Chapter 10 z Transform

Transcription

1 Chapter 0 Trasfor 熊红凯特聘教授 电子工程系上海交通大学 07

2 DT Fourier trasfor eables us to do a lot of thigs, e.g. Aale frequec respose of LTI sstes Modulatio Wh do we eed et aother trasfor? Oe view of Laplace Trasfor is as a etesio of the Fourier trasfor to allow aalsis of broader class of sigals ad sstes I particular, Fourier trasfor caot hadle large ad iportat classes of sigals ad ustable sstes, i.e. whe eet. ca ot

3 How do we aale such sigals/sstes? Recall the eigefuctio propert of LTI sstes: h H Eigefuctio for DT LTI H h assuig it coverges is a eigefuctio of a DT LTI sste We ow do ot restrict ourselves just to = e jω

4 Topic 0. Bilateral Trasfor 0. Properties of RoCs 0. Trasfor Properties 0.4 Iverse Trasfor 0.5 Aalsis of LTI Sstes usig the T 0.6 Uilateral Trasfor ad Applicatios

5 Topic 0. Bilateral Trasfor 0. Properties of RoCs 0. Trasfor Properties 0.4 Iverse Trasfor 0.5 Aalsis of LTI Sstes usig the T 0.6 Uilateral Trasfor ad Applicatios

6 7.. The Bilateral Trasfor Z{ } is a cople variable Now we eplore the full rage of Basic ideas Epress the cople variable i polar for as =re jω re j j re r e j re jω is the Fourier trasfor of ultiplied b a real epoetial r-, if it eists. For r=, the trasfor reduces to the Fourier trasfor.

7 I geeral, the trasfor of a sequece has associated it with a rage of values of for which coverges, ad this rage of values is referred to as the regio of covergece ROC j ROC re at which r depeds ol o r =, just lie the ROC i s-plae ol depeds o Res If the ROC icludes the uit circle, the the Fourier trasfor also coverges

8 Eaple : a u - right -sided a a arbitrar real or cople uber - a u This for for PFE ad iverse - trasfor 0 a a a If a,i.e., a This for to fid pole ad ero locatios That is, ROC > a, outside a circle

9 Eaple : -sided - left u a a u a 0 a a.,., If a e i a Sae as i previous eaple, but differet ROC Ke Poit ad e differece fro FT: Need both ad ROC to uiquel deterie. No such a issue for FT.

10 7.. Graphical Visualiatio of the ROC Eaple Eaple a u - right -sided a u - left -sided

11 Ecept for a scale factor, a coplete specificatio of a ratioal trasfor cosists of the pole-ero plot of the trasfor, together with its RoC 7.. Ratioal Trasfors Ma but b o eas all trasfors of iterest to us are ratioal fuctios of s e.g., Eaple ad ; i geeral, ipulse resposes of LTI sstes described b LCCDEs, where = N/D, N,D poloials i Roots of N= eros of Roots of D= poles of A cosistig of a liear cobiatio of cople epoetials for > 0 ad for < 0 e.g., as i Eaple # ad # has a ratioal trasfor.

12 Topic 0. Bilateral Trasfor 0. Properties of RoCs 0. Trasfor Properties 0.4 Iverse Trasfor 0.5 Aalsis of LTI Sstes usig the T 0.6 Uilateral Trasfor ad Applicatios

13 ROC j re at which r Propert : The ROC of cosist of a rig i the -plae cetered about the origi Proof: I order to ae the F.T of r coverged, it should be absolutel suable, i.e. the RoC ol depeds o the radius of Propert : The ROC does ot cotai a poles. If is ratioal, the its ROC is bouded b poles or eteds to ifiit.

14 Propert : If is of fiite duratio, the the ROC is the etire -plae, ecept possibl =0 ad/or = N whe N<N<0, the RoC is the etire -plae, ecludig = : 0 whe N>N>0, the RoC is the etire -plae, ecludig =0 0 whe N<0,N>0, the RoC is the etire -plae, ecludig =0 ad = 0 N

15 Propert 4: If is a right-sided sequece, ad if = r 0 is i the ROC, the all fiite values of for which > r 0 are also i the ROC N N r coverges faster tha r 0

16 Propert 5: If is a left-sided sequece, ad if = r 0 is i the ROC, the all fiite values of for which 0 < < r 0 are also i the ROC. If is two-sided, ad if = r 0 is i the ROC, the the ROC cosists of a rig i the -plae icludig the circle = r 0. What tpes of sigals do the followig ROC correspod to? right-sided left-sided two-sided

17 Eaple: 0, b b u b u b b b u b b b u b,, Fro:

18 b b, b b Clearl, ROC does ot eist if b > No -trasfor for b.

19 If is ratioal, the its ROC is bouded b poles or eteds to ifiit. I additio, o poles of are cotaied i the ROC. Suppose is ratioal, the If is right-sided, the ROC is to the right of the outerost pole. If is left-sided, the ROC is to the left of the ierost pole. If ROC of icludes the uit circle, the FT of eists.

