1、In chapter 4 and in chapter 5, we consider s=j and z= ej in the Fourier transform. Now , well consider s=+ j in the Laplace transform. And z= rej in the Z-transform.,本章 要求 ： 正确理解拉普拉斯变换及其定义式、熟练掌握由信号时域特点判断其拉氏变换收敛范围的定性分析方法（包括：有限持续期信号、左边信号、右边信号以及双边信号）。牢记常用典型信号的拉氏变换、正确理解拉氏变换的基本性质（特别注意单边拉氏变换和双边拉氏变化的区别、必须弄
2、清楚初值定理和终值定理的使用条件）。熟练掌握从基本变换对出发、灵活运用拉氏变换的性质求解信号拉氏变换的基本方法（包括正变换和反变换）、掌握采用部分分式展开法求解拉氏反变换的方法。熟练掌握运用拉氏变换分析LTI系统的方法。熟练掌握由系统函数H(s)判断系统因果性和稳定性的定性分析方法。对给定系统的数学模型或模拟框图或信号流图会求其H(s)和h(t)。了解由H(s)的零极点分布确定系统频响的几何方法，能熟练运用Mason公式化简信号流图、能应用拉氏变换分析具体电路。,9 The Laplace Transform,9. The Laplace Transform,9.1 The Laplace T
3、ransform,LTI,9 The Laplace Transform,(1) Definition,The Laplace transform of the signal x(t) is:,9 The Laplace Transform,Operator Lx(t) denotes the Laplace transform of x(t):,The transform relationship between x(t) and X(s):,9 The Laplace Transform,a. Especially, when s=j， above equation becomes The
4、 Fourier transform of signal x(t):,the relationship between the Fourier transformand the Laplace transform,9 The Laplace Transform,b. On the other hand,9 The Laplace Transform,Example 9.1 Compute the Laplace transform of the followingsignals.,(a),(b),9 The Laplace Transform,Solution: (a),(b),(2) Reg
5、ion of Convergence ( ROC ),ROC: Range of s for X(s) to convergeRepresentation: A. InequalityB. Region in S-plane,9 The Laplace Transform,9 The Laplace Transform,x(t)的拉普拉斯变换相当于x(t)e-t的傅利叶变换，在满足x(t)e-t绝对可积条件下的的取值范围称为拉普拉斯变换的收敛域。,9 The Laplace Transform,Example for ROC,9 The Laplace Transform,In s-plane
6、, the horizontal axis is Res or -axis; The vertical axis is Ims or j-axis.,9 The Laplace Transform,Example 9.2,Compute the Laplace transform of the followingsignal.,9 The Laplace Transform,Solution:,9 The Laplace Transform,ROC:,9 The Laplace Transform,and,have same Laplace transform representation,
7、buttheir ROC is different.,L,L,Note: for a signal x(t), we must give out the representation of Laplace transform and its ROC.,Example 9.3 9.4 Determine the Laplace transform of the following signals.,(a),(b),9 The Laplace Transform,9 The Laplace Transform,(a) solution,9 The Laplace Transform,(b) sol
8、ution,9 The Laplace Transform,9 The Laplace Transform,(3) The pole-zero plot of X(s),X(s) can be represented the ratio of two polynomials, the numerator polynomial;,the denominator polynomial;,9 The Laplace Transform,Definition: The zeros of X(s): the roots of the numerator polynomial N(s) is called
9、 the zeros of X(s).,The poles of X(s): the roots of the denominator polynomial D(s) is called the poles of X(s).,9 The Laplace Transform,9 The Laplace Transform,The representation of X(s) through its poles and zeros in the s-plane is referred to the pole-zeroplot of X(s).,Definition:,In the s-plane,
10、 use “X” to indicate the poles of X(s); and use “O” to indicate the zeros of X(s);,9 The Laplace Transform,On the other hand,If MN, s, X(s) , X(s) have (M-N) poles atinfinity.If MN, s, X(s) 0, X(s) have (N-M) zeros atinfinity.,9 The Laplace Transform,Example:,9 The Laplace Transform,Example:,-2 -1,R
11、es,Ims,s-plane,Note: The algebraic form of X(s) does not by itself identify the ROC for the Laplace transform. That is, a complete specification, to within a scalefactor, a Laplace transform consists of the pole- zero plot of the transform , together with its ROC.,9 The Laplace Transform,9 The Lapla
12、ce Transform,Example 9.5 Determine the Laplace transform of the following signal.,9 The Laplace Transform,Solution:,9 The Laplace Transform,-1 1 2,Res,Ims,s-plane,9.2 The Region of Convergence for Laplace Transform,Property 1: The ROC of X(s) consists of strips parallel to j-axis in the s-plane.Prop
13、erty 2: For rational Laplace transform, the ROC does not contain any poles.Property 3: If x(t) is of finite duration and is absolutely integrable, then the ROC is the entire s- plane,9 The Laplace Transform,9 The Laplace Transform,Example 9.6 Compute the Laplace transform of x(t).,Solution:,Property
