Z-Transform of a discrete time signal x(n) is defined as the power series

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2 Z-Trsform of discrete time sigl x is defied s the ower series x 3.. This reltio lso clled direct Z-Trsform. Z[ x ] 3.. x 3..3 Regio of covergece ROC of is the set of ll vlues of for which ttis fiite vlue. ROC of fiite-durtio sigl is the etire -le, excet ossible the oits =, d/or =. for for

3 Give 3..4 I the ROC of, <, but 3..5 To fid the ROC, exress s 3..6 If the first sum i 3..6 coverges, vlues of r must smll eough such tht the x-r, <, is bsolutely summble.

4 If the d sum i 3..6 is coverges, vlues of r must lrge eough such tht the x/r, <, is bsolutely summble. b c r < r < r is the commo regio where both sums i 3..6 is fiite. For r > r, there is o commo regio of covergece for the two sums, does ot exist. Oe side or uilterl -trsform is defied s

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6 The iverse Z-trsform rocess for trsformig from -domi to the time domi Let us begi with the s-trsform Multily both side with - d itegrte both side over close cotour withi the ROC of which eclose the origi ow, iterchge the order of itegrtio d summtio o the right hd side Cuchy itegrl theorem sttes tht C is y cotour tht ecloses the origi. Hece, the desired iversio formul is

7 3. Proerties of Z-Trsform Lierity If d the Timig shiftig If the

8 Sclig i the Z-domi If the Proof: ROC of is r < < r, the ROC of - is r < - < r or r < < r

9 Time reversl If the Proof: where l = -. The ROC is s r < - < r or r < < r

10 Differetitio i the -domi If the Proof: ote tht both trsforms hve the sme ROC.

11 Covolutio of two sequeces If the Proof: Covolutio of x d x is Z-trsform of x is By iterchgig the order of summtios d ly the time shiftig roerty, we get

12 Correltio of two sequeces If the Proof: Recll tht, By usig covolutio d time-reversl roerties, we get The ROC of R xx is t lest the itersectio of tht for d -.

13 Multilictio of two sequeces If the Proof: Z-trsform of x is Let us substitute the iverse trsform for i the d iterchge the order of summtio d itegrtio. Thus, we get Lets sy coverges for r l < v < r u d coverges for r l < < r u, the ROC for is t lest r l r l < < r u r u

14 Prsevl s reltio If x d x re comlex-vlued sequeces, the rovided tht r l r l << r u r u, where r l < < r u d r l < < r u re the ROC of d The iitil vlue theorem If x is cusl [i.e., x= for <], the Proof: x is cusl, hece s, - sice >.

15 3.3 RTIOL Z-TRSFORM Rtiol fuctio is rtio of two olyomils i Z - or Z. Tht is very imortt issues whe lyig d desigig LTI system. where d D re olyomil i Z -. If d b, we c void the egtive owers of by fctorig out terms b -M d - s follows:

16 rtiol -trsform c be ltertely writte i fctored form s where ote G hs M fiite eros d fiite oles. If > M there re dditiol eros t = the origi i the -le If < M there re dditiol oles t = Poles or eros my lso occur t =. i ero exists t = if =. ii ole exists t = if =. If the oles d eros t ero d ifiity, hs exctly the sme umber of oles s eros.

17 Poles d Zeros The roots of umertor i the rtiol fuctio will be eros of Z-trsform. The roots of deomitor i the rtiol fuctio will be oles of Z-trsform. = eros of. = oles of.

18 Some Commo Z-Trsform Pirs

19 Pole ero lot The oles d eros c be lotted o comlex -le. The loctio of oles showed by crosses x d loctio of eros by circles. By defiitio, the ROC -trsform should ot coti y oles.

20 Pole Loctio d Time-domi Behvior for Cusl Sigl The chrcteristic behvior of cusl sigls deeds o whether the oles of the trsform re cotied i the regio ІІ<, or i the regio ІІ>, or o the circle ІІ=. The circle ІІ= hs rdius of which is clled the uit circle. Cusl rel sigls with simle rel oles or simle comlex cojugte irs of oles, which re iside or o the uit circle re lwys bouded i mlitude. It should be stress tht everythig sid bout cusl sigls lies s well to cusl LTI systems, sice their imulse resose is cusl sigl. If ole of system is outside the uit circle, the imulse resose of the system becomes ubouded d cosequetly the system is ustble.

