信号与系统英文版教学课件：ch6 The Laplace Transform

1、1 With Laplace transform, we expand the application in which Fourier analysis can be used. The Laplace transform provides us with a representation for signals as linear combinations of complex exponentials of the form with s= + j The Laplace transform (拉普拉斯变换) is a generalization of the continuous-t

2、ime Fourier transform. INTRODUCTION21. The Laplace TransformLet s = + j, and using X(s) to denote this integral, we obtainFor some signals which have not Fourier transforms, if we preprocess them by multiplying with a real exponential signal , then they may have Fourier transforms.The Laplace transf

3、orm of x(t) 1) Development of The Laplace TransformWe will denote the transform relationship between x(t) and X(s) as3The Laplace transform is an extension of the Fourier transform; the Fourier transform is a special case of the Laplace transform when = 0.generallyThat is, the laplace transform of x

4、(t) can be interpreted as the Fourier transform of x(t) after multiplication by a real exponential signal. The real exponential may be decaying or growing in time, depending on whether is positive or negative.In specifying the Laplace transform of a signal, both the algebraic expression and the rang

5、e of values of s for which this expression is valid are required.The range of values of s for which the integral in X(s) converges is referred to as the region of convergence(ROC).4Example 6.1 Consider the signalFor convergence, we require that Res + 0, or Res , Thus,region of convergence (ROC ) (收敛

6、域)5Example 6.2 Consider the signalFor convergence, we require that Res + 0, or Res 0 will also be in the ROC; and the ROC of a right-sided signal is a right-half plane. RReIm5)Property 5 If x(t) is left sided, and if the line Res = 0 is in the ROC, then all values of s for which Res 0 will also be i

7、n the ROC; and the ROC of a left-sided signal is a left-half plane.LReIm156) Property 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 that includes the line Res = 0.LRReIm7) Property 7If the Laplace transform X(s) of x(t) is rational

8、, then its ROC is bounded by poles or extends to infinity. In addition, no poles of X(s) are contained in the ROC.8) Property 8 If the Laplace transform X(s) of x(t) is rational, then if x(t) is right sided, the ROC is the region in the s-plane to the right of the rightmost pole. If x(t) is left sid

9、ed, the ROC is the region in the s-plane to the left of the leftmost pole.16Example 6.6LetReIms-planeROC corresponding to a right-sided signal ROC corresponding to a left-sided signal ROC corresponding to a two-sided signal There are three possible ROCs that can be associated with this algebraic exp

10、ression, corresponding to three distinct signals.173. The Inverse Laplace TransformMultiplying both sides by , we obtain Changing the variable of this integration from to s and using the fact that is constant, so that ds = jd. Thus, the basic inverse Laplace transform equation is:The inverse Laplace

11、 transform equation states that x(t) can be represented as a weighted integral of complex exponentials. The formal evaluation of the integral for a general X(s) requires the use of contour integration (围线积分) in the complex plane. For the class of rational transforms, the inverse Laplace transform ca

12、n be determined by using the technique of partial-fraction expansion. 拉普拉斯变换与傅里叶变换的关系1）付氏变换与拉氏变换的形式相似，基本差别： 付氏变换时域与变换域变量皆为实数（ ） 拉氏变换时域变量为实数，变换域变量为复数（ ）2）物理意义 傅氏：将 分解成许多形式为 的指数项之和，每一对正、负 组成一个余弦振荡，振幅为 拉氏：将 分解成许多形式为 的指数项之和，每一对正、负 组成一个变幅的余弦振荡,振幅为 3）傅立叶变换是双边拉普拉斯变换中 的一种特殊情况，因此，求两者反变换的积分路径不同。 18拉普拉斯反变换的求法：

13、(一）部分分式展开法 19（二） 围线积分法（留数法） 拉氏反变换： 留数定理： 上式左边的积分是在s平面内沿一不通过被积函数极点的封闭曲线C进行的，右边则是在此围线C中被积函数各极点上留数之和。 为应用留数定理，在求拉氏反变换的积分线（ ）上应补足一条积分线以构成一个封闭曲线。 当然要求必须有 或或上式在满足以下两个条件（约当引理）时成立 时， 一致地趋近于零； 因子 的指数st的实部应小于 ， 即 20一般条件都能满足（ 除外）， 当 或条件满足 即 积分沿左半圆弧进行； 积分沿右半圆弧进行。因此围线中被积函数 所有极点的留数之和 留数的求取： 的极点即为 的极点 规则： 若 为一阶极点，

14、则（） 规则： 若 为p阶极点， 则（） 2122Example 6.7LetPerforming the partial-fraction expansion, we obtain -2ReImReIm-123Example 6.8LetCompute the x(t) with contour integration method.X(s) has two first-order poles: and a second-order pole:From the Residue Theorem, 24If ROC isThen If ROC isIf ROC is25Example 6.9Le

15、tX(s) has a pair of conjugate poles:FromSo26274. Geometric Evaluation of The Fourier Transform From The Pole-Zero PlotA general rational Laplace transform has the form:and it can be factored into the form: where i, j are zeros and poles of X(s), respectively. Im s1 Res-planeComplex plane representat

