Chapter 10: The Z-Transform Adapted from: Lecture notes from MIT, Binghamton University Dr. Hamid R. Rabiee Fall 2013
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1 Sigls & Systems Chpter 0: The Z-Trsform Adpted from: Lecture otes from MIT, Bighmto Uiversity Dr. Hmid R. Rbiee Fll 03
2 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Outlie Itroductio to the -Trsform Properties of the ROC of the -Trsform Iverse -Trsform Properties of the -Trsform System Fuctios of DT LTI Systems o Cuslity o Stbility Geometric Evlutio of -Trsforms d DT Frequecy Resposes First- d Secod-Order Systems System Fuctio Algebr d Block Digrms Uilterl -Trsforms Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
3 Lecture 5 Chpter 0 Lecture 6 Chpter 0 The -Trsform Motivtio: Alogous to Lplce Trsform i CT [ ] We ow do ot restrict ourselves just to = e jω H h[ ] H ssumig it coverges y[ ] H Eige fuctio for DT LTI The Bilterl -Trsform Z [ ] [ ] Z{ [ ]} 3 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
4 Lecture 5 Chpter 0 Lecture 6 Chpter 0 The ROC d the Reltio Betwee T d DTFT j re, r j re F [ ] re [ ] r j [ ] r e j ROC j re t which [ ] r depeds oly o r =, just like the ROC i s-ple oly depeds o Res Uit circle r = i the ROC DTFT e jω eists 4 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
5 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Emple # u - right -sided 5 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
6 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Emple # u - right -sided This form for PFE d iverse - trsform 0 - u If,i.e., Tht is, ROC >, outside circle This form to fid pole d ero loctios 6 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
7 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Emple #: u[ ]- left -sided 7 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
8 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Emple #: Sme s i E #, but differet ROC. -sided [ ]-left u u 0,.,., If e i 8 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
9 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Rtiol -Trsforms [] = lier combitio of epoetils for > 0 d for < 0 is rtiol N D Polyomils i chrcteried ecept for gi by its poles d eros 9 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
10 Lecture 5 Chpter 0 Lecture 6 Chpter 0 The -Trsform -depeds oly o r =, just like the ROC i s-ple oly depeds o Res Lst time: Z Z o Uit circle r = i the ROC DTFT j ROC re t which r o Rtiol trsforms correspod to sigls tht re lier combitios of DT epoetils j e eists 0 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
11 Some Ituitio o the Reltio betwee T d LT Lecture 5 Chpter 0 Lecture 6 Chpter 0 Let t=t Z st t s t e dt L{ t} st lim T e T 0 T lim T 0 The Bilterl -Trsform st T e { } C thik of -trsform s DT versio of Lplce trsform with st e Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
12 More ituitio o T-LT, s-ple - -ple reltioship e st Lecture 5 Chpter 0 Lecture 6 Chpter 0 j is i s -ple s j e jt uit circle i - pl LHP i s-ple, Res < 0 = e st <, iside the = circle. Specil cse, Res = - = 0. RHP i s-ple, Res > 0 = e st >, outside the = circle. Specil cse, Res = + =. A verticl lie i s-ple, Res = costt e st = costt, circle i -ple. Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
13 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Properties of the ROCs of -Trsforms The ROC of cosists of rig i the -ple cetered bout the origi equivlet to verticl strip i the s-ple The ROC does ot coti y poles sme s i LT. 3 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
14 Lecture 5 Chpter 0 Lecture 6 Chpter 0 More ROC Properties 3 If [] is of fiite durtio, the the ROC is the etire - ple, ecept possibly t = 0 d/or =. Why? N N Emples: CT couterprt 4 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
15 Lecture 5 Chpter 0 Lecture 6 Chpter 0 ROC Properties Cotiued 4 If [] is right-sided sequece, d if = r o is i the ROC, the ll fiite vlues of for which > r o re lso i the ROC. N N r coverges fster th r 0 5 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
![Side by Side Lecture 5 Chpter 0 Lecture 6 Chpter 0 5 If [] is left-sided sequece, d if = r o is i the ROC, the ll fiite vlues of for which 0 < < r o re lso i the ROC.](/docs-images/82/85448311/images/16-1.jpg "6 If [] is two-sided, d if = r o is i the ROC, the the ROC cosists of rig i the -ple icludig the circle = r o. Wht types of sigls do the followig ROC correspod to?")