21 Time-domi behvior- Sigle rel-ole cusl sigl. cusl rel sigl with oe rel oles hs the form with ROC > Oe ero t = d oe ole t =

22 Time-domi behvior- double rel-ole cusl sigl. cusl rel sigl with rel oles hs the form with ROC > Oe ero t = d oles t = =

23 Time-domi behvior- sigle rel-ole cusl sigl. The sigl is decyig if the ole is iside the uit circle =. The sigl is costt if the ole is o the uit circle =. The sigl is costt if the ole is o the uit circle = -. The sigl is divergig if the ole is outside the uit circle. Time-domi behvior- double rel-ole cusl sigl. The sigl is bouded if the oles re iside the uit circle =. The sigl is ubouded if the oles lie o or outside the uit circuit.

24 Time-domi behvior - Comlex-cojugte oles cusl rel sigl with comlex-cojugte oles hs the form There exist ero d ir of comlex- cojugte ole

25 The distce r of the oles from the origi determies the eveloe of the siusoidl sigl d their gle with the rel ositive xis ir of comlex-cojugte oles: i The mlitude of the sigl is growig if r>. ii The mlitude of the sigl is decyig if r<. iii The mlitude of the sigl is costt if r=. Summry: Cusl rel sigls with simle rel ole or simle comlex-cojugte irs of oles, which re iside or o the uit circuit re lwys bouded i mlitude. sigl with ole or comlex-cojugte irs of oleser the origi decy more ridly the ssocited with ole er but iside the uit circuit Time behvior of sigl strogly deed o the loctio of ole Zero lso ffect the behvior of sigl but ot s strogly s oles

26 The System Fuctio of Lier Time-Ivrit LTI System From covolutio roerty Y = h*x By exress this reltioshi i the -domi s Y = H Hece, we c determie the y by iverse the y if we ow H d We lso determie the uit imulse resod H if we ow x d observe the outut y d the evlutig the iverse -trsform of H get h domis. d h re equivlet descritios of system i two H is clled the system fuctio.

27 The System Fuctio of Lier Time-Ivrit LTI System For Lier costt-coefficiet differece equtios: by ly shiftig roerty

28 The System Fuctio of Lier Time-Ivrit LTI System If = for H become ll-ero system - FIR system/m movig vergesystem. If b = for M; H become ll-ole system - IIR system. Geerl form of the system fuctio cotis both oles d eros. Hece, it clled ole-ero system with oles d M eros. Due to the resece of oles, ole-ero system is IIR system.

29 Iverse of -trsform is give by x j C d The itegrl is cotour itegrl over closed th C tht ecloses the origi d lies withi the regio of covergece. For simlicity, C c be te s circle i the ROC of i the -le. There re three methods to evlute the iverse Z- trsform:. Direct evlutio by cotour itegrtio;. Exsio ito series of terms, i vribles d - ; 3. Prtil frctio exsio d tble loo-u;

30 Cuchy residue Theorem: Let f be fuctio of the comlex vrible d C be closed th i the -le. If the derivtive df dexists o d iside the cotour C d if f hs o oles t, the f f, if is iside C d j C, if is outside C More geerlly, if the +-order derivtive of d f hs o oles t, the d f f d, C! d j, if if f exists is iside C is outside C

31 I more geerlied form, the itegrd of the cotour itegrl is P where f hs o oles iside the cotour C d g is olyomil with distict simle roots The vlues,,, j C f g i i i, i,,,. iside C. The i d C d j i i i C d i j i i i i where i f g i P f i g re residues of the corresodig oles t The vlues of the cotour itegrl is equl to the sum of the residues of ll oles iside the cotour C.

32 i I cse tht the oles x j ll i oles isidec i C residue i of re simles, d i t i If hs o oles iside the cotour C for oe or more vlues of, the x for these vlues.

33 c be exressed by the followig ower series exsio: c x c c c c c Ex ROC : b ROC :.5

34 x This is cusl cse for which is cosidered for egtive ower series. b This is ti-cusl cse for which is cosidered for ositive ower series x x x

35 K K x x x x K K M M M M b b b b b b D M M b b b M d,,

36 M M b b b Distict Poles: x l l Multile-order Poles: If hs ole of multilicity l, the it cotis i its deomitor the fctor. i

37 Ex Solutio d d

38 Multile-order Poles: Ex Solutio - cotiued sigl cusl ti ROC if u sigl cusl ROC if u Z :, :, u Z u r Z cos * * *, u x * * j e j e

39 r M M b b M c r M If the oles of re distict, it c be exressed i rtil frctios: r

40 3.5 Oe-sided -trsform The two-sided -trsform requires the corresodig sigls be secified for the etire rge -<<. The systems re described by differece equtios with oero iitil coditios. Whe the iut is lied t fiite time, both iut d outut sigls re secified for. Therefore the two-sided -trsform cot be used. Rther oe-sided -trsform is used.

