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Signals and Systems Lecture 24

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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 Transform
1. 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)


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