Chapter 9 The Laplace Transform

Transcription

1 Chapr 9 Th Laplac Tranform 熊红凯特聘教授 hp://min.ju.du.cn 电子工程系上海交通大学 7

2 Topic 9. DEFINATION OF THE LAPLACE TRANSFORM 9. THE REGION OF CONVERGENCE FOR LAPLACE THANSFORMS 9. PROPERTIES OF THE LAPLACE TRANSFORM 9.4 THE INVERSE LAPLACE TRANSFORM 9.5 UNILATERAL LAPLACE TRANSFORM 9.6 ANALYSIS OF LTI SYSTEMS USING LAPLACE TRANSFORM

3 Topic 9. DEFINATION OF THE LAPLACE TRANSFORM 9. THE REGION OF CONVERGENCE FOR LAPLACE THANSFORMS 9. PROPERTIES OF THE LAPLACE TRANSFORM 9.4 THE INVERSE LAPLACE TRANSFORM 9.5 UNILATERAL LAPLACE TRANSFORM 9.6 ANALYSIS OF LTI SYSTEMS USING LAPLACE TRANSFORM

4 CT Fourir ranform nabl u o do a lo of hing,.g. Analyz frquncy rpon of LTI ym Sampling Modulaion Why do w nd y anohr ranform? On viw of Laplac Tranform i a an nion of h Fourir ranform o allow analyi of broadr cla of ignal and ym In paricular, Fourir ranform canno handl larg and imporan cla of ignal and unabl ym, i.. whn

5 Fourir Tranform of jω jω d Convrgnc condiion of h Fourir ranform : d < d < If h condiion ar no aifid,an anuaion facor numbr i Inroducd o ha σ σ i ral σ d <

6 ω σ ω σ ω σ σ j d d j j d if hn σ jω Th Fourir Tranform of σ -Laplac Tranform

7 ω ω σ π ω ω σ π ω σ ω σ d j d j j j 4 j j d j σ σ π Th Invr Fourir Tranform j d d ω Conidr Thn -Invr Laplac Tranform

8 Topic 9. DEFINATION OF THE LAPLACE TRANSFORM 9. THE REGION OF CONVERGENCE FOR LAPLACE THANSFORMS 9. PROPERTIES OF THE LAPLACE TRANSFORM 9.4 THE INVERSE LAPLACE TRANSFORM 9.5 UNILATERAL LAPLACE TRANSFORM 9.6 ANALYSIS OF LTI SYSTEMS USING LAPLACE TRANSFORM

9 Rgion of Convrgnc ROC -Th rang of valu of for which h ingral d convrg i.. find σ R.. d

10 Eampl: Eampl: α u α > if R α > α α α α d hn u, R[ ] α α u α > caual ignal α > α ani-caual ignal Th of h ignal i drmind by i prion and h rang of valu of h of h prion - h convrgnc domain ROC. α u, R[ ] α < α

11 Som ignal do no hav Laplac Tranform hav no ROC i dfind only in ROC; w do no allow impul in LT

12 Graphical Viualizaion of ROC I m [] S-plan α R [] α Rgion of Convrgnc Ai of Convrgnc α u, R[ ] > α α α u, R[ ] < α α

13 u u ] [, < < R Two-idd ignal Eampl: - -

14 Raional Laplac ranform. i raional funcion N --N,D polynomial in D. Zro om,.. N or, pol om,.. D or,. For raional Laplac ranform,h numbr of zro poin and ha of pol poin ar qual

15 Pol-zro plo of Im{ } jω -p -p z R{} σ Rgion of Convrgnc Ai of Convrgnc pol-zro plo of

16 Propry : Th ROC of coni of rip paralll o h jω-ai in h -plan. 的 ROC 在 平面上由平行于 jω 轴的带状区域组成 No: σ Sinc hi condiion only dpnd on h ral par of d <

17 Prory : If h Laplac ranform of i raional, hn i ROC i boundd by pol or nd o infiniy. In addiion, no pol of ar conaind in h ROC. 若 是有理的, 则其 ROC 被极点所界定或延伸到无限远, 且 ROC 内不包含任何极点. Eampl: Wih pol -, -, hr ar hr poibl ROC:

18 Propry : If i of fini duraion and i aboluly ingrabl, hn h ROC i h nir -plan. 若 是有限持续时间信号, 且绝对可积, 则其 ROC 是整个 平面 Eampl: T T

19 Propry 4: if i righ idd, and if h lin R{}σ i in h ROC, hn all valu of which R{}>σ will alo b in h ROC. 若 为右边信号, 则其收敛域将位于某个收敛轴 R{}σ 的右边 if i righ idd, and h Laplac Tranform of i raional, hn h ROC i h rgion in h -plan o h righ of h righmo pol. 若 是右边信号, 且 是有理的, 则其 ROC 位于最右边极点的右边 righ idd ignal:,wih<t Eampl: u u - T - -

