2. 1 Signals and Systems Lecture 26 Properties of Laplace Transform Analysis LTI System using LT System Function

3. 2 Chapter 9 The Laplace Transform §9.5.9 Integration in the Time Domain ROC 的变化： 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

4. 3 Chapter 9 The Laplace Transform ② R 与 部分重叠。 ③ R 与 部分重叠。

5. 4 Chapter 9 The Laplace Transform §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. 为真分式

6. 5 Chapter 9 The Laplace Transform 2. The Final-Value Theorem 的极点均在 jω 轴左侧，允许在 s=0 有一个一阶极点 终值不存在。

7. 6 Chapter 9 The Laplace Transform §9.5.11 运用基本性质求解拉氏变换 Example 1 Determine Example 2 Determine

8. 7 Chapter 9 The Laplace Transform Example 3 Determine

9. 8 Chapter 9 The Laplace Transform §9.7 Analysis and Characterization of LTI Systems Using the Laplace Transform ——System Function or Transfer Function

10. 9 Chapter 9 The Laplace Transform For a system with a rational system function, causal §9.7.2 Stability ( 稳定性） stable §9.7.1 Causality Causal

11. 10 Chapter 9 The Laplace Transform Example 9.20 Causal, unstable system noncausal, stable system anticausal, unstable system （反因果）

12. 11 系统因果、稳定 Chapter 9 The Laplace Transform 的极点均在 轴左侧， 且 如果 为有理函数 Stability of Causal System Consider the following causal systems ——Stable ——unstable

13. 12 Chapter 9 The Laplace Transform Causal For a system with a rational system function, causal stable

14. 13 Chapter 9 The Laplace Transform §9.7.3 LTI Systems Characterized by Linear Constant-Coefficient Differential Equations ROC

15. 14 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.

16. 15 Chapter 9 The Laplace Transform Example 9.24 Consider a RLC circuit in Figure 9.27 + RL C - + - Figure 9.27

17. 16 Chapter 9 The Laplace Transform Example 9.25 Consider an LTI system with input, Output. (a)Determine the system function. (b)Justify the properties of the system. (c)Determine the differential equation of the system.

18. 17 Chapter 9 The Laplace Transform Example Consider a causal LTI system, b——unknown constant Determine the system function and b.

19. 18 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

20. 19 Chapter 9 The Laplace Transform Example 9.27 已知一因果稳定系统， 为有理函数，有一极点 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。 2. 3. 为一因果稳定系统的单位冲激响应。 4. 至少有一个极点。 5. 为有限长度信号。

21. 20 Chapter 9 The Laplace Transform 6. 在 s=-2 处有极点在 s=+2 处有极点 7. 无法判断正确与否。

22. 21 Chapter 9 The Laplace Transform 例 设信号 是系统函数为 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？

23. 22 Chapter 9 The Laplace Transform §9.8 System Function Algebra and Block Diagram Representations （方框图） §9.8.1 System Functions for Interconnections of LTI Systems 1. Series interconnection 2.Parallel interconnection 3.Feedback interconnection

24. 23 Chapter 9 The Laplace Transform Example 9.28 Consider the causal LTI system + - + + Example 9.29 Consider the causal LTI system

25. 24 Chapter 9 The Laplace Transform Example 9.30 Consider the causal LTI system Example 9.31 Consider the causal LTI system (a) direct form (b) cascade form (c) parallel form

26. 25 Chapter 9 The Laplace Transform §9.8.3 系统的模拟 1. 加法器 2. 标量乘法器 3. 积分器 一 基本的模拟单元 a 1/s

27. 26 Chapter 9 The Laplace Transform 二 方框图模拟 Example 9.29 Consider the causal LTI system S1S1 S2S2

28. 27 Chapter 9 The Laplace Transform 交换 S 1 和 S 2 的连接顺序 S1S1 S2S2 输入相同 输出相同 系统等价为：

29. 28 Chapter 9 The Laplace Transform 三 信号流图模拟 两个基本约定： 1. 假定所有的环路均相互接触； 2. 假定每一前向通路与所有的环路相互接触；

30. 29 Example 9.29 Consider the causal LTI system Chapter 9 The Laplace Transform 11/s -3 2 1 公共点

31. 30 Chapter 9 The Laplace Transform Example 9.31 Consider the causal LTI system -61/s 1 公共点 -3 -2 4 2

32. 31 (a) direct form (b) cascade form (c) parallel form Example Consider the causal LTI system Chapter 9 The Laplace Transform

33. 32 Chapter 9 The Laplace Transform §9.9 The Unilateral Laplace Transform （单边拉氏变换） Defining If is causal,

34. 33 Chapter 9 The Laplace Transform Example 9.33 Example

35. 34 Chapter 9 The Laplace Transform Example 9.36 Consider the unilateral transform §9.9.2 Properties of the Unilateral Laplace Transform Causal Signals:

36. 35 Chapter 9 The Laplace Transform 1. Differentiation in the time-domain Example Consider the signal determine the unilateral Laplace Transform of

37. 36 Chapter 9 The Laplace Transform ① The Initial-Value Theorem ② The Final-Value Theorem

38. 37 Chapter 9 The Laplace Transform 2. Integration in the time-domain

39. 38 Chapter 9 The Laplace Transform §9.9.3 Solving Differential Equations Using the Unilateral Laplace Transform 时域解 经典解法 零输入、零状态解法 频域解 复频域解 双边拉氏变换 初始状态为零 单边拉氏变换 初始状态不为零

40. 39 Chapter 9 The Laplace Transform Example 9.38 Suppose a causal LTI system with initial conditions: Let the input to this system be, Determine the full response of the system. Full response

41. 40 Homework: P729 9.32 9.33