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Signals and Systems Lecture 24

Dive into the Laplace Transform theory with examples and properties including ROC, poles, zeros, and inverse transforms. Explore different ROC scenarios and Laplace pair relationships. Includes exercises for further practice.

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Signals and Systems Lecture 24

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  1. Signals and SystemsLecture 24 The Laplace Transform ROC of Laplace Transform Inverse Laplace Transform

  2. Chapter 9 The Laplace Transform

  3. Chapter 9 The Laplace Transform Defining §9.1 The Laplace Transform ——Laplace Transform 1. The relationship

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

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

  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 pole-zero plot does not exist. entire S plane

  10. Chapter 9 The Laplace Transform Zeros: Poles: ROC of X(s) 1. The direction of signals 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 3: If is of finite duration and is absolutely integrable, then the ROC is the entire s-plane. ① when Property 2: For rational Laplace transforms, the ROC does not contain any poles.

  12. Chapter 9 The Laplace Transform

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

  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 has no Laplace transform. Example 9.7 If b>0, If b≤0,

  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 ROC of X(s) 1. The direction of signals 2. The position of poles

  20. Chapter 9 The Laplace Transform 1. 2. 3.

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

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

  23. Problem Set • P724 9.21 (a) (b) • P725 9.22 (a) (b)

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