DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018

Transcription

1 DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Assoc. Prof. Dr. Bur Kelleci Sprig 8

2 OUTLINE The Z-Trsform The Regio of covergece for the Z-trsform The Iverse Z-Trsform Geometric Evlutio of the Fourier Trsform from the Pole-Zero Plot Properties of Z-Trsform Alysis d Chrcteritio of LTI Systems usig Z-Trsform The Uilterl Z-Trsform Sigls d Systems - Wee

3 THE Z-TRANSFORM For discrete-time lier time-ivrit system with impulse respose h, the respose y of the system to complex expoetil iput of the form is y H where H h For =e jw, this summtio correspods to the discrete-time Fourier trsform of h. Whe is ot restricted to uity, the summtio is referred to s the -trsform of h Sigls d Systems - Wee

4 THE Z-TRANSFORM The -trsform of geerl discrete-time sigl x is defied s x where is complex vrible. For coveiece, the reltioship betwee x d - trsform is idicted s Z x Sigls d Systems - Wee

5 THE Z-TRANSFORM To explore the reltioship betwee -trsform d discrete-time Fourier trsform, let s write complex vrible i polr form. jw re jw jw re x re jw re x r (re jw ) is the Fourier trsform of the sequece x multiplied by rel expoetil r -. jw r - my be decyig or growig with icresig depedig o the vlue of r. e Sigls d Systems - Wee 5

6 THE Z-TRANSFORM For cotiuous-time, the Lplce trsform reduces the Fourier trsform o the imgiry xis. I cotrst, the -trsform reduces to the Fourier trsform whe the mgitude of the trsform vrible is uity. The -trsform reduces the to the Fourier trsform i the complex -ple correspodig to circle with rdius of uity. Uit Circle wp wp Im wp/ w jw e Re w wp/ Sigls d Systems - Wee 6

7 THE Z-TRANSFORM For covergece of the -trsform, the Fourier trsform of xr - must coverge. Obviously, this sequece will coverge for some r vlues. The rge of vlues for which () coverges is referred s regio of covergece (ROC). If the ROC icludes the uit circle, the Fourier trsform lso coverges. Sigls d Systems - Wee 7

8 x u u For covergece The EAMPLE Uit Circle regio of covergece if the rge of Im vluesfor which Re Sigls d Systems - Wee 8

9 EAMPLE u u x vluesfor which the rge of covergece if regio of The Re Im Uit Circle Sigls d Systems - Wee 9

10 THE Z-TRANSFORM AND EAMPLE The -trsform of Exmple d re the sme but they differ oly by their regio of covergece. Sigls d Systems - Wee / d must coverge,which requires tht For covergece both sums si / / / / / / / / e e e e e j e j u e j u e j u x j j j j j j j j p p p p p p p p p Re Im Uit Circle

11 THE REGION OF CONVERGENCE FOR THE Z-TRANSFORM Property : The ROC of () cosists of rig i the -ple cetered bout the origi. Property : The ROC does ot coti y poles Property : If x is of fiite durtio, the the ROC is the etire -ple, except possibly = d/or = Property : If x is right-sided sequece, d if the circle =r is i the ROC, the ll fiite vlues of for which >r will lso i the ROC. Property 5: If x is left-sided sequece, d if the circle =r is i the ROC, the ll fiite vlues of for which < <r will lso i the ROC. x r N N N N x x x Sigls d Systems - Wee

12 THE REGION OF CONVERGENCE FOR THE Z-TRANSFORM Property 6: If x is two-sided, d if the circle =r is i the ROC, the the ROC will cosist of rig i the -ple tht icludes the circle =r. Property 7: If the -trsform () of x is rtiol, the its ROC is bouded by poles or exteds to ifiity. Property 8: If the -trsform () of x is rtiol, d if x is right-sided. the the ROC is the regio i the -ple outside the outermost pole. Property 9: If the -trsform () of x is rtiol, d if x is left-sided. the the ROC is the regio i the -ple iside the iermost pole. Sigls d Systems - Wee

13 THE INVERSE Z-TRANSFORM x c be recovered from its -trsform evluted cotour =re jw i the ROC with r fixed d w vryig over p itervl. x pj d The evlutio of the iverse -trsform requires the itegrtio roud couterclocwise closed circulr cotour cetered t the origi d with rdius r. The ltertive method is expdig the lgebric expressio ito prtil-frctio expsio d recogiig the sequeces ssocited with the idividul terms. Sigls d Systems - Wee

14 EAMPLE 6 5 u u x x x x Sigls d Systems - Wee

15 THE INVERSE Z-TRANSFORM Aother very useful procedure for determiig the iverse - trsform relies o power-series expsio of () The coefficiets i this power series re the sequece vlues x Sigls d Systems - Wee 5 x

16 GEOMETRIC EVALUATION OF THE FOURIER TRANSFORM FROM THE POLE-ZERO PLOT Sice -trsform reduces to the Fourier trsform for =, we cosider the vectors from the poles d eros to the uit circle. The mgitude of the frequecy respose t frequecy w is the rtio of the legth of the vector v to the legth of the vector v. The phse of the frequecy respose is the gle of v with respect to the rel xis mius the gle of v H Sigls d Systems - Wee 6