14、4: If x(t) is right sided, and if the line Res=0 is in the ROC, then all valuesof s for which Res0 will also in the ROC.,9 The Laplace Transform,Property5: If x(t) is left sided, and if the line Res=0 is in the ROC, then all values of s for which Res0 will also in the ROC.,9 The Laplace Transform,Pr
15、operty 6: If x(t) is two sided, and if the line Res=0 is in the ROC, then the ROC will consist of a strip in the s-plane thatincludes the line Res=0 .,9 The Laplace Transform,9 The Laplace Transform,Example 9.7 Determine the Laplace transform of x(t).,9 The Laplace Transform,Property7: If the Laplac
16、e transform X(s) of x(t) is rational, then its ROC is bounded by poles or extends to infinity. In addition, no poles of X(s) are contained in the ROC.Property8: If the Laplace transform X(s) of x(t) is rational, then if x(t) is right sided, theROC is the region in the s-plane to the right of the rig
17、htmost pole. If x(t) is left sided, the ROC is the region in the s- plane to the left of the leftmost pole.,9 The Laplace Transform,9 The Laplace Transform,Example 9.8,Determine the inverse Laplace transform of X(s).,9.3 The Inverse Laplace Transform,So,9 The Laplace Transform,Appendix ：Partial Frac
18、tion Expansion （补充：部分分式展开法求拉氏反变换）,Consider a fraction polynomial:,Discuss three cases of D(s)=0, for distinct real roots, conjugation complex roots,and same roots.,9 The Laplace Transform,Case 1: Distinct real roots:,thus,9 The Laplace Transform,Calculate A1 :Multiply two sides by (s-p1):,Let s=p1,
19、so,Generally,9 The Laplace Transform,9 The Laplace Transform,Example 9.9 9.10 9.11 Determine the inverse Laplace transform of X(s).,(a),(b),(c),9 The Laplace Transform,Case 2: Conjugation complex roots:,9 The Laplace Transform,p1 is a complex root of D(s).,thus,9 The Laplace Transform,Calculate A1 :
20、,9 The Laplace Transform,9 The Laplace Transform,Other computation is same to case (1) 【distinct real roots】.,9 The Laplace Transform,Example: Determine the inverse Laplace transform.,9 The Laplace Transform,Solution:,Case 3: Same root:,9 The Laplace Transform,9 The Laplace Transform,thus,9 The Lapl
21、ace Transform,For first order poles:,Multiply two sides by (s-p1)r :,For r-order poles:,9 The Laplace Transform,9 The Laplace Transform,So,9 The Laplace Transform,using,We can obtain x(t).,9 The Laplace Transform,Example: We have known the system function of a LTI system:,Determine its unit step res
22、ponse.,9 The Laplace Transform,Solution:,9 The Laplace Transform,In above three cases, we emphasize nm. that is, in the representation of X(s),If nm, how can we do? Such as the following example.,9 The Laplace Transform,Example: Determine x(t) of X(s).,9 The Laplace Transform,Solution:,如果mn, 则必须将有理分
23、式化为多项式加真分式的形式，然后进行部分分式展开。,9 The Laplace Transform,9 The Laplace Transform,Example: Determine x(t) of X(s).,9 The Laplace Transform,Solution:,Then, we can obtain x(t).,9.4 Geometric Evaluation of the Fourier Transform form the Pole-Zero Plot,For a certain value of s1, then,Re,9.4 Geometric Evaluation
24、 of the Fourier Transform form the Pole-Zero Plot,零点向量：零点到s1的向量； 极点向量：极点到s1的向量。,X(s1)的模等于各零点向量模的乘积除以各极点向量摸的乘积再乘以系数M。 X(s1)的相位等于各零点向量相位的和减去各极点向量相位的和。,9.4 Geometric Evaluation of the Fourier Transform form the Pole-Zero Plot,即，傅立叶变换X(j1)的模等于各零点向量模的乘积除以各极点向 量模再乘以系数M。 X(j1)的相位等于各零点向量相位的和减去各极点向量相位的和。,If,
25、零点向量：零点到虚轴上某点j1的向量； 极点向量：极点到虚轴上某点j1 的向量。,9.4 Geometric Evaluation of the Fourier Transform form the Pole-Zero Plot,9.4.1 first-order systems(一阶系统）,The differential equation for a first-order system is often expressed in thd form,The coefficient is the time constant of the system.,Such as RC circuit.
