信号与系统第九章课件

1、1,Chapter 9 The Laplace Transform,2,9.1 The Laplace Transform,Defining,Laplace Transform,When ,1. The relationship,3,2. Region of Convergence（收敛域）,Dirichlet Condition 1 :,ROC：对给定的 ，使其拉氏变换存在的 对应的 S平面上的区域。,4,Example 9.1,pole-zero plot 零极点图,5,Example 9.2,6,Particularly,The Fourier transform of does not

2、 exist.,7,Example 9.3,8,Example 9.3,9,Example 9.4,entire S plane,does not exist.,10,1. 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 j-axis in the s-plane,Depends only on ,11,Property 2: For rational Laplace

3、 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., when,12,13,Example,pole:,zero:,zeros:,14,15,ROC:,16,Example 9.7,If b0,If b0,has no Laplace transform.,17,Example 9.8,left sided,two sided,right sid

4、ed,18,none,19,9.3 The Inverse Laplace Transform,defining,20,Solution:,21,22,9.4 Geometric evaluation of the Fourier transform 几何求值from the Pole-Zero plot,Pole vector:,Zero vector:,23,Example 9.12,24,9.4.1 First-Order System,time constant (时间常数） controls the speed of response of first-order systems,2

5、5,9.4.2 Second-Order System,26,27,9.4.3 All-Pass Systems （全通系统）,First-Order System,零极点相对于j轴对称,全通系统：零极点个数相同，且相对于j轴对称。,28,9.5 Properties of the Laplace Transform,9.5.1 Linearity of the Laplace Transform,29,Example 9.13,30,9.5.2 Time Shifting,Example,Poles:,31,9.5.3 Shifting in s-Domain,32,When,33,34,9

6、.5.4 Time Scaling,35,36,When,9.5.5 Conjugation,37,9.5.6 Convolution Property,38,Example,39,9.5.7 Differentiation in the Time Domain,40,41,42,9.5.8 Differentiation in the s-Domain,43,more generally,44,Solution:,45,Poles:,46,9.5.9 Integration in the Time Domain,ROC的变化：, 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

7、,47, R与 部分重叠。, R与 部分重叠。,48,9.5.10 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,49,50,2. The Final-Value Theorem,的极点均在j轴左侧，允许在s=0有一个一阶极点,终值不存在。,51,9.5.11 运用基本性质求解拉氏变换,52,53,S域微分,时域积分,54,9.7 Analysis

8、 and Characterization of LTI Systems Using the Laplace Transform,System Function or Transfer Function,9.7.1 Causality,55,For a system with a rational system function,9.7.2 Stability (稳定性）,Stable system:,when,converges,56,Example 9.20,Causal , unstable system,noncausal , stable system,anticausal , un

9、stable system （反因果）,57,如果 为有理函数,Stability of Causal System,Consider the following causal systems,Stable,unstable,58,Stable system,Stable system,Unstable system,Unstable system,59,高阶因果系统的稳定性,稳定的必要条件：ai 均不为零，且同号。,任意的ai 为零，或符号不同，系统均不稳定。,下列因果系统：,不稳定,不稳定,可能稳定,60,9.7.3 LTI Systems Characterized by Linear

10、Constant-Coefficient Differential Equations, Causal, Anticausal,Stable System.,Unstable System.,61,Linear Constant-Coefficients Differential Equations,ROC,A ratio of polynomials in s,62,系统稳定,63, causal, stable,64,Solution,Causal,is the eigenfunction of system.,65,The system is unstable.,66,Causal,Ex

11、ample 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,67,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,68,1. 的傅立叶变换收敛。,2.,69,3. 为一因果稳定系统的单位冲激响应。,4. 至少有一个极点

12、。,5. 为有限长度信号。,ROC不变,70,6.,在s=-2处有极点,7.,无法判断正确与否。,71,72,9.8 System Function Algebra and Block Diagram Representations （方框图）,9.8.1 System Functions for Interconnections of LTI Systems,1. Series interconnection,73,Parallel interconnection,Feedback interconnection,74,Example 9.28 Consider the causal LTI

13、 system,75,Example 9.29 Consider the causal LTI system,76,Example 9.30 Consider the causal LTI system,77,Example 9.31 Consider the causal LTI system,78,9.8.2 LTI 系统的信号流图表示,一 定义,节点：表示信号和变量。,支路：表示节点间信号的传输路径和方向。,支路增益：转移函数 。,源点：只有输出支路的节点。,阱点：只有输入支路的节点。,前向通路：从源点到阱点沿箭头方向连通的路径 （每个接点只通过一次）。,环路：起点即终点的通路（每个接点

14、只通过一次）。,79,通路增益：通路中所有支路增益的乘积。,环路增益：环路中所有支路增益的乘积。,80,二 信号流图的基本性质,信号只能沿箭头方向流动。,节点信号为连接到接点的所有输入支路信号的叠加， 并传输到每一输出支路。,3. 对于给定系统，信号流图表示不唯一。,4. 如果信号流图倒置，系统的转移函数不变。,81,三 Mason 增益公式（源点到阱点的增益）,信号流图的特征行列式,=1 -（所有不同的环路增益之和）,+（每两个互不接触的环路增益的乘积之和）,-（每三个互不接触的环路增益的乘积之和）,+（每四个互不接触的环路增益的乘积之和）,82,k前向通路的序号,gk第k条前向通路的增益,

15、k去掉第k条前向通路后,余下的子流图的特征行列式,83,84,85,86,9.8.3 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,87,二 方框图模拟,Example 9.29 Consider the causal LTI system,88,交换S1和S2的连接顺序,输入相同,输出相同,系统等价为：,89,三 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；,90,Example 9.29 Consider the causal LTI system,91,Example 9.31 Conside

16、r the causal LTI system,92,Example Consider the causal LTI system,1. 直接模拟,93,2. 级联型模拟,94,3. 并联型模拟,95,9.9 The Unilateral Laplace Transform （单边拉氏变换）,Defining,If is causal,96,Example 9.33,97,Example 9.33,Example,It does not exist bilateral Laplace Transform.,Example,It does not exist bilateral Laplace

17、Transform.,98,Example 9.36 Consider the unilateral transform,9.9.2 Properties of the Unilateral Laplace Transform,Causal Signals:,99,1. Differentiation in the time-domain,100,Example Consider the signal determine the unilateral Laplace Transform of,Solution 1,Solution 2,101, The Initial-Value Theorem, The Final-Value Theorem,102,2. Integrat