Chapter 4. The Z-Transform

Transcription

1 // eview Chter. The -Trsfor Go io Schoo of E.E. idi Uiv. htt://see.idi.ed.c/tech/tds/ The discrete-tie Forier trsfor roch for reresetig discrete sigs sig coe eoeti seqece. Advtges for LTI sste It descries sstes i the freqec doi sig the freqec resose fctio. The cottio of the sisoid sted-stte resose is gret fciitted the se of. esose to ritrr sote se seqece c esi e coted i the freqec doi tiig the trsfor d the freqec resose. Shortcoigs to the FT. There re sef sigs i rctice sch s for which the DTFT does ot eist.. The trsiet resose of sste de to iiti coditios or de to chgig its cot e coted sig the DTFT roch. Etesio of the DTFT To ddress the ove two roes -trsfor is roosed. Biter two-sided versio rovides other doi i which rge css of seqece d sstes c e ed. Uiter oe-sided versio c e sed to oti sste resose with iiti coditios or chgig its.

2 // The iter -trsfor Coets ] is coe vrie. The set of ves for which eists is ced the regio of covergece OC d is give For soe ositive ers - d +. ] C j C is cotercocwise cotor ecircig the origi d ig i the OC. d. The coe vrie is ced the coe freqec give e where is the ttetio d w is the re freqec;. Sice the OC is defied i ters of the gitde the she of the OC is oe rig. ote tht - e eq to d/or + cod ossi e ifiit;. If + <- the the OC is sce d the T does ot eist; The FT is seci cse of the T The fctio = or e is circe of it rdis i the - e d is ced the it circe. If the OC cotis the it circe the we c evte o the it circe. e e F ] e Therefore the discrete-tie Forier trsfor e viewed s seci cse of the -trsfor. Ees Ee. Positive-tie seqece Ee. egtive-tie seqece Ee. Two-sided seqece Core their OCs eros d oes.

3 // Proerties of the OC. The OC is ws oded circe sice the covergece coditio is o the gitde ;. The OC for right-sided seqeces < = is ws otside of circe of rdis -. if >= is cs seq.. The OC for eft-sided seqeces > = is ws iside of circe of rdis +. if <= is tics seqece. The OC for two-sided seqeces is ws oe rig - <<+ if it eists. Proerties of the OC 5. The OC for fiite-drtio seqeces< d > = is the etire -e. If < the =ifiit is ot i the OC. If > the = is ot i the OC; 6. The OC cot icde oe sice coverges ifor i there; 7. There is t est oe oe o the odr of OC of rtio ; 8. The OC is oe cotigos regio the OC does ot coe i ieces. Iortt roerties of the -trsfor. Lierit ] ; OC : OC OC. Se shiftig ] ; OC :. Freqec shiftig. Fodig OC ] ; OC : OC sced ] / ; OC : Iverted OC Iortt roerties of the -trsfor 5. Coe cojgtio * * * ] ; OC : OC 6. Differetitio i the -doi d ] ; OC : OC d 7. tiictio ] j 8. Covotio v / v v dv tiictio r OC : OC OC C * ] OC : OC OC

4 // Soe coo -trsfor irs OC Trsfor Seqece cos cos ] cos cos si ] si w w w w w w OC Trsfor Seqece Soe coo -trsfor irs Ees: Covotio: E. E.5 E.6 Usig -trsfor roerties d the - trsfor te deterie the -trsfor of cos.5 Iversio of the -trfor Fro the defiitio of the iverse -trsfor cottio reqires cotor evtio of coe itegr tht i geer is coicted rocedre. The ost rctic roch is to se the rti frctio esio ethod. It es se of the -trsfor te. The - trsfor however st e rtio fctio. This reqireet is geer stisfied i digit sig rocessig.

