1. Signals and Systems Lecture 24
The Laplace Transform
ROC of Laplace Transform
Inverse Laplace Transform

2. Chapter 9
The Laplace Transform

3. Chapter 9 The Laplace Transform
Defining
——Laplace Transform
1. The relationship

4. Chapter 9 The Laplace Transform
2. Region of Convergence（收敛域）
Dirichlet Condition 1 :
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。

5. Chapter 9 The Laplace Transform
Example 9.1
pole-zero plot
零极点图
Example 9.2
pole-zero plot

6. Chapter 9 The Laplace Transform
Particularly,
The Fourier transform of does not exist.

7. Chapter 9 The Laplace Transform
Example 9.3

8. Chapter 9 The Laplace Transform
Example 9.3

9. Chapter 9 The Laplace Transform
Example 9.4
entire S plane
pole-zero plot
does not exist.

10. Chapter 9 The Laplace Transform
Zeros:
Poles:
1. The direction of signals
ROC of X(s)
2. The position of poles
§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
——Depends only on σ

11. 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.
① when

12. Chapter 9 The Laplace Transform

13. Chapter 9 The Laplace Transform
Example
零极点 抵消
pole-zero plot
pole:
zero:
zeros:

14. Chapter 9 The Laplace Transform
Property 4:
If is right sided,
finite

15. Chapter 9 The Laplace Transform
Property 5:
If is left sided,
Property 6:
If is two sided,
ROC:

16. Chapter 9 The Laplace Transform
Example 9.7
If b>0,
If b≤0,
has no Laplace transform.

17. Chapter 9 The Laplace Transform
Property 7:
If the Laplace transform of is rational,
① is right sided,
② is left sided,
Example 9.8
left sided
right sided
two sided

18. Chapter 9 The Laplace Transform
Basic Laplace Pairs
Poles
ROC
none

19. Chapter 9 The Laplace Transform
1. The direction of signals
2. The position of poles
ROC of X(s)

20. Chapter 9 The Laplace Transform1. 2. 3.

21. Chapter 9 The Laplace Transform
§9.3 The Inverse Laplace Transform
defining
Example 9.9
Determine the inverse Laplace transform for all possible ROC.

22. Summary
The Laplace Transform and its RoC
Inverse Laplace Transform

23. Problem Set
P (a) (b)
P (a) (b)