17 GEOMETRIC EVALUATION OF THE FOURIER TRANSFORM FROM THE POLE-ZERO PLOT Sigls d Systems - Wee 7

18 PROPERTIES OF Z-TRANSFORM Property Sigl Z-Trsform ROC x R x R R x Lierity x bx b Time Shiftig x Sclig i the -domi e j w x e jw x Time Reversl Time Expsio x r x At lest theitersectio of R d R x x r for some iteger r r Cojugtio x Covolutio x x First x x Differece Accumultio Differetitio i the - domi x x d R, except for thepossibledditio or deletio of set of poits the origi R Scled versio of poits R R (i.e., R the for i R) Iverted R (i.e., R the set of R - where is i R), where is i R) (i.e., the set of poits R - At lest theitersectio of R d R At lest theitersectio of R d d Iitil Vlue Theorems if x for, the x lim At lest the itersectio of R, R d Sigls d Systems - Wee 8

19 SOME COMMON Z-TRANSFORM PAIRS Sigl Trsform ROC All u u m m All, except (if m ) or u u u u cosw u cos cosw u siw si cosw cosw r cosw r cosw r siw r siw r cosw r r u r u (if m ) w w r r Sigls d Systems - Wee 9

20 ANALYSIS AND CHARACTERIZATION OF LTI SYSTEMS USING Z-TRANSFORM The -trsform plys importt role i the lysis d represettio of discrete-time LTI systems. From the covolutio property H H() is referred s trsfer fuctio or system fuctio. () is the -trsform of the iput sigl. Y() is the -trsform of the output sigl. Y My properties of system c be tied directly to chrcteristics of the poles, eros d regio of covergece of the trsfer fuctio. Sigls d Systems - Wee

21 ANALYSIS AND CHARACTERIZATION OF LTI SYSTEMS Cuslity USING Z-TRANSFORM A cusl LTI system hs impulse respose h tht is ero for <, therefore it is right-sided. A discrete-time LTI system is cusl if d oly if the ROC of its trsfer fuctio is the exterior of circle, icludig ifiity. A discrete-time LTI system with rtiol trsfer fuctio H() is cusl if d oly if: The ROC is the exterior of circle outside the outermost pole with H() expressed s rtio of polyomils i, the order of the umertor cot be greter th the order of the deomitor. (limit of H() s must be fiite i order to meet iitil-vlue theorem) Sigls d Systems - Wee

22 ANALYSIS AND CHARACTERIZATION OF LTI SYSTEMS Stbility USING Z-TRANSFORM The stbility of discrete-time LTI system is equivlet to its impulse respose beig bsolutely summble. I this cse the Fourier trsform of h coverges d cosequetly, the ROC of H() must iclude the uit circle. A LTI system is stble if d oly if the ROC of its trsfer fuctio H() icludes the uit circle, = A cusl LTI system with rtiol trsfer fuctio H() is stble if d oly if ll of the poles of H() lie iside the uit circle. I other words they must ll hve mgitude smller th. Sigls d Systems - Wee

23 EAMPLE Cosider the followig system. H 8 Without eve owig the ROC for this system, we c coclude tht the system is ot cusl, becuse the umertor of H() is higher order th the deomitor. Sigls d Systems - Wee

24 EAMPLE Cosider cusl system which hs pole t =. For this system to be stble, its pole must be iside the uit circle. Therefore <. H This is cosistet with the coditio for the bsolute summbility of the correspodig impulse respose h u Sigls d Systems - Wee

25 LTI SYSTEMS CHARACTERIZED BY LINEAR CONSTANT- COEFFICIENT DIFFERENCE EQUATIONS. Lierity d time-shiftig property c be used to obti - trsform of the systems defied by differece equtios. The trsfer fuctio for system stisfyig lier costtcoefficiet differece equtio is lwys rtiol. Sice the differece equtio by itself does ot provide iformtio bout ROC, dditiol costrit such s stbility d/or cuslity should be defied. Sigls d Systems - Wee 5 N M M N M N M N b Y H b Y b Y x b y

26 THE UNILATERAL Z-TRANSFORM The Uilterl -trsform is used to lye cusl systems specified by lier costt-coefficiet differece equtios with oero iitil coditios. x The uilterl -trsform c be thought of s the bilterl trsform of xu. If y sequece tht is ero for <, the uilterl d bilterl -trsforms will be ideticl. Sigls d Systems - Wee 6

27 PROPERTIES OF THE UNILATERAL Z-TRANSFORM Property Sigl Uilterl -Trsform x x x Lierity x bx b Time dely x x Time dvce x x j e w x e jw x Sclig i the -domi Time Expsio Cojugtio Covolutio (ssumig tht x d x re ideticlly ero for <) x x x m m for y m m x x x First Differece x Accumultio Differetitio i the - domi x x x d d x Iitil Vlue Theorems x lim Sigls d Systems - Wee 7

28 EAMPLE Cosider the followig differece equtio with x=u d with the iitil coditio y-=b The first-term (=) is iterpreted s the uilterl trsform of the ero-iput respose. The ero-stte respose (b=) correspods to the respose of the cusl LTI system defied by the differece equtio d the coditio of iitil rest. Sigls d Systems - Wee 8 / Y Y Y x y y b b

29 EAMPLE (CONT.) For y vlues of d b, we c expd Y() usig prtil frctio expsio d ivert the result to obti y. For exmple, if =8 d b= Sigls d Systems - Wee 9 for / 8 / u y Y

30 EAMPLE (CONT.) A Simuli model is geerted for this system y x y Sigls d Systems - Wee

31 EAMPLE (CONT.) =8 d b= 8 7 Clcultio Simultio Sigls d Systems - Wee

32 EAMPLE (CONT.) Zero-stte respose 8 7 Clcultio Simultio Sigls d Systems - Wee

33 EAMPLE (CONT.) Zero-iput respose.5. Clcultio Simultio Sigls d Systems - Wee