1 Signals and Systems Lecture 25 The Laplace Transform ROC of Laplace Transform Inverse Laplace Transform

2 Appendix Partial Fraction Expansion Consider a fraction polynomial: Discuss two cases of D(s)=0, for distinct root and same root. Chapter 9 The Laplace Transform

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

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

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

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

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

8 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 Chapter 9 The Laplace Transform

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

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

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 §9.5.2 Time Shifting Example pole-zero plot

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

18 Chapter 9 The Laplace Transform

19 Chapter 9 The Laplace Transform §9.5.4 Time Scaling When

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 §9.5.7 Differentiation in the Time Domain Example Determine

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

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 §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

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 § 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

32 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。

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

34 Chapter 9 The Laplace Transform Example 3 Determine

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

36 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

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

38 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

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

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

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 RL C Figure 9.27

43 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.

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= Determine Example 9.26 An LTI system: 1. The system is causal. 2. is rational and has only two poles: s=-2 and s= Determine

46 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。 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。

47 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。

48 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？

49 Problem Set P P