The Laplace Transform

Transcription

1 The Laplace Transform Previous basis funcions: 1, x, cosx, sinx, exp(jw). New basis funcion for he LT => complex exponenial funcions LT provides a broader characerisics of CT signals and CT LTI sysems Two ypes of LT Unilaeral (one-sided): good for solving differenial equaions wih iniial condiions. Bilaeral LT (wo-sided): good for looking a he sysem characerisics such as sabiliy, causaliy, and frequency response 1

2 I. Laplace Transform So far, signals represened using superposiions of complex sinusoids, exp(jω) Now, le s consider complex exponenials as basis exp(s)=exp(s+jω)=exp(s)cos(ω)+jexp(s)sin(ω) Why? LT can be used o analyze a larger class of CT problems involving signals ha are no absoluely inegrable Remember, he FT does no exis for signals ha are no absoluely inegrable The properies of LT are very much similar o hose of FT Two ypes of LT Unilaeral, or one sided: good for solving differenial equaions wih iniial condiions Bilaeral, or wo sided: good for looking a he sysem characerisics such as sabiliy, causaliy, and frequency responses 2

3 Laplace Transform e s = e Re{ e Im{ e s s s cos( w) + je s sin( w), s = s + } = exponenially damped cosine } = exponenially dampled sine jw Real and imaginary pars of he complex exponenial e s, where s = s + jw. 3

4 Who is Laplace? Pierre-Simon, marquis de Laplace (23 March March 1827) was a French mahemaician and asronomer whose work was pivoal o he developmen of mahemaical asronomy and saisics. He summarized and exended he work of his predecessors in his five volume Mécanique Célese (Celesial Mechanics) ( ). This work ranslaed he geomeric sudy of classical mechanics o one based on calculus, opening up a broader range of problems. In saisics, he so-called Bayesian inerpreaion of probabiliy was mainly developed by Laplace. He formulaed Laplace's equaion, and pioneered he Laplace ransform which appears in many branches of mahemaical physics, a field ha he ook a leading role in forming. The Laplacian differenial operaor, widely used in mahemaics, is also named afer him. He resaed and developed he nebular hypohesis of he origin of he solar sysem and was one of he firs scieniss o posulae he exisence of black holes and he noion of graviaional collapse. He is remembered as one of he greaes scieniss of all ime, someimes referred o as a French Newon or Newon of France, wih a phenomenal naural mahemaical faculy superior o any of his conemporaries. 4

5 5 Transfer Funcion ò ò ò ò = \ = = - = * = = = d e h s H d e h e d e h d x h x h x H y h e x s s s s s ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )} ( { ) ( ) ( and an impulse response funcion ) ( Given ) ( Transfer funcion

6 Bilaeral Laplace Transform X ( s) x( ) = = ò 1-2pj x( ) e ò s + j s - j -s d X ( s) e x ( ) «X ( s) s ds Signal x() is expressed as a weighed superposiion of complex exponenials exp(s) In pracice, we usually do no evaluae his inegral direcly (i requires a lecure on conour inegraion) 6

7 Convergence ò - x( ) e - s d < We mus have he above as a necessary condiion for convergence of he LT The range of s for which he LT converges is ermed he region of convergence (ROC) LT exiss for signals ha do no have a FT: i.e., wihin a cerain range of s, we can ensure ha x()exp(-s) is absoluely inegrable. 7

8 Example FT of x()=exp()u() does no exis. Why? If s>1, x()exp(- s)=exp((1-s))u() is absoluely inegrable. So he LT exiss. The Laplace ransform applies o more general signals han he Fourier ransform does. (a) Signal for which he Fourier ransform does no exis. (b) Aenuaing facor associaed wih Laplace ransform. (c) The modified signal x()e -s is absoluely inegrable for s > 1. 8

9 s-plane Represen he complex frequency s graphically in erms of a complex plane = s-plane Horizonal axis represens he real par Verical axis represens he imaginary of s If s=0, hen X(jω)=X(s) s=0. Tha is replace jw by s. Lef side of he s-plane Righ side of he s-plane 9

10 Poles and Zeros Zeros ( o ) X ( s) = b Õ N Õ M k = 1 M k = 1 ( s ( s - - c d k k ) ) Poles x The s-plane. The horizonal axis is Re{s} and he verical axis is Im{s}. Zeros are depiced a s = 1 and s = 4 ± 2j, and poles are depiced a s = 3, s = 2 ± 3j, and s = 4. 10

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13 II. Unilaeral LT Since mos of signals and sysems we deal are causal signals and sysems, use unilaeral LT is good enough; for noncausal signals and sysems, use bilaeral LT. Thus ULT is limied o causal signals: ha is signals are zeros for ime <0 In pracice, LT means ULT Unilaeral LT is used o analyze he behavior of he causal sysem described by a differenial equaions wih iniial condiions Unilaeral LT ò -s - x( e X ( s) = 0 ) d 0 - implies ha we do no include disconinuies and impulses ha occur a =0 Ani-causal (noncausal) signals are no covered in his class. 13

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15 III. Some Properies of Unilaeral LT Lineariy Scaling Time Shif S-domain Shif Convoluion Differeniaion in he s-domain Differeniaion in he Time Domain General form of he differeniaion propery Inegraion Propery Iniial and Final Value Theorem 15

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17 Examples 17

18 Examples 18

19 Examples 19

20 Examples Solving differenial equaions 20

21 Examples 21

22 Examples Transfer funcions 22

23 Examples Three Types of he Sysems 23

24 Some Signals and Their ROCs 24

25 Bilaeral LT: Causaliy The relaionship beween he locaions of poles and he impulse response in a causal sysem. (a) A pole in he lef half of he s-plane corresponds o an exponenially decaying impulse response. (b) A pole in he righ half of he s-plane corresponds o an 25 exponenially increasing impulse response. The sysem is unsable in his case.

26 Bilaeral LT: Sabiliy The relaionship beween he locaions of poles and he impulse response in a sable sysem. (a) A pole in he lef half of he s-plane corresponds o a righ-sided impulse response. (b) A pole in he righ half of he s-plane corresponds o an lef-sided impulse 26 response. In his case, he sysem is noncausal.

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28 Bilaeral LT: Boh Sable & Causal A sysem ha is boh sable and causal mus have a ransfer funcion wih all of is poles in he lef half of he s-plane, as shown here. 28

29 Inverse Sysems 29