Transform
9.3 The Inverse Laplace Transform So

17. The calculation for inverse Laplace transform:
9 The Laplace Transform
The calculation for inverse Laplace transform:
(1) Integration of complex function by equation.
(2) Compute by Fraction expansion.
General form of X(s):
Important transform pair:
Example