信号与系统课件chapter 6

1、chapter 6the laplace transform6.0 introductionn with laplace transform, we expand the application in which fourier analysis can be used.sten the laplace transform provides us with a representation for signals as linear combinations of complex exponentials of the form with s= + jn the laplace transfo

2、rm (拉普拉斯变换) is a generalization of the continuous-time fourier transform.6.1 the laplace transformlet s = + j, and using x(s) to denote this integral, we obtaindteetxtjt)(dtetxsxst)()(for some signals which have not fourier transforms, if we preprocess them by multiplying with a real exponential sig

3、nal , then they may have fourier transforms.tethe laplace transform of x(t) the laplace transform is an extension of the fourier transform; the fourier transform is a special case of the laplace transform when = 0.example 6.1consider the signal).()(tuetxtsesesdtedtetuesxtststssttt111)()(lim)(0)(0)(f

4、or convergence, we require that res + 0, or res , thus,1)(sssxreregion of convergence (roc ) (收敛域)example 6.2consider the signal).()(tuetxttststsstttessesdtedttueesxlim)(0)(0)(111)()(for 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

5、right-half planeright-half plane. 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 in the roc; and the roc of a left-sided signal is a left-half planeleft-half plane.property 6: if x(t) is two sided, and if the line res = 0 i

6、s in the roc, then the roc will consist of a strip in the s-plane that includes the line res = 0. rreimlreimlrreimproperty 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 roc.property 8:

7、 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 sided, the roc is the region in the s-plane to the left of the leftmost pole.example 6.6let)2)(1(1)(sssxreims-planeroc correspondi

8、ng to a right-sided signal roc corresponding to a left-sided signal roc corresponding to a two-sided signal 6.3 the inverse laplace transformdesxsxtj)(21)(1 -fmultiplying both sides by , we obtain tedesxtxst)(21)(changing the variable of this integration from to s and using the fact that is constant

9、, so that ds = jd. jjstdsesxjtx)(21)(thus, the basic inverse laplace transform equation is:the inverse laplace 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 i

10、ntegration（围线积分）（围线积分） in the complex plane. for the class of rational transforms, the inverse laplace transform can be determined by using the technique of partial-fraction expansion. tetx)(example 6.7let . 1re2,)2)(1(1)(ssssxperforming the partial-fraction expansion, we obtain )2(1) 1(1)(sssx. 12,

11、)2)(1(1)()()(2ssstuetuetxltttre, 1,11)(sstuelttre. 2,21)(2sstuelttrewhat if ? . 1re2,4)2)(1(1)(ssssssxexample 6.8let . 0re,) 1)(3(2)(2ssssssxcompute the x(t) with contour integration method. 3, 021ssx(s) has two first-order poles: and a second-order pole:. 14 , 3sfrom the residue theorem, 1,)(re3,)(

12、re0 ,)(re)(stststesxsesxsesxstx 320132)(0 ,)(re20sesssessxesxsstsststtstsststesesssesxsesxs323121312)(33,)(rettsststsstsststeteessstessssesssdsdesxsdsdesxs432132364)3(2)(1! 111,)(re1222112)(232112132)(,3tuetetxthustt . 0re1,) 1)(3(2)(2ssssssx . 3re,) 1)(3(2)(2ssssssxwhat if ?or ?)(2321121)(32)(3tuet

13、etutxtt)(232112132)(3tuetetxttorthen6.4 geometric evaluation of the fourier transform from the pole-zero plota general rational laplace transform has the form:)()()(sdsnsxpjjriissmsx11)()()(and it can be factored into the form: where i, j are zeros and poles of x(s), respectively. s1 re ims-plane1s1

14、spjjriijjmjx11)()()(,)(111pjjriibamjxripjjijx111)(argcomplex plane representation of the vectors s1, , and representing the complex numbers s1, and respectively. 1s1szero vectors（零点矢量）pole vectors（极点矢量）lets take an example to show how to evaluate the fourier transform from the pole-zero plot: . 1,)2

15、)(1(1)(ssssxregivengeometrically, from figure, we can write2122221)1)(2(1)(bbjx2111tan2tan)(argjx -2 -1reim s-plane21bj12bj21)(arg, 0)(arg)2,2,(,212121jxjxhoweverjxbb:0: , 21b,21jx0)(arg, 0, 021jx, 12b)(arg,2,221jx,21bb0jx)(argjx- |x(j)| 0 1/20 all-pass function：a laplace transform with all of its p