20 Propry 5: if i lf idd, and if h lin R{}σ i in h ROC, hn all valu of which R{}<σ will alo b in h ROC. 若 为左边信号, 则其收敛域将位于某个收敛轴 R{}σ 的左边 if i lf idd, and h Laplac Tranform of i raional, hn h ROC i h rgion in h -plan o h lf of h lf mo pol. 若 是左边信号, 且 是有理的, 则其 ROC 位于最左边极点的左边 Lf idd ignal:,wih >T Eampl: u u T - -

21 Propry 6: if i wo idd, and if h lin R{}σ i in h ROC, hn ROC will coni of a rip in h -plan ha includ h lin R{}>σ. 若 为双边信号, 且 R{}σ 位于 ROC 内, 则其收敛域是包括收敛轴 R{}σ 的带状区域 Two idd ignal: Eampl: u u - -

22 Laplac Tranform v Fourir Tranform Whn jω loca in ROC, jω whn jω loca ouid of ROC, jω don ' i. jω

23 Eampl: δ, All u, R[ ] >

24 Topic 9. DEFINATION OF THE LAPLACE TRANSFORM 9. THE REGION OF CONVERGENCE FOR LAPLACE THANSFORMS 9. PROPERTIES OF THE LAPLACE TRANSFORM 9.4 THE INVERSE LAPLACE TRANSFORM 9.5 UNILATERAL LAPLACE TRANSFORM 9.6 ANALYSIS OF LTI SYSTEMS USING LAPLACE TRANSFORM

25 Many paralll propri of h CTFT, bu for Laplac ranform w nd o drmin implicaion for h ROC Linariy ROC a la h inrcion of ROC of and ROC can b biggr du o pol-zro cancllaion R R a b a b Eampl: ω in ω u R[ ] > ω co ω u R[ ] > ω

26 Eampl: ] [ ] [ > > R R ] [ > R No: h ROC can b biggr du o pol-zro cancllaion a -

27 Tim Shif R R Eampl: Aum: α u α α u?

28 Eampl: Aum: u Compu: u? u?

29 Shifing in h -Domain R Eampl: α ω α α ω α ω α ω ω α α > R > R ] [, co ] [, in u u ] [ R R

30 Conjugaion R R if i ral valud * *

31 Diffrniaion in h Tim-Domain R d d conaining R

32 Diffrniaion in h S-Domain R d d R or: d d

33 Eampl: u d d u! d d u!! n n n n n u n u α α α α α > R ] [,! u n n n

34 Ingraion in h Tim Domain R τ dτ R R[ ] >

35 Eampl: ' '' τ τ τ τ τ τ τ τ 4τ τ '' τ j jτ τδ 4 τδ τδ τ τ 4τ τ /

36 Inrgaraion in h S-Domain R λ dλ If whn, and lim α Eampl: u α u α α u - dλ ln λ λ α i α

37 Convoluion Propry R R H Y h y z z R R Eampl:

38 Th Iniial and Final-Valu Thorm mu b propr fracion 真分式!.<,., conain no impul or high ordr ingularii Iniial valu Thorm: lim lim Final-valu horm: lim lim

39 Th Taylor panion of a u n! n n! u n n n [ ] n n n / lim

40 Erci: Drmin h Laplac Tranform of h following ingal u u

41 Eampl:ingl id priodic ignal, < < T n nt and d T n T nt d T T nt T d d T [ nt ] T - T d nt T n

42 p Eampl:on id ampling ignal p p If n δ nt hn p n n n nt δ nt nt δ nt nt δ nt d d n nt nt

43 Topic 9. DEFINATION OF THE LAPLACE TRANSFORM 9. THE REGION OF CONVERGENCE FOR LAPLACE THANSFORMS 9. PROPERTIES OF THE LAPLACE TRANSFORM 9.4 THE INVERSE LAPLACE TRANSFORM 9.5 UNILATERAL LAPLACE TRANSFORM 9.6 ANALYSIS OF LTI SYSTEMS USING LAPLACE TRANSFORM

44 ω ω σ π ω ω σ π ω σ ω σ d j d j j j j j d j σ σ π j d d ω Conidr Thn -Invr Laplac Tranform Th Invr Fourir Tranform No: w hould choo any valu of σ in h RoC o ma h ingraion convrgd

45 Th invr Laplac ranform quaion a ha can b rprnd a a wighd ingral of compl ponnial Th formal valuaion of h ingral for a gnral rquir h u of conour ingraion 围线积分 in compl plan For h cla of raional ranform, h invr Laplac ranform can b drmind by uing h chniqu of parial-fracion panion.