20 Topic 0. Bilateral Trasfor 0. Properties of RoCs 0. Trasfor Properties 0.4 Iverse Trasfor 0.5 Aalsis of LTI Sstes usig the T 0.6 Uilateral Trasfor ad Applicatios

21 Liearit a b a b ROC cotais the itersectio of R ad R, which a be ept, also ca be larger tha the itersectio Eaple: 0 cos, j0 j0 e e cos j0 j0 cos u e e u 0 0 si si, 0 0 u cos 0 R R

22 Tie Shift RoC=R, ecept for the possible additio or deletio of the origi or ifiit. Eaple: suppose u To deterie the trasfor of the followig sigal: j j g G G G j j g g j j ROC of G is that of, ecludig = But if = is eros of, the RoC of G is idetical to that of

23 Tie Reversal Tie Epasio for the

24 Scalig i the _Doai

25 Cojugatio Cosequece, if is real, the Thus, if has a pole/ero at = 0, it ust also have a pole/ero at the cople cojugatio poit = 0 *

26 The Covolutio Propert ROC cotais the itersectio of R ad R, which a be ept, also ca be larger tha the itersectio * j g j j u j u j g j j j j Eaple:

27 Differetiatio i the Doai ROC R d d ROC R Eaple: If log a a Let The followig: d log d

28 d a d a a u a a a u a a a a a u a 即 : a u thus a u a log Siilarl: a u log a

29 The Iitial-Value Theore If is a casual sequece, i.e. whe <0, =0, the 0 li Hits to prove the iitial-value theore: accordig to the defiitio of the Z trasfor Fial-value theore If is a casual sequece, i.e. whe <0, =0, ad the RoC cotais the uit circle, the li li Hits to prove the fial-value theore: Z{+-}=Z{+}-Z{} =-0- ad rewrite Z{+-} accordig to the defiitio of the Z Trasfor

30 Topic 0. Bilateral Trasfor 0. Properties of RoCs 0. Trasfor Properties 0.4 Iverse Trasfor 0.5 Aalsis of LTI Sstes usig the T 0.6 Uilateral Trasfor ad Applicatios

31 7.4. The Iverse Trasfor ROC }, { j j re r F re d e re re r j j j } { F d e r re j j d j d d jre d re j j d j for fied r: We should choose a value of r i the RoC to ae the itegratio coverged

32 The sbol O deotes the itegratio aroud a couterclocwise circular cotour cetered at the origi ad with radius r. The iverse trasfor equatio states that ca be represeted as a weighted itegral of cople epoetials The foral evaluatio of the itegral for a geeral requires the use of cotour itegratio 围线积分 i cople plae There are two alteratives to obtai a sequece fro its trasfor: Partial fractioal epasio: Power series epasio:

33 Cotour Itegratio Method c s d j Re Where C is the couterclocwise circular cotour cetered at the origi ad withi the ROC of, is the pole of - at the left-had side of C. If is the s-th order pole of -, the s s s d d s s } {! Re

34 Eaple: 5 7

35 whe >, is a causal sequece, G Let As show i the figure, the poles at the left-had side of C are, 5 5 u / C The Thus:

36 Whe /< <, is a double-side sequece 5 G whe 0, there is oe pole at the left-had side of C: 5 / C

37 Whe =-, 5 G ad 0 The poles at the left-had side of C are Both of the are st order poles

38 Whe =-, 5 G At the left-had side of C, besides the st order pole, there is a d order pole: ! 5 d d Siilarl, whe =-, u u u u I suar,

39 For <0, i order to avoid calculatig the residual at =0, we ca calculate the cotour itegratio alog the clocwise circular cotour C, which is cetered at the origi ad withi the ROC of j c' Re s d the pole of - at the left-had side of C, i.e. the right-had side poles of C As i the previous eaple, whe <0, there is ol oe pole at the right-had side of C: =, the 5 5 Re s 0 C' C /

40 Whe </, is a ati-causal sequece 5 G As show i the figure, =0 is the -th order pole at the left-had side of C, whereas the poles at the right-had side of C are, The Re 5 Re s s u / c

41 PFE Partial fractioal epasio Method Suppose b0 b b a a a 0... b... a r r r It has M st order poles at Ad a s-th order pole at,,,...m i

42 M 0 B s j j j i A A 0 0 A A i s i j s j j d d j s } {! B s!... u a u a a!... where Notes:

43 Eaple: C C C A C A j j j j C A B, B, B=,4,6

44 A,C Substitutig i C C Equatig coefficiets, oe obtais 6 C C, a u a a!...! u u The

45 Power-Series Epasio is irratioal: Eaple: arctg arctg arctg u

46 is ratioal N M Perforig log divisio approach to obtai a power series of for, the coefficiets i this power series are the sequece of. R is a right-sided sequece, arrage the uerator ad deoiator with a order of the power of decreasig R is a left-sided sequece, arrage the uerator ad deoiator with a order of the power of icreasig

47 Eaple: ' ' u Notes: Power-series epasio a coe to ope/ifiit epressio

48 Topic 0. Bilateral Trasfor 0. Properties of RoCs 0. Trasfor Properties 0.4 Iverse Trasfor 0.5 Aalsis of LTI Sstes usig the T 0.6 Uilateral Trasfor ad Applicatios