26、,9.4 Geometric Evaluation of the Fourier Transform form the Pole-Zero Plot,9.4 Geometric Evaluation of the Fourier Transform form the Pole-Zero Plot,The Bode plot of the frequency response:,9.4 Geometric Evaluation of the Fourier Transform form the Pole-Zero Plot,讨论： 1）增加，H(j)的模单调下降， H(j)的相位从0下降到/2。
27、 2）当1/时， H(j)的模下降了3dB，而此时 H(j)的相位是/4.,所以把1/称为3dB点或者转折频率点。 称为系统的时间常数。,一阶连续系统，表现为低通滤波器特性。,9.4 Geometric Evaluation of the Fourier Transform form the Pole-Zero Plot,9.4.2 second-order systems,The linear constant-coefficient differential equationfor a second-order system is,Such as RLC circuit.,9.4 Geom
28、etric Evaluation of the Fourier Transform form the Pole-Zero Plot,9.4 Geometric Evaluation of the Fourier Transform form the Pole-Zero Plot,讨论： 1）1,两个极点都在实轴上，如图9.19a，由0变到，则|H(j)|单调下降，而H(j)的相位由0变到。近似为一个低通滤波器特性。 2） 01,两个极点是共轭复数，h(t)有振荡，此时频率响应如图9.20，此时对某个频率点的较窄的频率范围内，有一个陡峭的尖峰，越小，尖峰越大。这可以用来对一些正弦信号进行选频性放
29、大，所以，此时可以做一个选频放大器。 或者说近似为带通滤波器特性。,9.4 Geometric Evaluation of the Fourier Transform form the Pole-Zero Plot,Re,0,Im,9.4.2 all-pass systems(全通系统）,全通系统：对于所有，频率响应的模都是一个常数。,极点与零点关于虚轴对称。,9.4 Geometric Evaluation of the Fourier Transform form the Pole-Zero Plot,9.4 Geometric Evaluation of the Fourier Tran
30、sform form the Pole-Zero Plot,Exercise: 10.9,9.5 Properties of the Laplace Transform,(1) Linearity,9 The Laplace Transform,Example 9.13,Find the ROC for x(t). We have known,9 The Laplace Transform,极点（1）抵消了,收敛域扩大了,Solution:,(2) Time shifting,9 The Laplace Transform,9 The Laplace Transform,Example: De
31、termine the Laplace transform of the impulse trains.,9 The Laplace Transform,Solution:,说明：上式中每一个冲激的收敛域均为Res, 但无穷多个冲激线性组合后，总的收敛域变为Res0，即收敛域缩小了。这是由于线性组合过程中产生了新的极点s=0，因此性质（1）【线性组合】中关于收敛域的概念只适合于有限项的线性组合。,9 The Laplace Transform,(3) Shifting in the s-domain,(4) Time scaling,Figure 9.24 (b) (c ) error,9 T
32、he Laplace Transform,Specially:,(5) Conjugation,(6) Convolution property,9 The Laplace Transform,9 The Laplace Transform,Example:,若有极点被抵消的情况，则收敛域会扩大。,9 The Laplace Transform,Example: we have known the system function of a causal and LTI system:,Determine the output y1(t) and y2(t).,And its input sig
33、nal is :,9 The Laplace Transform,Solution: Because the system is causal, its ROC is Res-2.,9 The Laplace Transform,Discussion: As for the above example, if the input is x(t)=e-4t, H(-4) is not convergent, So, y(t) does not exist.,9 The Laplace Transform,的拉普拉斯变换不存在。证明如下：,说明：,9 The Laplace Transform,所
34、以，上面例题中，不能用下面的方法求y1(t).,(7) Differentiation in the time domain,9 The Laplace Transform,9 The Laplace Transform,(8) Differentiation in the s-domain,Example 9.14 Find the Laplace transform of x(t).,9 The Laplace Transform,9 The Laplace Transform,Important formula:,9 The Laplace Transform,In the same w
35、ay, we can obtain,9 The Laplace Transform,Example 9.15 Find the inverse Laplace transform of X(s).,(9) Integration in the time domain,9 The Laplace Transform,Note: if there is no intersection between R and Res0, the Laplace transform of x(-1)(t) does not exist.,(10) The initial- and final-value theo
36、rems,If x(t)=0 for t0 and x(t) contains no impulses or higher-order singularities at t=0, then,If x(t)=0 for t0 and x(t) has a finite limit as t, then,9 The Laplace Transform,The initial-value theorem:,The final-value theorem:,9.5.11 table of properties,Table 9.1,9 The Laplace Transform,9.6 Some Laplace Transform Pairs,Table 9.2,9 The Laplace Transform,9 The Laplace Transform,Determine the Laplace transform of the following signals.,(a),(b),9 The Laplace Transform,Solution:,9 The Laplace Transform,Solution:,