46 n n m m n n n n m m m m p p p b z z z a n m b b b a a a B A < P whr m z z n p p ar h zro of ar h pol of ricly propr raional funcion 真分式 polynomial P Impul funcion and i diffrnial 冲激函数及其各阶导数

47 Rcall α α α > R ] [,! u n n n α ω α α ω α ω α ω ω α α > R > R ] [, co ] [, in u u

48 n b n n n A p p p p p p Suppo : n pol ar all diinc n 个极点都是单极点 < R > R i p i i p i i i p u p u p i i

49 How o drmin h conan i. To quaing cofficin of qual powr of 对应项系数平衡相等. p i, n i i p i

50 Eampl:

51 oluion: oluion:

52 ] R[ < and R 5 6 u u R 5 6 u u 5 6 u u δ δ For h varid ROC: 5 6

53 : i h -h ordr pol p D E p p p D p A i i p n i n i i p u n p i > R ] [,!

54 Tha i:! p whr d d i p i i i p p p d d d d How o drmin h conan i

55 Eampl: d d d d d d

56 And hu If R > u u u u u u

57 ha compl pol pair α ω ω α α α ω ω α ω α α > R > R ] [, co ] [, in u u

58 Eampl: If ] R[ > in ' u δ δ

59 Eampl: 4 L hn 4 No: by ploiing h propri of h Laplac ranform

60 whr

61 To quaing h cofficin of qual powr of Thu, 4 4 co u u

62 Eampl: ] R[ > whr nt T n No:

63 δ δ ] [ n n n n n n n n δ δ δ

64 Informaiv Invr Laplac Tranform by man of Conour Ingraion

65 If z m i h -h ordr pol of n z z m z m z n m z z n z z z z dz d z z } [ {! ] [ R hn

66

67 Topic 9. DEFINATION OF THE LAPLACE TRANSFORM 9. THE REGION OF CONVERGENCE FOR LAPLACE THANSFORMS 9. PROPERTIES OF THE LAPLACE TRANSFORM 9.4 THE INVERSE LAPLACE TRANSFORM 9.5 UNILATERAL LAPLACE TRANSFORM 9.6 ANALYSIS OF LTI SYSTEMS USING LAPLACE TRANSFORM

68 Dfinaion No: d d > πj ROC i alway on h righ id of h righmo pol Th unilaral Laplac ranformaion only conidr h ignal a >, bu ndn b, a < If i caual ignal< 时,,h unilaral Laplac ranform i h am a Laplac ranform.

69 α α > R ] [, u α Laplac ranform: α α α α α α > R ] [, d d Eampl: Unilaral Laplac ranform:

70 Propri Convoluion For all < Diffrniaion in h im domain d d

71 n n n n n m m m n n n Solving Linar Conan Cofficin Diffrnial Equaion wih Nonzro Iniial Condiion Convring diffrnial quaion ino algbraic quaion Dircly olv h compl rpon, and a h am im find h zro-inpu rpon and zro a rpon. Applicaion

72 For an LTI ym rprnd by LCCDE N a d y M d b d d N Wih iniial condiion: {, ',, } y y Th inpu aifi: whn <, y hn

73 N b p p p b y a Y a ] [ ] [ ] [ B M Y A A B A M Y M p p p y Y a ] [ A M B ZIR Rpon for zro inpu ZSR Rpon for zro a, iniially a r

74 Eampl: y y y 6 wih u y, y To drmin y, y, y zi z 6 Y y y Y y Y i.. Y [ y y y ] 6 y y' y 6 Y

75 , y y y Y zi zi y u Y z z u y y y z zi Subiuing, and in

76 Topic 9. DEFINATION OF THE LAPLACE TRANSFORM 9. THE REGION OF CONVERGENCE FOR LAPLACE THANSFORMS 9. PROPERTIES OF THE LAPLACE TRANSFORM 9.4 THE INVERSE LAPLACE TRANSFORM 9.5 UNILATERAL LAPLACE TRANSFORM 9.6 ANALYSIS OF LTI SYSTEMS USING LAPLACE TRANSFORM

77 Sym funcion of LTI ym.dfiniion h y y h y Y H h d Sym Funcion Y H / H Y

78 . H can fully dcrib a ym. Phyical maning A baic ignal, Saifi: A fairly broad cla of ignal can b rprnd by a "linar combinaion of Th rpon of h ym o i impl and h rpon of any ignal can b prd by h "linar combinaion" of h rpon of

79 Conidr inpu : H d h d h d h h y * τ τ τ τ τ τ τ τ < <, α α α H y ; i calld ignfuncion H i calld ignvalu

80 Conidr any inpu: a H a h a h y ] * [ * H giv h chang of h ampliud and pha of any compl frquncy componn hrough LTI ym. H h y d j π d H j π