41 Oe-sided -trsform of sigl x is defied by Or x x 3.5.

42 Does ot coti iformtio bout the sigl x for egtive vlues of time i.e.,for < Uique oly for cusl sigl, bcuse oly these sigls re ero for < Oe sided -trsform of x is ideticl to two-sided -trsform of the sigl xu. The ROC of,is lwys the exterior of circle sice xu is cusl. Therefore, it is ot ecessry refer to to their ROC whe we del with oe-sided -trsform.

43 lmost ll roerties for the two-sided - trsform lso ly i the oe-sided - trsform excet shiftig roerty.

44 Cse : Time Dely If The, > 3.5. x x x

45 If cse x is cusl, the Proof : From defiitio 3.5. By chgig the idex from l to = -l, the result 3.5. is obtied. x l l l l l x l x x l l l x ssume l=-

46 If we write 3.5. s follows: > We should shift x by smles to the right i order to obti x-> from x. Z x x x x...

47 Cse : Time dvce If The > Proof: Where we chge the idex of summtio from to l=+ x x x l l Z l x x x ssume l=+

48 From 3.5., we obti l l By combiig the lst two reltios, we c obti l We shift the x by smles to the left to obti x+,>. The smles x, x,, x- t the egtive xis. We remove the their cotributio to the, d multily wht remis by to comeste for the shiftig of the sigl by smles. l x l x l x l l l

49 Imortt theorem useful i the lysis of sigls d systems. x If The lim lim x The limit i exists if the ROC of - icludes the uit circle.

50 Oe-sided -trsform is very efficiet tool for the solutio of differece equtios. Reduce the differece equtio reltig to two time-domi sigls to equivlet lgebric equtio reltig their oe sided -trsform. The sigl i the time domi is obtied by ivertig the resultig -trsform.

51 The well ow Fibocci sequece of iteger umbers is comutig ech term s the sum of the two revious oes. The first few terms of the sequece re,,,3,5,8 Determie the closed form exressio for the th term of the Fibocci sequece.

52 Solutio: Let y be the th term of the Fibocci sequece. Clerly, y stisfies the differece equtio y y y With iitil coditios y y y y y y b From 3.5.8b, we hve y- =. The, y- = Thus, we determie y,, which stisfied with iitil coditios y- = d y- =.

53 Tig oe sided -trsform d shiftig roerty of or We hve y- = d y- = y y y Y Y Y y y y Y

54 We c ivert the by the rtil frctio exsio method. The oles re The corresodig coefficiets re & Therefore, or equivletly Y u y u y

55 Exmle Determie the ste resose of the system y y x - < α < Whe the iitil coditio is y- = Solutio: Tig oe sided -trsform of both sides, Y Y Substitute y- d We obti the result. Y y d solvig for Y

56 Usig rtil frctio exsio d iverse trsformig the result : u u y u

57 3.6. Resose of System with Rtiol System Fuctios H B Y Let us ssume tht the iut sigl x hs rtiol -trsform of the form Q If the system is iitilly relxed, i.e. y-=y-=y-3= =y- =, the outut Y H B Q Suose tht the system cotis simle oles,,, d cotis oles q, q,, q L, where q m for ll =,,, d m =,,, L. ssume tht the eros of B d do ot coicide with the oles { } d {q }, so tht there is o ole-ero ccelltio.

58 L q Q Y The outut, L u q Q u y Iverse trsform of Y, turl resose Forced resose The scle fctors { } d {Q } re fuctios of oles { } d {q }. For exmle, if is the Y is, d cosequetly the outut is ero. Whe d H hve oe or more oles i commo or whe d/or H coti multile-order oles, the Y will hve multile-order oles. Cosequetly, Y cotis fctors of the form /- l -, =,,, m, where m is the ole-order.

