Signals and Systems Lecture 25

1. Signals and SystemsLecture 25 The Laplace Transform ROC of Laplace Transform Inverse Laplace Transform

2. Chapter 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.

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

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

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

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

7. Chapter 9 The Laplace Transform 9.3 The Inverse Laplace Transform So

8. Chapter 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 9.9 9.10 9.11

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

10. Chapter 9 The Laplace Transform Pole vector: Zero vector: §9.4 Geometric evaluation of the Fourier transform 几何求值from the Pole-Zero plot

11. Chapter 9 The Laplace Transform Example 9.12 §9.4.1 First-Order System τ——time constant (时间常数） controls the speed of response of first-order systems

12. Chapter 9 The Laplace Transform §9.4.2 Second-Order System

13. Chapter 9 The Laplace Transform §9.4.3 All-Pass Systems （全通系统） First-Order System 零极点相对于jω轴对称 全通系统：零极点个数相同，且相对于jω轴对称。

14. Chapter 9 The Laplace Transform §9.5 Properties of the Laplace Transform §9.5.1 Linearity of the Laplace Transform

15. Chapter 9 The Laplace Transform Example 9.13

16. Chapter 9 The Laplace Transform Example pole-zero plot §9.5.2 Time Shifting

17. Chapter 9 The Laplace Transform ROC的边界平移 §9.5.3 Shifting in s-Domain

18. Chapter 9 The Laplace Transform

19. Chapter 9 The Laplace Transform When §9.5.4 Time Scaling

20. Chapter 9 The Laplace Transform

21. Chapter 9 The Laplace Transform §9.5.5 Conjugation

22. Chapter 9 The Laplace Transform §9.5.6 Convolution Property

23. Chapter 9 The Laplace Transform Example 不存在傅立叶变换

24. Chapter 9 The Laplace Transform Example Determine §9.5.7 Differentiation in the Time Domain

25. Chapter 9 The Laplace Transform §9.5.8 Differentiation in the s-Domain

26. Chapter 9 The Laplace Transform more generally,

27. Chapter 9 The Laplace Transform Example Determine Solution:

28. Chapter 9 The Laplace Transform Example Determine

29. Chapter 9 The Laplace Transform ① R与 无公共部分，积分的拉氏变换不存在。 的积分不存在拉氏变换 §9.5.9 Integration in the Time Domain ROC的变化：

30. Chapter 9 The Laplace Transform ② R与 部分重叠。 ③ R与 部分重叠。

31. Chapter 9 The Laplace Transform 为真分式 §9.5.10 The Initial- and Final-Value Theorems 初值定理和终值定理 1. The Initial-Value Theorem Contains no impulses or higher order singularities at the origin.

32. Chapter 9 The Laplace Transform 的极点均在jω轴左侧，允许在s=0有一个一阶极点 2. The Final-Value Theorem 终值不存在。

33. Chapter 9 The Laplace Transform Example 1 Determine Example 2 Determine §9.5.11 运用基本性质求解拉氏变换

34. Chapter 9 The Laplace Transform Determine Example 3

35. Chapter 9 The Laplace Transform ——System Function or Transfer Function §9.7 Analysis and Characterization of LTI Systems Using the Laplace Transform

36. Chapter 9 The Laplace Transform causal stable Causal §9.7.1 Causality For a system with a rational system function, §9.7.2 Stability (稳定性）

37. Chapter 9 The Laplace Transform Example 9.20 Causal , unstable system noncausal , stable system anticausal , unstable system （反因果）

38. Chapter 9 The Laplace Transform 如果 为有理函数 的极点均在 轴左侧， 且 系统因果、稳定 Stability of Causal System Consider the following causal systems ——Stable ——unstable

39. Chapter 9 The Laplace Transform causal For a system with a rational system function, stable Causal

40. Chapter 9 The Laplace Transform ROC §9.7.3 LTI Systems Characterized by Linear Constant-Coefficient Differential Equations

41. Chapter 9 The Laplace Transform Example Consider a causal LTI system whose input and output related through an linear constant-coefficient differential equation of the form Determine the unit step response of the system.

42. Chapter 9 The Laplace Transform Example 9.24 Consider a RLC circuit in Figure 9.27 R L + C + Figure 9.27 - -

43. Chapter 9 The Laplace Transform Example 9.25 • Consider an LTI system with input , • Output . • Determine the system function. • Justify the properties of the system. • Determine the differential equation of the system.

44. Chapter 9 The Laplace Transform Example Consider a causal LTI system , b——unknown constant Determine the system function and b.

45. Chapter 9 The Laplace Transform Example 9.26 An LTI system: 1. The system is causal. 2. is rational and has only two poles: s= - 2 and s=4. 3. 4. Determine Example 9.26 An LTI system: 1. The system is causal. 2. is rational and has only two poles: s=-2 and s=-4. 3. 4. Determine

46. Chapter 9 The Laplace Transform 已知一因果稳定系统， 为有理函数，有一极点 Example 9.27 在s=-2处，原点（s=0）处没有零点，其余零极点未知， 判断下列说法是否正确。 1. 的傅立叶变换收敛。 3. 为一因果稳定系统的单位冲激响应。 4. 至少有一个极点。 2. 5. 为有限长度信号。

47. Chapter 9 The Laplace Transform 7. 6. 在s=+2处有极点 在s=-2处有极点 无法判断正确与否。

48. Chapter 9 The Laplace Transform 是系统函数为 例 设信号 的因果全通系统的输出。 1. 求出至少有两种可能的输入 都能产生 。 2. 若已知 问输入 是什么？ 3. 如果已知存在某个稳定（但不一定因果）的系统， 它若以 作输入，则输出为 ，问这个输入 是什么？系统的单位冲激响应是什么？

49. Problem Set • P728 9.28 • P729 9.31