16、ion of the vectors s1, a, and s1a representing the complex numbers s1, a and s1 a respectively. 28Lets take an example to show how to evaluate the Fourier transform from the pole-zero plot:Given -2 -1ReIm s-planeGeometrically, from Figure, we can write|X(j)| is the reciprocal of the product of the l

17、engths of the two pole vectors(极点矢量); arg X(j) is the negative of the sum of the angles of the two vectors. zero vectors(零点矢量)295. Properties of The Laplace Transform1) Linearity2) Time ShiftingIfand thenNote: ROC is at least the intersection of R1 and R2, which could be empty, also can be larger th

18、an the intersection. Ifthen303) Shifting in the s-Domain Ifthen4) Time Scaling IfthenConsequence: if x(t) is real and if X(s) has a pole or zero at s = s0 , then X(s) also has a pole or zero at the complex conjugate point s = s0*. 5) Conjugation When x(t) is real: Consequence:316) Convolution Proper

19、ty Ifand then7) Differentiation in the Time Domain Ifthen8) Differentiation in the s-Domain 329) Integration in the Time Domain 10) The Initial- and Final-Value Theorems(初值和终值定理) Initial-value theorem :Final -value theorem :Conditions : x(t)=0 for t0 and that x(t) contains no impulses or higher orde

20、r singularities at the origin.33Example 6.10 Consider the signalWe knowAnd from the time shifting property,So thatHere, the pole at s = 0 is removableExample 6.11 Determine the Laplace transform of Sawtooth 0 T t Ex(t) = + +E0 T t0 T t 0 T tSolution A: 34Solution B: 0 T t 0 t 0 T t = *3536Example 6.

21、12 Determine the Laplace transform ofSinceFrom the differentiation in the s-domain property,In fact, by repeated application of this property, we obtain37Example 6.13 Use the initial-value theorem to determine the initial-value ofIn the Time Domain:386. Analysis and Characterization of LTI Systems U

22、sing The Laplace Transform 1) System functionWe know, in the time domain, the input and the output of an LTI system are related through Convolution by the impulse response of the system. Thus y(t) = h(t) *x(t) supposeFrom Convolution Property Y(s) = H(s) X(s) system function(transfer function)For ,H

23、(s) is the frequency response of the LTI system.39The ROC associated with the system function for a causal system is a right-half plane.An ROC to the right of the rightmost pole does not guarantee that a system is causal. For a system with a rational system function, causality of the system is equiv

24、alent to the ROC being the right-half plane to the right of the rightmost pole.2) Relating Causality to the System function For a causal LTI system, the impulse response is zero for t0. The ROC associated with the system function for a anticausal system is a left-half plane.40Example 6.14 Consider a

25、 system with impulse response Since h(t) = 0 for t 0, this system is causal. The system function: It is rational and the ROC is to the right of the rightmost pole, consistent with our statement. 41Example 6.15 Consider the system functionFor this system, the ROC is to the right of the rightmost pole

26、. The impulse response associated with the systemis nonzero for 1 t 1. 77 Thus,unilateral Laplace transform provide us with information about signals only for Example 6.25 Consider the unilateral Laplace transform Taking inverse transforms of each term results in782) Properties of the Unilateral Lap

27、lace Transform Time scaling:Convolution: assuming that x1(t) and x2(t) are identically zero for t 0. Differentiation in the time domain :79 Proof of this property for first-derivative of x(t): Similarly, the unilateral Laplace transform of second-derivative of x(t) can be obtained by repeating using

28、 the property:803) Solving Differential Equations Using the Unilateral Laplace TransformExample 6.26 Consider the system characterized by the differential equation with initial conditionsLet x(t) = u(t). Determine the output y(t).Applying the unilateral transform to both sides of the differential eq

29、uation, we obtain 81or equivalently,Thus, we obtainzero-state response zero-input response The unilateral Laplace transform is of considerable value in analyzing causal systems which are specified by linear constant-coefficient differential equations with nonzero initial conditions (i.e., systems th

30、at are not initially at rest).824) Representation of Circuits in s-domainThe relations between I and V in the time domain for R,L,C are:Respectively.Apply unilateral Laplace transform to each equation to obtain For a circuit, if we obtain the representation for the basic elements in the circuit in t

31、he s-domain, then we also obtain the circuit in the s-domain. 83An inductor with inductance L and initial current may be taken as an inductor with inductance L and zero initial current cascaded with a impulse source voltage with area . A capacitor with capacitance C and initial voltage may be taken

32、as a capacitor with capacitance C and zero initial voltage cascaded with a step source voltage with step . IR(s)VR (s)R+-IL(s)VL (s)sL+-+IC(s)VC (s)+-+Representation of the three basic elements in the s-domain with initial state being equalized as a source voltage84IR(s)VR (s)R+-VL (s)sL+-VC (s)+-Representation of the three basic elements in the s-domain with initial state being equalized as a source currentAnother expression of the relation between current and voltage of three basic elements