59 3.6. Resose of Pole Zero System with oero Iitil Coditios The oe-sided -trsform: M b y Y Y M y H y b Y where Sice x is cusl, + = ero iutresose Y s resose ero iut Y i

60 L i i s D u q Q u y u D y y y y ' ', where sice Thus

61 3.6.3 Trsiet d Stedy-Stte Resoses Theturlresose of cuslsystem, where{ { y r },,,...,re theoles of the system d } rescle fctors. u If - - for ll, the y r trsietresose decys toeros rochesifiity

62 The forcedresose of thesystem, y fr where{q d{q Q L q u },,,..., Lre theoles i the forcig fuctio } rescle fctors. If ll theoles of theiut siglflliside theuit circle, y eros rochesifiity sice theiut siglis trsietsigl. fr willdecys to Whe thecusliut siglis siusoidl, theoles fllo theuit circle,d cosequetly, the forcedresose is lso siusoidl thtersists for ll - - stedy- stteresose

63 3.6.4 Cuslity d Stbility Coditio of cuslltisystem : h ROC : exteriorof circle Cosequetly, LTI systemis cusl if d oly if the ROC of the system fuctio is the exteriorof circle of rdius r,icludig the oit.

64 Coditioof LTI systemtobebibo stble sice Whe, H Therefore, LTI system isbibo stble if fuctioicludestheuit circle. Isummry, cusl LTI system isbibo stble if d oly if ll theoles of H re iside theuit circle. H H h h h h h d oly if theroc of thesytem

65 3.6.5 Pole-Zero Ccelltios Zero ositio sme with ole ositio C occurs i. I system fuctio - the order of the system is reduced by ii. I the roduct of the system fuctio with -trsform of the iut sigl -the ole of the system is suressed by the eros of the iut sigl, or vice-vers. o-exct ole-ero ccelltio : Whe the ero is locted very er the ole but ot exctly the sme ositio, the term i the resose hs very smll mlitude Cosequetly, oe should ot ttemt to stbilie iheretly ustble system by lcig ero i the iut sigl t the loctio of the ole.

66 3.6.6 Multile-Order Poles d Stbility The BIBO stbility requires ht the system oles be strictly iside the uit circle. If the system oles re ll iside the uit circle d the excittio sequece x cotis oe or more oles tht coicide with the ole of system, the outut Y will coti multile-order oles i terms of b u where b m- d m is the order of the ole If <, these terms decy to ero s roches ifiity becuse the exoetil fctor domites the term b. Cosequetly, o bouded iut sigl c roduce ubouded outut sigl If the system oles re ll iside the uit circle.

67 3.6.7 The Schur-Coh Stbility Test Theoles of the system re theroots of the deomitor olyomil of H,... Stbility criteri lie outside theuit circle. -rocedure todetermieif y of theroots of i olyomil of degreem is m m m m Thereciroclor reverseolyomil B B m m m m m m m of degreemis

68 Ithe Schur - Coh stbility test,tocomute : i. Set ii. dreflectiocoefficiet,k m K iii. comute thelower - degreeolyomils ccordig to therecursive equtio m K m m B m m,m, -, -,..., wherek m m m The Schur - Coh stbility test sttestht theolyomilhsll its roots isidetheuitcircle if doly if thecoefficiet K K m for ll m,,..., m stisfy thecoditio

69 . d The systemisbibo stbleif This systemhs eros t the origidolest The systemfuctiois Cosidercusltwo - ole systemdescribedby the secod- order differece equtio 4 4, b b Y H x b y y y Stbility of Secod-Order System

70 or K K K K or lso. Cosequetly, d Thesystemis stbleif doly if Coh stbility test, - from therecursive equtio from Schur or Coditio : Theroots of qudrticequtiostisfy thereltios

71 Thestbility coditio defie regioi thecoefficietle Thesystemis stbleif doly if theoit - -- stbility trigle,,. lies iside the trigle

72 smleresose is thedifferece of decyig exoetilsequeces. Therefore,theuit smleresose is theuit Cosequetly, where thesystem fuctio, rereld, Sice 4 i. Thebehvior of thesystem : u b h b b H Relddistictoles

73 ii. Reldequloles - / The system fuctio is H dhece theuit smle resose of the systemis h b b 4 u Observed thth is theroduct of rmsequece dreldecyig exoetilsequece.

74 si cos j e b e e r re b b r r re re re H j j j j j j j d where Sice theoles recomlex cojugte,thesystem fuctio 4 iii. cojugteoles Comlex -

75 Cosequetly, theuit smleresose of system withcomlex cojugteoles is br h si e j j e j br si u si u Hece, h hs oscilltory behvior withexoetilly decyig eveloe wher The gle of theoles determiesthe frequecy of oscilltiod the distce r of theoles from the origidetermiestherteof decy.whe r is close touity, the decyis slow. Whe r is close to the origi, the decyis fst..