16、oles and zeros located on both sides of the j- axis symmetrically. and all the poles are on the left of the j- axis. all the zeros are on the right of the j- axis. *zzpp12216.5 properties of the laplace transform6.5.1 linearity6.5.2 time shiftingifand 111),()(rrocsxtxlt222),()(rrocsxtxltthen212121),

17、()()()(rrcontainingrocwithsbxsaxtbxtaxltnote: roc is at least the intersection of r1 and r2, which could be empty, also can be larger than the intersection. ifrrocsxtxlt),()(thenrrocsxettxstlt),()(006.5.3 shifting in the s-domain ifrrocsxtxlt),()(then 00),()(0srrocwithssxtxelttsre6.5.4 time scaling

18、ifrrocsxtxlt),()(thenrrocwithsxtxlt,1)(consequence: 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*. 6.5.5 conjugation .),()(*rrocwithsxtxlt when x(t) is real: )()(*sxsxrrocwithsxtxlt),()(consequence:6.5.6 convolutio

19、n property if111),()(rrocsxtxltand 222),()(rrocsxtxltthen212121),()()()(rrcontainingrocwithsxsxtxtxlt6.5.7 differentiation in the time domain ifrrocsxtxlt),()(thenrcontainingrocwithssxdttdxlt),()(6.5. 8 differentiation in the s-domain rrocdssdxttxlt,)()(6.5.9 integration in the time domain .0,)()(sr

20、containingrocwithssxdxtltre6.5.10 the initial- and final-value theorems（初值和终值（初值和终值定理）定理） initial-value theorem :)()0(limssxxsfinal -value theorem :)()(limlim0ssxtxstexample 6.9 consider the signal).1()()(tututxwe know0,1)(sstultre, 0,1) 1(sestusltreand from the time shifting property,so that.,111)(

21、planesentirerocseesssxssexample 6.10 determine the laplace transform of).()(tutetxtsince,1)(sstuelttrefrom the differentiation in the s-domain property,.,11)(2sssdsdtutelttrein fact, by repeated application of this property, we obtain,)(1)()!1(1sstuentnlttnreexample 6.11use the initial-value theorem

22、 to determine the initial-value of)()3(cos)()(2tutetuetxtt2201441252)()0(2323limlimssssssssxxss. 1,)2)(102(1252)()3(cos)(222sssssstutetueltttreexample 6.12determine the laplace transform of the causal periodic signal x(t) which is depicted in the following figure:)2()()()(000ttxttxtxtxtstsesxesxsxsx

23、2000)()()()(1)(20tstseesxstesx1)(0 0 t 2t t)(tx 0re11)(0sesxst 0 t 2t t)(txconsider:consider: whats the lt of the periodic signal in the following figure?6.6 analysis and characterization of lti systems using the laplace transform the laplace transforms of the input and the output of an lti system a

24、re related through multiplication by the laplace transform of the impulse response of the system. y(s) = h(s) x(s) the 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

25、with a rational system function, causality of the system is equivalent to the roc being the right-half plane to the right of the rightmost pole.system function(transfer function)example 6.13consider a system with impulse response ).()()21(tuethtjsince h(t) = 0 for t 0, this system is causal. the sys

26、tem function: , 1,211)(sjsshreit is rational and the roc is to the right of the rightmost pole, consistent with our statement. example 6.14consider the system function . 1,1)(sseshsrefor this system, the roc is to the right of the rightmost pole. the impulse response associated with the system),1()(

27、) 1(tuethtit is nonzero for 1 t 1. )()(2tueetxtt thus,unilateral laplace transform provide us with information about signals only for .0texample 6.23consider the unilateral laplace transform .23)(2sssx212)(sssxtaking inverse transforms of each term results in)()()(2)(2tuetttxt6.9.1 properties of the

28、 unilateral laplace transform time scaling:0),(1)(aasaatxulxconvolution: assuming that x1(t) and x2(t) are identically zero for t 0. )()()()(2121sstxtxulxxdifferentiation in the time domain :)0()()(xsstxdtdulx).0()()()(101knkknnulnnxssstxdtdx proof of this property for first-derivative of x(t):0)()(

29、dtedttdxdttdxstul0)(tdxest0)()(0dtetxsetxstst)0()(xssx similarly, the unilateral laplace transform of second-derivative of x(t) can be obtained by repeating using the property:).0()0()()0()0()()(222xsxssxxsssdttxdxxul6.9.2 solving differential equations using the unilateral laplace transformexample 6.24consider the system characterized by the differential equation ),()(2)(3)(22txtydttdydttydwith initial conditions.)0(,)0(yylet x(t) = u(t). determine the output y(t).applying the unilate