5 // 5 Cetr Ide Whe is rtio fctio of - it c e eressed s s of sie first-order fctors sig the rti frctio esio. The idivid seqeces corresodig to these fctors c the e writte dow sig the -trsfor te. ethod: Give Eress it s if rt ooi c rt rtio Proer This c e otied erforig ooi divisio if >= sig the decov fctio Perfor rti frctio esio o the roer rtio rt of to oti c Where is the -th oe of d is the reside t. It is ssed tht the oes re distict for which the resides re give For reeted oes the esio hs ore geer for. If oe hs tiicit r the its esio is give r r r r Write s c Fi se the retio fro Te to coete

6 // Ees E.7 Fid the iverse -trsfor of Sice the OC is ot secified there re three ossie OCs. ight-sided seqece Left-sided seqece Two-sided seqece Igir Prt e Prt t Ieettio A t fctio reside is vie to cote the reside rt d the direct or ooi ters of rtio fctio i -. Let B A c e rtio fctio i which the ertor d the deoitor oois re i scedig owers of -. The c] = reside The retred co vector cotis the resides; Co vector cotis the oes octios; ow vector c cotis the direct ter. If = =+r- is oe of tiicit r the the esio icdes the ter of the for r r Siir ]=residec c e sed to covert the rti frctio esio c to oois with coefficiets i row vectors d. 6

7 // 7 Ees E.8 Chec E.6 E.9 Cote the iverse -trsfor of E. Deterie the iverse -trsfor of So tht the restig seqece is cs d cotis o coe ers Pooi coefficiets=porootroot root].6.8. Sste reresettio i the -doi Siir to the freqec resose fctio e we c defie the -doi fctio ced the sste fctio. owever ie e eists for sstes tht ot e BIBO ste. Defiitio : The sste fctio is give h h h h ] h OC OC OC : Sste fctio fro the differece eqtio reresettio A B After fctoritio we oti -trsfor Where s re the sste eros d s re the sste oes. Ths c so e rereseted i the -doi sig oeero ot. This fct is sef i desigig sie fiters roer ceet of oes d eros. To deterie eros d oes of rtio : t fctio: roots: ooi--root o: root-- ooi

8 // e: ot the oes d eros give the ertor row vector d the deoitor row vector. Siir e: ots the eros i co vector d the oes i the co vector. Igir Prt Trsfer fctio reresettio If the OC of icdes it circe =e the we c evte o the it circe restig i freqec resose fctio or trsfer fctio e. e e e e e e e e j w e Iterrettio istrtio e Prt e or ] w] e e costt ier oier t ieettio w]=freq etrs the -oit freqec vector w d the -oit coe freqec resose vector of the sste. It is evted t oits eq sced rod the er hf of the it circe. w] = freq whoe Uses oits rod the whoe it circe for cottio. =freqw It retrs the freqec resose t freqecies desigted i vector w or etwee d i. etioshis etwee sste reresettio Eress i / cross ti d te iverse Te -trsfor sove for / Differece Eqtio Te DTFT sove for / Sstitte =e e Te iverse - trsfor Te -trsfor Te iverse DTFT h Te Forier trsfor The dshed ths eist o if the sste is ste 8

9 // 9 Stiit d Csit Theore -doi LTI stiit A LTI sste is ste if d o if the it circe is i the OC of Theore -doi cs LTI stiit A cs LTI sste is ste if d o if the sste fctio hs its oes iside the it circe. Sotios of the differece eqtios Lier costt coefficiet differece eqtios Prticr d hoogeeos sotio ero-it iiti coditio d the ero-stte resoses -trsfor Trsiet d sted-stte resoses The oe-sided -trsfor The oe-sided -trsfor of seqece is give ] ] ] The the se shiftig roert is give ] ] Sove differece eqtios with oero iiti coditios or with chgig its Sject to these iiti coditios: i i d i i Ee... d s t

10 // 9 Fors of the sotios oogeeos d rticr rts: rt rticr rt hoogeeos The hoogeeos rt is de to the sste oes d the rticr rt is de to the it oes. Fors of the sotios Trsiet d sted-stte resose: resose stte sted resose trsiet The trsiet resose is de to oes tht re iside the it circe whie the sted-stte resose is de to oes tht re o the it circe. ote tht whe the oes re otside the it circe the resose is tered oded resose. Fors of the sotios ero-it or iiti coditio d ero-stte resoses: resose it ero resose stte ero IC IC I S 8 } {

11 // t Ieettio =fiteric ic is eqivet iiti-coditio it rr ic = fitic d re the fiter coefficiet rr d re the iiti-coditio rrs fro the iiti coditios o d resective i the for efereces d Assiget Tetoo: 8 Chiese referece oo: 6 Eercises: :.d;.;. :.5;.8; ] ]