Lecture 4 Outie: Z Trsfors ouceets: HW 7 to e oste Friy, ue Jue Wi e ore HW, y e short, o te HW8s Lst ecture Jue 6 wi icue course review Fi ex Jue t 3:306:30 i this roo; i fterwrs ore fi ex ouceets ext wee o css oy iter Z Trsfor its ROC Rtio Z Trsfors ZTrsfor Proerties Pirs ROC for Sie Sis Iversio of ZTrfors Systes ysis usi ZTrfors
Review of Lst Lecture ( crifictio) LTI Systes Feec t y Crifictio: Stiity i LTI Systes Syste is IO ste if oue iuts yie oue oututs Equte IO stiity with stiity; exist other efiitios (uifor, sytotic, ) syste is (IO) ste Û its iuse resose is soutey itere Þ iies H(jw) exists. H(s) is ste Û H(s) stricty roer jwîroc (etis i Reer) orer Lowss Systes y w w y w t Feec i LTI Systes ( t) x( t) ( s w s w ) Y ( s) w ( s) C stiie uste syste, e resose ore ie, itite isturces e( t) ( t) x ( t) S y( t) G( s) ± h( t) H( s) Exes: ey, ifier x(t) H ( s) x(t) R S e(t) r(t) L C G( s) x ( t) y( t) G( s) H( s) s K y(t) y(t) w ± LC R Y C L ( s) w ( s) s w s w Dey >>0 stiies Poe t s(/k K ), Uste if K K > oe>0
iter ZTrsfor Discrete Tie; ysis very siir to Lce Geeries DiscreteTie Fourier Trsfor wys exists withi Reio of Coverece Use to stuy systes/sis without DTFTs Def: Z Retio with DTFT: Reio of Coverece (ROC): vues of such tht () exists Dees oy o r (vs. s i Lce) Circes iste of es e t Z u[] Exe: jw { x[ ] } x[ ], e re ; exists if x[ ] r < å å { x[ r } DTFT ] ( jw e ) DTFT{ x[ ]}, r Circes i e Sest r: () exists > True for soe rîr Potte o e
Rtio Z Trsfors uertor Deoitor re oyois C fctor s rouct of oois s re eros (where ()0), s re oes (where () ) ROC cot icue y oes If () re, s s re re eros of () re re or occur i coexcojute irs. Se for the oes. /, eros, oes, ROC fuy secify () Exe: sie exoeti 0 0 [ ] [ ] [ ] u u x
ROC for Sie Sis Rihtsie: x[]0 for < for soe ROC is outsie circe ssocite with rest oe Exe: RH exoeti: x[] u[] Leftsie: x[]0 for > for soe ROC is isie circe ssocite with sest oe Exe: LH exoeti: x[] u[] Twosie: either riht or eft sie C e writte s su of RH LH sie sis ROC is circur stri etwee two oes Exe: sie exoeti: x[], <
Extrct the Stricty Proer Prt of (s) If <, is stricty roer, rocee to ext ste If ³, erfor o ivisio to et, where Ivert D(s) to et tie si ( ; ) Foows fro trsfor te The seco ter is stricty roer Perfor rti frctio exsio: Ivert rti frctio exsio teryter (chec ROC) For rihtsie sis: Iversio of Rtio Lce Trsfors 0 0, D 0 D / Õ Õ å å Oti coefficiets vi resiue etho å å u u u x ] [ ] [ ) )...( ( ] [ ] [ [ ] [ ] [ ] [ ] [ ] 0
LTI Systes ysis usi Trsfors LTI ysis usi covoutio roerty h[] x [] y[] H y [ ] x[ ]* h[ ] Y [ ] H[ ] [ ] ROCÌROC x ÇROC h Equivece of systes se s for Lce Systes G[] H[] i Pre: T[]G[]H[] Systes G[] H[] i Series: T[]G[]H[] H[] ce the trsfer fuctio of the syste Cusity stiity i LTI iscrete systes Syste is cus if h[]0, <0, so if h[] rihtsie; cus systes hve ROC outsie circe ssocite with rest oe. Syste is IO ste if oue iuts yie oue oututs Equte IO stiity with stiity; exist other efiitios cus syste is (IO) ste Û iuse resose h[] is soutey sue Þ H(e jw ) exists. H() is ste Û H() stricty roer oes isie uit circe, i.e. rîroc (etis i Reer)
i Poits Ztrsfor eeries DTFT, siir ysis s Lce. Icues the ROC which is efie y circes i e Exressi the trsfor i rtio for ows iverse vi rti frctio extio so ows trsfor chrcteritio vi oeero ot OeSie sis hve sie ROCs outsie/isie circe. Two sie sis hve circur stris s ROCs Covoutio other roerties of trsfors ow us to stuy iut/outut retioshi of LTI systes cus syste with H() rtio is ste if & oy if oes of H() ie isie uit circe circe ( oes hve <) ROC efie iicity for cus ste LTI systes Hve ret hoiy weee