49 7.5. DT Sste Fuctio h h Y = H, ROC at least the itersectio of the ROCs of H ad, ca be bigger if there is pole/ero cacellatio. e.g. H Y H h The Sste, a a, Fuctio ROC all H + ROC tells us everthig about sste a

50 7.5. CAUSALITY h right-sided ROC is the eterior of a circle possibl icludig = : H h ROC outside N a circle N If N 0, the the rer h N, but does ot at iclude. Causal N 0 No ters with >0 =>= A DT LTI sste with sste fuctio H is causal the ROC of H is the eterior of a circle icludig =

51 Causalit for Sstes with Ratioal Sste Fuctios N M a a a a b b b b H N N N N M M M M if, at No poles 0 0 A DT LTI sste with ratioal sste fuctio H is causal a the ROC is the eterior of a circle outside the outerost pole; ad b if we write H as a ratio of poloials the D N H degree degree D N

52 7.5. Stabilit LTI Sste Stable h icludes the uit circle = ROC of H Frequec Respose He jω DTFT of h eists. A causal LTI sste with ratioal sste fuctio is stable all poles are iside the uit circle, i.e. have agitudes <

53 H Eaple: Deterie h, ad the causalit ad stabilit of the sste u u h, Causal, ustable u u h, No-causal, ustable u u h, No-causal, stable

54 7.5.4 LTI Sstes Described b LCCDEs Use the tie-shift propert M N b a 0 0 N M b Y a 0 0 N M a b H H Y 0 0 Ratioal ROC: Depeds o Boudar Coditios, left-, right-, or two-sided. For Causal Sstes ROC is outside the outerost pole

55 Sste Fuctio vs Differece Equatio Suppose whe the iput is u the ero-state respose of the sste is 9 4 u To deterie h ad the differece equatio of the sste

56 Y H H u h 9 4 Y

57 H 后向差分 H 前向差分

58 Eaple: The iput-output relatioship i tie doai for the causal sste could be represeted b a d order costcoefficiet differece equatio, if the iput is u The sste respose ero state respose is g 5 0 u the If the sste is with ero-iitial state, deterie the d order differece equatio If the iitial state of the sste is -=,-=, deterie the ero-iput respose of the sste with the d order differece equatio obtaied i step If the iitial state of the sste is -=,-=4,, the iput is u u 5, deterie the full respose of the sste

59 the G H G

60 H 系统特征根为 =, =5 The characteristic values are 设零输入响应为 The ero-iput respose ca be Substitutig The c c 5 i c c 5 c 将 代入 i c 5 c c 4 5 则 i 55

61 Sice the sste could be described b a d order LCCDE, it is a LTI sste, satisfig liear ero-iput, ad liear ero-state properties: whe 当, 4时 i 55 whe 当 u u 5 时 g g 5 s i s

62 Eaple: For a, whe Whe u,, =0 a u 4 Deterie: the coefficiet a, Whe for all, =, deterie

63 4 4 4 a Y H a Y whe = 4 = H , a H H whe the

64 Topic 0. Bilateral Trasfor 0. Properties of RoCs 0. Trasfor Properties 0.4 Iverse Trasfor 0.5 Aalsis of LTI Sstes usig the T 0.6 Uilateral Trasfor ad Applicatios

65 7.6. Uilateral -Trasfor UZT Note: If = 0 for < 0, the UZT of = BZT of u ROC alwas outside a circle ad icludes = For causal LTI sstes 0 H The calculatio of the iverse UZT is basicall the sae as that for BZT, with the costrait that the RoC for a UZT ust alwas be the eterior of a circle. Differs fro BZT

66 7.6. Properties of Uilateral -Trasfor Ma properties are aalogous to properties of the BZT e.g Covolutio propert for <0 = <0 = 0 But there are iportat differeces. For eaple, tie-shift UZ Iitial coditio Y 0 0 Y 0

67 u More geeral 0 u u Proof: 0 0 u u u

68 7.6. Use of UZTs i Solvig Differece Equatios with Iitial Coditios For a LSI sste Assue =0 whe <0, the iitial coditio of the sste is {-,-,,-}, the N M b a B b M l a A Y a M N l l N A B A M Y B M Y A Zero-iput respose Zero-state respose

69 Eaple:, UZT of Differece Equatio u ZIR ZSR Y { } UZ Y Y ZIR ZSR Output purel due to the iitial coditios, Output purel due to the iput.

70 β = 0 Sste is iitiall at rest: ZSR α = 0 Get respose to iitial coditios ZIR H H H H Y Y u

71 Eaple: with -=,-=-/ =u deterie,, s i Y Y Y Y Y

72 Substitutig -=,-=-/, ad I Y, oe obtais 4 Y 4 u Y u Y s i 4 u s i

73 Hoewor BASIC PROBLEMS WITH ANSWER: 0. BASIC PROBLEMS: 0., 0., 0.4, 0.6, 0.4a ADVANCED PROBLEMS: 0.46