Sigls & Systems Chpter 0: The Z-Trsform Adpted from: Lecture otes from MIT, Bighmto Uiversity Hmid R. Riee Arm Sepehr Fll 00
Lecture 5 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 Stility Geometric Evlutio of -Trsforms d DT Frequecy Resposes First- d Secod-Order Systems System Fuctio Alger d Block Digrms Uilterl -Trsforms Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 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 [ ] X [ ] Z{ [ ]} 3 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 The ROC d the Reltio Betwee T d DTFT jω re, r j ω X jω re [ ] re F{ r [ ] } [ ] r e j ω ROC jω 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 Xe jω eists 4 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Emple # [] u[] - right -sided X This form for PFE d iverse - trsform - u 0 [ ] If <,i.e., > Tht is, ROC >, outside circle This form to fid pole d ero loctios 5 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Emple #: [] u[ ]- left - sided X { u [ ] } 0, If <, i. e., < Sme X s i E #, ut differet ROC. 6 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Rtiol -Trsforms [] lier comitio of epoetils for > 0 d for < 0 X is rtiol X N D Polyomils i chrcteried ecept for gi y its poles d eros 7 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 The -Trsform ROC { } Z [ ] [ ] [ ] X Z jω re t which [] r < -depeds oly o r, just like the ROC i s-ple oly depeds o Res Lst time: o Uit circle r i the ROC DTFT X e jω eists o Rtiol trsforms correspod to sigls tht re lier comitios of DT epoetils 8 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Some Ituitio o the Reltio etwee T d LT Let tt Z st t X s t e dt L{ } t st lim T e T 0 3 [ ] T lim T 0 [] st T e The Bilterl -Trsform [] X [] { []} C thik of -trsform s DT versio of Lplce trsform with st e 9 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 More ituitio o T-LT, s-ple - -ple reltioship e st jω is i s - ple s jω e jωt 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. 0 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Properties of the ROCs of -Trsforms The ROC of X cosists of rig i the -ple cetered out the origi equivlet q to verticl strip i the s-ple The ROC does ot coti y poles sme s i LT. Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 More ROC Properties 3 If [] is of fiite durtio, the the ROC is the etire - ple, ecept possily t 0 d/or. Why? N X N [ ] Emples: CT couterprt Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 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. [ ] r N coverges fster th N [] r 0 3 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Side y Side 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 p icludig the circle r o. Wht types of sigls do the followig ROC correspod to? right-sided left-sided two-sided 4 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Emple # [] 0, > p [ ] [ ] [ ] u u [ ] From > [ ] [ ] u u,, < > 5 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Emple # cotiued X, < < Clerly, ROC does ot eist if > No -trsform for. 6 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Iverse -Trsforms for fied r: X X re [] r re { [ ] r }, re jω jω jω F { X re } X re π π π jω jω jω [] X re r e dω d π jre 3 dω dω e ROC j jω dω d jω jω πj [] X d πj 7 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Emple # p 6 5 3 6 5 3 B A X Prtil Frctio Epsio Alger: A, B 3 4 3 4 3 4 X X 3 4 Note, prticulr to -trsforms: [ ] [ ] ] [ Whe fidig poles d eros, epress X s fuctio of. Whe doig iverse -trsform usig PFE, epress X s fuctio of - X s fuctio of. 8 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Emple # Cotiued ROC III: > 3 4 - right - sided sigl ROC II: 4 < < 3 [] u[] [] u[] 3 4 u 3 [ ] u[ ] [ ] [ ] - two - sided sigl ROC I: < 4 4 [ ] u [ ] 3 - left - sided sigl [] u[ ] 9 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Iversio y Idetifyig Coefficiets i the Power Series X [] [] - coefficiet of Emple #3: X 3 3-4 [ 3] [ ] [] 4 [ ] 3-0 for ll other s A fiite-durtio DT sequece 