第九章讲义——拉普拉斯变换.ppt

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1、 Signals and SystemsChapter 9The Laplace TransformLiu Ke,School of Automation Engineering9.0 IntroductionRepresent the signals and LTI systems Defect:cant be used in unstable systemGeneralization of Fourier transformLaplace transformHow can represent signals and systems?Fourier transform:Analysis th

2、e LTI systems in other domainFourier transformAnalysis the system in frequency domainLaplace transformAnalysis the system in S domainConvolutionAnalysis the system in time domainThe contents of the chapterThe Laplace transform and Inverse Laplace transform The ROC for Laplace transform Geometric eva

3、luation of the Fourier transform from the Pole-Zero plotProperties of the Laplace transform Analysis and characterization of LTI system using the Laplace transform System function algebra and block diagram representations The unilateral Laplace transform9.1 The Laplace transformConception Laplace tr

4、ansform ROC Pole-Zero plotThe relationship between Laplace transform and Fourier transformThe representation of the Laplace transformFourier transform:Laplace transform:The relationship between the Laplace transform and the Fourier transformv v If If Fourier transform is a particular form of Laplace

5、 transformLaplace transform is Fourier transform ofExample 9.1Example 9.2Region of convergence(ROC)vROCthe range of values of S for which integral in convergesvLaplace transform includes:(1)the algebraic expression(2)ROCvThe representation of ROCComplex plane(S-plane)Example 9.3Example 9.4Pole-Zero

6、plotvLaplace transform maybe a ratio of polynomialsnumeratordenominatorvPoles the roots of D(S)Zeros the roots of N(S)vThe representation of poles and zerosPole-Zero plotvAnother representation of Laplace transformPole-Zero plot and ROCvThe order of pole or zero Example 9.5SummarizationThe represent

7、ation of Laplace transformThe relationship between Laplace transform and Fourier transformROCPole-Zero plot9.2 The ROC for Laplace transformThe characteristic of ROCThe relationship between the ROC and the signalsvProperty 1 the ROC of X(S)consists of strips parallel to the jw-axis in the s-plane vP

8、roperty 2 for rational Laplace transforms,the ROC does not contain any poles vProperty 3 if x(t)is of finite duration and is absolutely integrable,then the ROC is the entire s-plane Example 9.6vProperty 4 if x(t)is right sided,and if the line Res=is in the ROC,then all values of S for which Res will

9、 also be in the ROCvProperty 5 if x(t)is left sided,and if the line Res=is in the ROC,then all values of S for which Res will also be in the ROCvProperty 6 if x(t)is two sided,and if the line Res=is in the ROC,then the ROC will consist of a strip in the s-plane that include the line Res=Example 9.7v

10、Property 7 if the Laplace 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 ROCvProperty 8 if the Laplace transform X(S)of x(t)is rationalif x(t)is right sided,the ROC is the region in the s-plane to the right of

11、 the rightmost poleif x(t)is left sided,the ROC is the region in the s-plane to the left of the leftmost poleExample 9.89.3 The inverse Laplace transformThe representation of the inverse Laplace transformThe usual method of determine the inverse rational Laplace transformThe representation of the in

12、verse Laplace transformThe usual method of determine the inverse rational Laplace transformvPartial-fraction expansionExample 9.9Example 9.10Example 9.119.4 Geometric evaluation of the Fourier transform from the Pole-Zero plotReview the relationship between Fourier transform and Laplace transformA s

13、imple method geometric evaluation Review the relationship between the Laplace transform and Fourier transformv Fourier transform is a particular form of Laplace transformv Laplace transform is Fourier transform ofThe basic knowledge of geometric evaluationv Examplev v 9.5 Properties of the Laplace t

14、ransformLinearityTime shiftingShifting in the s-domainTime scalingConjugationConvolution propertyDifferentiation in the time domainDifferentiation in the s-domainIntegration in the time domainThe initial-and final-value theoremsExample 9.13vLinearityvTime shiftingvShifting in the s-domainvTime scali

15、ngvConjugation vConvolution propertyvDifferentiation in the time domain vDifferentiation in the S-domain Example 9.14Example 9.15vIntegration in the time domain vThe initial-and final-value theorems Example 9.16Answer:v v 9.7 Analysis and characterization of LTI system using the Laplace transformThe

16、 relationship between the properties of system and H(S)How can get the X(S)or H(S)or Y(S)Several important properties of systemvCausality The ROC associated with the system function for a causal system is a right-half planeFor a system with a rational system function,causality of the system is equiv