16 Side by Side Lecture 5 Chpter 0 Lecture 6 Chpter 0 5 If [] is left-sided sequece, d if = r o is i the ROC, the ll fiite vlues of for which 0 < < r o re lso i the ROC. 6 If [] is two-sided, d if = r o is i the ROC, the the ROC cosists of rig i the -ple icludig the circle = r o. Wht types of sigls do the followig ROC correspod to? right-sided left-sided two-sided 6 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
17 Lecture 5 Chpter 0 Lecture 6 Chpter 0 b, b Emple # 0 7 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
18 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Emple # 0, b b u b u b b b u b b b u b,, From: 8 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
19 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Emple # cotiued b b, b b Clerly, ROC does ot eist if b > No -trsform for b. 9 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
20 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Iverse -Trsforms for fied r: ROC }, ] [ { j j re r 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 0 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
21 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Emple # ? Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
22 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Emple # Prtil Frctio Epsio Algebr: A =, B = Note, prticulr to -trsforms: Whe fidig poles d eros, epress s fuctio of. Whe doig iverse -trsform usig PFE, epress s fuctio of B A ] [ 3 4 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
23 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Emple # Cotiued ROC III: 3 ROC II: u u right u - sided sigl u - two- sided sigl ROC I: 4 4 u 3 - left - sided sigl u 3 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
24 Iversio by Idetifyig Coefficiets i the Power Series Lecture 5 Chpter 0 Lecture 6 Chpter 0 - coefficiet of Emple #3: Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
25 Iversio by Idetifyig Coefficiets i the Power Series Lecture 5 Chpter 0 Lecture 6 Chpter 0 - coefficiet of Emple #3: for ll other s A fiite-durtio DT sequece 5 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
26 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Emple #4: coverget for,i.e., b 6 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
27 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Emple #4: b,i.e., for coverget u e i.,., for coverget 3 3 u 7 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
28 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Properties of -Trsforms Time Shiftig 0?, -Domi Differetitio? 8 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
29 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Properties of -Trsforms Time Shiftig 0, 0 The rtiolity of uchged, differet from LT. ROC uchged ecept for the possible dditio or deletio of the origi or ifiity o > 0 ROC 0 mybe o < 0 ROC mybe d -Domi Differetitio sme ROC Derivtio: d d d d d 9 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
30 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Covolutio Property d System Fuctios Y = H, ROC t lest the itersectio of the ROCs of H d, c be bigger if there is pole/ero ccelltio. e.g. H Y,, H h H + ROC tells us everythig bout the system ROC ll TheSystem Fuctio 30 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
31 Lecture 5 Chpter 0 Lecture 6 Chpter 0 CAUSALITY h[] right-sided ROC is the eterior of circle possibly icludig = : H h N N If N 0, the the rerm h[ N ] ROC outside circle, but does t ot iclude. Cusl N 0 No m terms with m>0 =>= ROC A DT LTI system with system fuctio H is cusl the ROC of H is the eterior of circle icludig = 3 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
32 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Cuslity for Systems with Rtiol System Fuctios A DT LTI system with rtiol system fuctio H is cusl the ROC is the eterior of circle outside the outermost pole; d b if we write H s rtio of polyomils the N M b b b b H N N N N M M M M if, No poles t 0 0 D N H degree degree D N 3 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
33 Stbility Lecture 5 Chpter 0 Lecture 6 Chpter 0 LTI System Stble h ROC of H icludes the uit circle = Frequecy Respose He jω DTFT of h[] eists. A cusl LTI system with rtiol system fuctio is stble ll poles re iside the uit circle, i.e. hve mgitudes < 33 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
34 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Geometric Evlutio of Rtiol -Trsform Emple #: - A first -order ero Emple #: - A first - order pole Emple #3: M R i P j i j 34 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
35 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Geometric Evlutio of Rtiol -Trsform Emple #: Emple #: Emple #3: All sme s i s-ple -order ero - A first - A first - order pole, j P j i R i M j P j i R i M R i P j j i M 35 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
36 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Geometric Evlutio of DT Frequecy Resposes First-Order System oe rel pole,, u h H,, j j j e H e H e H 36 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
37 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Secod-Order System Two poles tht re comple cojugte pir = re jθ = * H H e r cos, 0 r, 0 r j e j re j e Clerly, H peks er ω = ±θ j re j, h r si si u 37 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
38 Lecture 5 Chpter 0 Lecture 6 Chpter 0 DT LTI Systems Described by LCCDEs Use the time-shift property ROC: Depeds o Boudry Coditios, left-, right-, or two-sided. For Cusl Systems ROC is outside the outermost pole M k k N k k k b k y 0 0 N k M k k k k k b Y 0 0 N k k k M k k k b H H Y 0 0 Rtiol 39 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
39 Lecture 5 Chpter 0 Lecture 6 Chpter 0 System Fuctio Algebr d Block Digrms Feedbck System cusl systems Emple #: H Y H H H egtive feedbck cofigurtio 40 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
40 Lecture 5 Chpter 0 Lecture 6 Chpter 0 System Fuctio Algebr d Block Digrms Feedbck System cusl systems egtive feedbck cofigurtio Emple #: H H H Y H 4 H y y 4 - D Dely 4 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
41 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Emple #: H 4 4 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
42 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Emple #: Cscde of two systems 4 4 H 43 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
43 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Uilterl -Trsform Note: 0 If [] = 0 for < 0, the UZT of [] = BZT of []u[] ROC lwys outside circle d icludes = 3 For cusl LTI systems, H H 44 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
44 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Properties of Uilterl -Trsform Covolutio property for [<0] = [<0] = 0 UZ But there re importt differeces. For emple, time-shift Derivtio: Y y[ ] [ ] Y [ ] 0 y Iitil coditio m 0 m m0 45 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
45 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Use of UZTs i Solvig Differece Equtios with Iitil Coditios UZT of Differece Equtio Y y[ ] [ ] Output purely due to the iitil coditios, Output purely due to the iput. y y[ ], { y[ ]} UZ Y Y u ZIR ZSR ZIR ZSR 46 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
46 Lecture 5 Chpter 0 Lecture 6 Chpter 0 Emple cotiued β = 0 System is iitilly t rest: ZS R α = 0 Get respose to iitil coditios ZI R H H H H Y Y ] [ ] [ u y 47 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Chapter 10: The Z-Transform Adapted from: Lecture notes from MIT, Binghamton University Hamid R. Rabiee Arman Sepehr Fall 2010
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