0 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Emple #4: X [ ] u [ ] L coverget for X [ ] u[ ] coverget 3 for 3 L <,i.e., L > <, i. e., < Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Properties of -Trsforms 0 Time Shiftig [ ] X, 0 The rtiolity of X uchged, differet from LT. ROC uchged ecept for the possile dditio or deletio of the origi or ifiity o > 0 ROC 0 mye o < 0 ROC mye dx -Domi Differetitio [ ] sme ROC Derivtio: d X dx d dx d [] [ ] [] Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Covolutio Property d System Fuctios Y HX, ROC t lest the itersectio of the ROCs of H d X, c e igger ifthere is pole/ero ccelltio. e.g. H, > X, Y ROC ll H h[] The System Fuctio H ROC tells us everythig out system 3 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 CAUSALITY h[] right-sided ROC is the eterior of circle possily icludig : H h N [] N If N < 0, the the rerm h [ N ] ROC outside circle, ut 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 4 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Cuslity for Systems with Rtiol System Fuctios y y y H N N N N M M M M 0 0 L L A DT LTI system with rtiol system fuctio H is cusl N M N N if, No poles t 0 y y the ROC is the eterior of circle outside the outermost pole; d if we write H s rtio of polyomils p y the D N H the degree degree D N 5 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Stility LTI System Stle 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 stle ll poles re iside the uit circle, i.e. hve mgitudes < 6 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Geometric Evlutio of Rtiol -Trsform Emple #: X - A first - order ero Emple #: X - A first - order pole X, X X X Emple #3: X M X M R i P j βi α R i P j R P M β i j β α X α i j i j j All sme s i s-ple 7 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Geometric Evlutio of DT Frequecy Resposes First-Order System oe rel pole H, > h u, < [ ] [ ] H υ υ jω jω jω e, H e, H e υ υ ω υ υ υ υ 8 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Secod-Order System Two poles tht re comple cojugte pir re jθ * H, 0< r <, 0 θ π r cosθ r H e jω jω jθ jω e re e re jθ Clerly, H peks er ω ±θ, h [] r [ θ ] u [] si siθθ 9 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Demo: DT pole-ero digrms, frequecy respose 8 30 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 DT LTI Systems Descried y LCCDEs Use the time-shift property N M k y k 0 k 0 [ k] k [ k] N M k k k Y k k 0 k 0 X Y H X M k k Rtiol k 0 H N k 0 k k ROC: Depeds o Boudry Coditios, left-, right-, or two-sided. ROC is outside the outermost pole For Cusl Systems 3 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 System Fuctio Alger d Block Digrms Feedck System cusl systems Emple #: Y H H X H H egtive feedck cofigurtio - D Dely H 4 y 4 [ ] y[ ] [ ] 3 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Emple #: Cscde of two systems 4 4 H y 4 4 33 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Uilterl -Trsform Note: χ [ ] 0 If [] 0 for < 0, the χ X UZT of [] BZT of []u[] ROC lwys outside circle d icludes 3 For cusl LTI systems, H H 34 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Properties of Uilterl -Trsform Covolutio property for [<0] [<0] 0 UZ But there re importt differeces. For emple, time-shift [ ] [ ] UZ χ χ Derivtio: Iitil coditio ] [ ] [ y χ Y [ ] [ ] [ ] [ ] 0 0 y Y [ ] [ ] 443 m m m χ 0 35 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Use of UZTs i Solvig Differece Equtios with Iitil Coditios [] y[ ] [ ] y α y[ ] β, [ ] αu[ ] UZT of Differece Equtio { y[ ]} 6 UZ 4 7448 Y α Y β β α Y 4 43 4 44 4 4 43 ZIR ZSR Output purely due to the iitil coditios, ZIR Output purely due to the iput. ZSR 36 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems
Lecture 5 Chpter 0 Emple cotiued β 0 System is iitilly t rest: ZSR X H X H Y 443 443 α H H X H α 0 Get respose to iitil coditios ZIR Y β Y ] [ ] [ u y β ] [ ] y[ 37 Shrif Uiversity of Techology, Deprtmet of Computer Egieerig, Sigls & Systems