The z-transform. Dept. of Electronics Eng. -1- DH26029 Signals and Systems

Transcription

1 0 Th -Trsform Dpt. of Elctroics Eg. -- DH609 Sigls d Systms

2 0. Th -Trsform Lplc trsform - for cotios tim sigl/systm -trsform - for discrt tim sigl/systm 0. Th -trsform For ipt y H H h with ω rl i.. DTFT of h -trsform of h wh is ot rstrictd to ity Dpt. of Elctroics Eg. -- DH609 Sigls d Systms

3 -trsform of grl discrt-tim sigl Nottio : Z For compl vril lt r r r r { r } r { r } Dpt. of Elctroics Eg. -- DH609 Sigls d Systms

4 is qivlt to th Forir Trsform Rgio of covrgc ROC : rg of vls of for which covrgs. Not If th ROC iclds th it circl th th Forir trsform lso covrgs. Dpt. of Elctroics Eg. -- DH609 Sigls d Systms

5 Dpt. of Elctroics Eg. DH609 Sigls d Systms -5- Rgio of Covrgc ROC Pol-ro plot d rgio of covrgc for 0<< E. 0. 0

6 Dpt. of Elctroics Eg. DH609 Sigls d Systms E.0.

7 Dpt. of Elctroics Eg. DH609 Sigls d Systms ROC? E. 0. & &

8 Dpt. of Elctroics Eg. -8- DH609 Sigls d Systms

9 Dpt. of Elctroics Eg. DH609 Sigls d Systms -9- si / / / / ROC 는? 0 0 E. 0. &

10 0. Th Rgio of Covrgc for th -Trsform Proprty : Th ROC of cosists of rig i th -pl ctrd ot th origi. r r Fig.0.6 Proprty : Th ROC dos ot coti y pols. why? is ifiit t pol Proprty : If is of fiit drtio th th ROC is th tir -pl cpt possily =0 d/or =. cs cs cs N N N N N : 0 0 N : 0 0 or ROC ROC ROC Dpt. of Elctroics Eg. -0- DH609 Sigls d Systms

11 Dpt. of Elctroics Eg. DH609 Sigls d Systms -- ROC : tir -pl cpt for = 0 d =. ROC : tir -pl cpt for =. ROC : tir compl pl cpt for = 0. E. : fiit sqc {}

12 Proprty : If is right-sidd sqc d if th circl =r 0 is i th ROC th ll fiit vls of for which >r 0 will lso i th ROC. N N : positivor gtiv Proprty 5 : If is lft-sidd sqc d if th circl =r 0 is i th ROC th ll vls of for which 0< <r 0 will lso i th ROC. Proprty 6 : If is two sidd d if th circl =r 0 is i th ROC th th ROC will cosist of rig i th -pl tht iclds th circl =r 0 rfr to Fig.0.8 p.75 or 0 r with r r0 r is s.smml if r0 is lso for N. r N N : positivor gtiv or 0 with r r0 r is s.smml if r0 is lso for. N Dpt. of Elctroics Eg. -- DH609 Sigls d Systms

13 Dpt. of Elctroics Eg. DH609 Sigls d Systms -- Fig.0.0 p75 0 p.755 d Fig.0. ROC For > o commo ROC For < c d ROC : Fig. 0. t p.755 E. 0.7

14 Proprty 7 : If th -trsform of is rtiol th its ROC is odd y pols or tds to ifiity Proprty 8 : : rtiol : right sidd => ROC : th rgio i th -pl otsid th otrmost pol. Not if is csl ROC? iclds Proprty 9 : : rtiol : lft sidd => ROC : th rgio i th -pl isid th irmost oro pol. Not if is ticsl ROC? iclds 0 Dpt. of Elctroics Eg. -- DH609 Sigls d Systms

15 0. Th Ivrs -Trsform r F - r r F r r F r r r d r r d r d d r d d d d From th Rsid thorm Rs{ } : itgrtio rod cotr clockwis closd circlr cotor ctrd t th origi d with rdis r Dpt. of Elctroics Eg. -5- DH609 Sigls d Systms

16 Dpt. of Elctroics Eg. DH609 Sigls d Systms Altrtiv mthod for th ivrs -trsform : prtil-frctio psio ROC : /< </ ROC : </ E

17 Aothr procdr for th ivrs -trsform : powr-sris psio of E Dpt. of Elctroics Eg. -7- DH609 Sigls d Systms

18 Dpt. of Elctroics Eg. DH609 Sigls d Systms -8- log divisio y If E. 0.

19 0. Gomtric Evltio of th Forir Trsform From th Pol-Zro Plot 0.. First-Ordr Systms H h H v v H H Fig. 0. p. 76 Dpt. of Elctroics Eg. -9- DH609 Sigls d Systms

20 Dpt. of Elctroics Eg. DH609 Sigls d Systms -0- v v v H H 0 t ros ro: dol loctios : Pol cos 0 & 0 cos & si si r r r r H r r r H r h 0.. Scod-Ordr Systms Fig. 0. p. 766