81 Sym Funcion v Diffrnial Equaion Eampl: 6 d d y d d y d d Taing LP of boh id 6 6 Y Y H Y Y No:W only obain h algbraic prion of H wihou ROC by h diffrnial quaion. Bcau h diffrnial quaion ilf can no fully characriz h LTI ym, ohr conrain mu b addd, uch a caualiy, abiliy, and o on. Known diffrnial quaion, ing ym funcion

82 Eampl: u - Compu H and diffrnial quaion. ] R[ ] R[ > > Y d d y y d d y d d Y H For LTI ym ] [ u y Th zro-a rpon of i

83 Eampl: For an LTI ym wih h am iniial condiion, Wih h inpu δ, h full ym rpon y u Wih h inpu u, h full ym rpon y u To drmin h ym impul rpon h and i diffrnial quaion. No: Full rpon zro-inpu rpon zro-a rpon

84 H u h d d y y d d y d d H Y z And Thn Y Y u y z zi Y Y u y z zi Bing

85 Eampl: For an LTI ym wih zro iniial condiion, i.. iniial r 4 u y δ a u. whn,. For all,, y To drmin: a Cofficin a and h diffrnial quaion of h ym. b Th ym rpon o h inpu 4 u

86 a 4, a Y a a Y H So H-, Subiuing i in, on obain a7/4 and H y H y

87 ba iniial r H Y H u u y K K K K H Y δ

88 Sym Funcion v Caualiy and Sabiliy Caualiy. Dfiniion - for any ym. <,h for LTI ym. Sym wih raional ym funcion H,caualiy Th ROC of H i on h righ id of h righmo pol For a ym wih a raional ym funcion, caualiy of h ym i quivaln h ROC bing h righ-half plan o h righ of h righmo pol. No: If H i no raional, h abov quivalnc i no ncarily ru.. Th caualiy of h LTI ym Th ROC of h ym funcion H i locad o h righ of h righmo pol and conain

89 Sabiliy. Dfiniion for any ym h d <. --- for LTI ym. Th abiliy of LTI ym Th ROC of h ym funcion H includ h jω ai An LTI ym i abl if and only if h ROC of h ym funcion Hinclud h nir jωai

90 Caualiy & Sabiliy A caual ym wih raional ym funcion H i abl if and only if all h pol of H li in h lf-half of h -plan

91 Eampl: 6 H,i caualiy, abiliy, and calcula h 5 5 H If ROC i R[]> caual, unabl ym 5 5 u h If ROC i -<R[]< Non-caual, abl ym 5 5 u u h If ROC i R[]<- Non-caual, unabl ym 5 5 u h Thr ar poibl RoC, i.. diffrn ym

92 Chcing if All Pol Ar In h Lf-Half Plan N H D Pol ar h roo of D n a n- n- aa Mhod #: Mhod #: Calcula all h roo and! Rouh-Hurwiz Wihou having o olv for roo. Fir - ordr Scond - ordr Third - ordr Polynomial a a a a a a Condiion o ha all roo ar in h LHP a > a and a >, a > >, a >, a > a < a a

93 Eampl: Suppo w hav h following informaion abou an LTI ym: Th ym i caual, Th ym funcion i raional and hav only wo pol a -,4 If, hn y 4 Th valu of impul rpon a i 4, i.. h 4 To drmin h ym funcion of h ym H Soluion: From h fir fac, w now ha h ROC of h ym i drawn in h righ figur. Th ym i caual and unabl

94 A H i raional, w hav h ym funcion of h following form 8 4 p p H From fac, H H y H H mu hav a zro a, i.. p q Finally, from fac 4 and iniial-valu horm, w obain ha 4 8 lim lim q H h W conclud ha

95 A, If h numraor ha highr dgr han h dnominaor, h limi will divrg If h numraor ha lowr dgr han h dnominaor, h A non-zro limi i only whn hy hav h am dgr! q h lim 4 8 i.. 4 Thu: H 4 8

96 Sym Funcion v Bloc Diagram Rprnaion Th baic lmn of h bloc diagram: Addr, conan cofficin muliplir and ingraor. Known ym funcion, for ym diagram Eampl: Y H a ay Y [ a Y ] y a

97 Eampl: b H b a a l z y z b z a ', z z b y a No: Th inpu of h ingraor i h diffrnial of h oupu ignal

98 Eampl: H Y Y Y Y [ Y Y ] No: cond-ordr ym mu hav wo ingral bloc y y y Dirc form

99 Eampl: H y Cacad form

100 Eampl: H y Paralll form

101 Eampl: H l z hn y z z ', y z z z Dirc form Eci: Draw i cacad and paralll form

102 . Known ym bloc diagram, ing ym funcion 4 6 y z z z Y 6 Z 4 Z Y Z 4 6 and Z Z Z Y Y Y i Y 4 6 Y 4 6 hn H

103 BASIC PROBLEMS: 9., 9., 9., 9., 9.7 PROBLEMS!

104 Q & A Many Than