The z-transform. Laplace

Transcription

1 The -Trsform Qote of the Dy Sch is the dvtge of well-costrcted lgge tht its simplified ottio ofte ecomes the sorce of profod theories. Lplce

2 Some Specil Fctios First cosider the delt fctio or it smple fctio:,, This llows ritrry seqece or cotiostime fctio ft to e epressed s: d t f t f k k k or,, t t t

3 Covoltio, Uit Step These re referred to s discrete-time covoltio, d is deoted y: k k Also cosider the it step fctio:,, ote lso: k k k

4 -Trsform The -trsform is the most geerl cocept for the trsformtio of discrete-time series. The Lplce trsform is the more geerl cocept for the trsformtio of cotios time processes. For emple, the Lplce trsform llows yo to trsform differetil eqtio, d its correspodig iitil d odry vle prolems, ito spce i which the eqtio c e solved y ordiry lger. The switchig of spces to trsform clcls prolems ito lgeric opertios o trsforms is clled opertiol clcls. The Lplce d trsforms re the most importt methods for this prpose. 4

5 The Z Trsforms The oe-sided -trsform of fctio : Also kow s ilterl -trsform. The two-sided -trsform of fctio : Bilterl - trsform The Lplce trsform of fctio ft: F st s f t e dt 5

6 Reltioship to Forier Trsform ote tht epressig the comple vrile i polr form revels the reltioship to the Forier trsform: i. e. Z i re re i re i, or i i re r e, d if r, e i e i which is the Forier trsform of. 6

7 The -trsform d the DTFT The -trsform is fctio of the comple vrile It is coveiet to descrie o the comple -ple If we plot =e j for = to we get the it circle Im j e Uit Circle r= Re Copyright C 5 Güer Arsl 7

8 Regio of CovergeceROC The power series for the -trsform is clled Lret series:... The set of ll vles of for which this series covergesttis the fiite vle is kow s its regio of covergece. Ech vle of r represets circle of rdis r.... Im Re The regio of covergece is mde of circles. Emple: -trsform coverges for vles of.5<r< ot ll seqece hve -trsform Emple: cos o Does ot coverge for y r o ROC, o -trsform 8

9 Poles d Zeros Whe is rtiol fctio, i.e., rtio of polyomils i, the:. The roots of the mertor polyomil re referred to s the eros of, d. The roots of the deomitor polyomil re referred to s the poles of. ote tht o poles of c occr withi the regio of covergece sice the -trsform does ot coverge t pole. Frthermore, the regio of covergece is oded y poles. 9

10 Clerly, hs ero t = d pole t =. Right-Sided Epoetil Seqece Emple For Covergece we reqire Hece the ROC is defied s Iside the ROC series coverges to Regio otside the circle of rdis is the ROC Right-sided seqece ROCs eted otside circle Geometric series forml Regio of covergece

11 Clerly, hs ero t = d pole t =. Left-Sided Epoetil Seqece Emple We c ltertively write s Hece the ROC is defied s Iside the ROC series coverges to Regio Iside the circle of rdis is the ROC Left-sided seqece ROCs eted Iside circle Geometric series forml Regio of covergece

12 Copyright C 5 Güer Arsl Two-Sided Epoetil Seqece Emple ROC : ROC : 6 Im oo Re Clerly, hs ero t =,/ d pole t = ½,-/. : ROC

13 Fiite Legth Seqece otherwise If 8 4 is the series of ve powers of. Therefore the ROC for fiite-legth right sided seqece is the etire -ple ecept t = otherwise If is the series of +ve powers of. Therefore the ROC for fiite-legth left sided seqece is the etire -ple ecept t =

14 4 Fiite Legth Seqece cotd.. otherwise is the series of ve powers & +ve powers of. Therefore the ROC for fiite-legth left sided seqece is the etire -ple ecept t = & = Regio otside the circle of rdis is the ROC

15 Properties of The ROC of Z-Trsform The ROC is rig or disk cetered t the origi. The ROC cot coti y poles. The ROC for right-hded seqece eteds otwrd from the otermost pole possily icldig =. The ROC for left-hded seqece eteds iwrd from the iermost pole possily icldig =. The ROC of two-sided seqece is rig oded y poles. The ROC mst e coected regio. The ROC for fiite-legth seqece is the etire -ple ecept possily = d =. A -trsform does ot iqely determie seqece withot specifyig the ROC. 5

16 Wht is the -trsform of delt fctio? 6,, ZT Wht is the -trsform of it step fctio?,, ZT Regio otside the circle of rdis is the ROC Z > ROC is the etire -ple

17 Review 7 Rocis etire ple ZT ROC is ZT ZT ROC is ZT ROC is elsezt does ot eit. If ROC is ZT Geometric series forml