17、alent to the ROC being the right-half plane to the right of the rightmost poleExample 9.17The system is causal,so the ROC of H(S)is a right-half planeExample 9.18The system is not causal,the ROC of H(S)is not a right-half planeExample 9.19ROC is a right-half plane,but the system is not causal,unless

18、 the system is rationalvStabilityAn LTI system is stable if and only if the ROC of its system function H(S)includes the entire jw-axis i.e.ReS=0 Example 9.20A causal system with rational system function H(S)is stable if and only if all of the poles of H(S)lies in the left-half of the s-plane i.e.,al

19、l of the poles have negative real partsExample 9.21LTI system characterized by linear constant-coefficient differential equationsvSome examplevThe relationship between system behavior and the system functionThe method of determine h(t)by differential equationsvUse partial fraction expansionv Example

20、 9.23Example 9.24Example 9.25Example 9.26An LTI system include the information as below:1.the system is causal 2.the system function is rational and has only two poles,at s=-2 and s=4 3.If x(t)=1,then y(t)=0 4.the value of the impulse response at t=0+is 4Example 9.27A stable and causal system,H(S)is

21、 rational,contains a pole at s=-2,and does not have a zero at the origin,please determine the statements below:P723-14P728-27P729-349.8 System function algebra and block diagram representationsReview the system functions for interconnections of LTI systemSignal-pass plotThe example of block diagram

22、representationsParallel interconnectionh h1 1(t)(t)h h2 2(t)(t)+x(tx(t)y(ty(t)y y1 1(t)(t)y y2 2(t)(t)Series combinationh h1 1(t)(t)h h2 2(t)(t)x(tx(t)y(ty(t)y y1 1(t)(t)Feedback interconnectionh h1 1(t)(t)h h2 2(t)(t)+x(tx(t)y(ty(t)e(t)e(t)z(t)z(t)+-Signal-pass plot（信流图）信流图）vSome concepts 节点节点表示一个信

23、号。只有信号输出的表示一个信号。只有信号输出的节点称为节点称为源点源点，只有信号输入的节点是，只有信号输入的节点是阱点阱点，既有信号输入、又有信号输出的节点称为既有信号输入、又有信号输出的节点称为混混合节点合节点。节点所代表的信号等于输入该节点节点所代表的信号等于输入该节点的全部信号的和，与输出支路无关。的全部信号的和，与输出支路无关。支路支路节点之间的有向线段称为支路节点之间的有向线段称为支路 通路通路一条或几条同方向的支路组成通一条或几条同方向的支路组成通路。路。通路的转移函数等于各支路转移函数的通路的转移函数等于各支路转移函数的乘积。乘积。不闭合的通路称为不闭合的通路称为开路开路，闭合的

24、通路，闭合的通路称为称为环路环路。信流图分析信流图分析梅森规则（相关术语梅森规则（相关术语1 1）接触接触没有公共节点的通路称为不接触通路没有公共节点的通路称为不接触通路切断支路切断支路切断某条支路意味着取消这条支路而仍然切断某条支路意味着取消这条支路而仍然保存两端的节点（保存两端的节点（不移去节点不移去节点）移去节点移去节点移去某节点意味着截断与该节点连接的全移去某节点意味着截断与该节点连接的全部支路（部支路（移去该节点和与该节点相连接的所有支路移去该节点和与该节点相连接的所有支路）移去通路移去通路移去某一条通路意味着移去该通路上的所移去某一条通路意味着移去该通路上的所有节点有节点信流图分析

25、信流图分析梅森规则（相关术语梅森规则（相关术语2 2）信流图行列式信流图行列式信流图分析信流图分析梅森规则梅森规则v梅森规则应用举例梅森规则应用举例Example 9.28Example 9.29Example 9.30vDirect formvCascade formvParallel formExample 9.31vDirect formvCascade formvParallel formP723-179.9 The unilateral Laplace transformRepresentation of the unilateral Laplace transformSome pr

26、opertiesSolving differential equations using the unilateral Laplace transformRepresentation of the unilateral Laplace transformExample 9.32Example 9.33Example 9.34Example 9.35Example 9.36Properties of the unilateral Laplace transformvThe same as bilateral Laplace transformLinearity、Shifting in the s

27、-domainTime scaling、ConjugationDifferentiation in the s-domainThe initial-and final-value theoremsvThe similar to BLTConvolution vThe difference between BLT and ULTDifferentiation in the time domainIntegration in the time domainSolving differential equations using the unilateral Laplace transformExample 9.38ExampleExampleExampleExample (18 points)XXO2-1-3Example(15 points)+xnyn+-