z-transform A generalization of the DTFT defined by

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Myrtle Dean
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1 The DTFT provides frequecy-domi represettio of discrete-time sigs d LTI discrete-time systems Becuse of the covergece coditio, i my cses, the DTFT of sequece my ot exist As resut, it is ot possie to mke use of such frequecy-domi chrcteritio i these cses A geeritio of the DTFT defied y X ( e ) x[ e eds to the -trsform -trsform my exist for my sequeces for which the DTFT does ot exist oreover, use of -trsform techiques permits simpe geric miputios 3 Cosequety, -trsform hs ecome importt too i the ysis d desig of digit fiters For give sequece g[, its -trsform ) is defied s g G ( ) [ where Re() + jim() is compex vrie 4 If we et r e reduces to, the the -trsform G ( r e ) g[ r e The ove c e iterpreted s the DTFT of the modified sequece { g[ r } Forr (i.e., ), -trsform reduces to its DTFT, provided the tter exists 5 The cotour is circe i the -pe of uity rdius d is ced the uit circe Like the DTFT, there re coditios o the covergece of the ifiite series g [ For give sequece, the set R of vues of for which its -trsform coverges is ced the regio of covergece (ROC) 6 From our erier discussio o the uiform covergece of the DTFT, it foows tht the series G ( r e ) g[ r e coverges if { g[ r summe, i.e., if g [ r } is soutey <

2 7 I geer, the ROC R of -trsform of sequece g[ is ur regio of the - pe: Rg < < Rg + where R g < Rg + ote:the -trsform is form of Luret series d is ytic fuctio t every poit i the ROC 8 Exmpe-Determie the -trsform X() of the cus sequece x[ α µ [ d its ROC ow X ( ) α µ [ α The ove power series coverges to X ( ), for α < α ROC is the ur regio > α Exmpe-The -trsform µ() of the uit step sequece µ[ c e otied from X ( ), for α < α y settig α : µ( ) for <, ROC is the ur regio < ote: The uit step sequece µ[ is ot soutey summe, d hece its DTFT does ot coverge uiformy Exmpe-Cosider the ti-cus sequece y[ α µ [ ] 9 Its -trsform is give y Y ( ) m m α α m m m α α m α, for α < α α ROC is the ur regio < α ote: The -trsforms of the two sequeces α µ[ d α µ [ ] re idetic eve though the two pret sequeces re differet Oy wy uique sequece c e ssocited with -trsform is y specifyig its ROC

3 The DTFT e ) of sequece g[ coverges uiformy if d oy if the ROC of the -trsform ) of g[ icudes the uit circe The existece of the DTFT does ot wys impy the existece of the -trsform Exmpe- The fiite eergy sequece h si ω [ c π < < LP, hs DTFT give y, ω ωc H LP ( e ), ωc < ω π which coverges i the me-squre sese 3 4 Te 6.: Commoy Used - Trsform Pirs However, h LP [ does ot hve -trsform s it is ot soutey summe for y vue of r Some commoy used -trsform pirs re isted o the ext side 5 6 Rtio s Rtio s 7 I the cse of LTI discrete-time systems we re cocered with i this course, pertiet -trsforms re rtio fuctios of Tht is, they re rtios of two poyomis i : P p + p +... ( ) ( ) + p p + ) D( ) ( ) d + d d + d The degree of the umertor poyomi P() is d the degree of the deomitor poyomi D() is A terte represettio of rtio - trsform is s rtio of two poyomis i : p + p +... ( ) + p + p ) d + d d + d 8 3

Rtio s Rtio s A rtio -trsform c e tertey writte i fctored form s p ) d ( ) ) ) ( ξ ( λ p d ( ξ ) ( λ ) At root ξ of the umertor poyomi ξ ), d s resut, these vues of re kow s the eros of ) At root λ

4 Rtio s Rtio s A rtio -trsform c e tertey writte i fctored form s p ) d ( ) ) ) ( ξ ( λ p d ( ξ ) ( λ ) At root ξ of the umertor poyomi ξ ), d s resut, these vues of re kow s the eros of ) At root λ of the deomitor poyomi λ ), d s resut, these vues of re kow s the poes of ) 9 Rtio s Cosider ( ) p G ( ξ) ( ) d ( λ) ote) hs fiite eros d fiite poes If > there re dditio eros t (the origi i the -pe) If < there re dditio poes t Rtio s Exmpe-The -trsform µ ( ) for >, hs ero t d poe t Rtio s Rtio s A physic iterprettio of the cocepts of poes d eros c e give y pottig the og-mgitude og ) s show o ext side for. 4 )

ROC of -trsform is importt cocept Without the kowedge of the ROC, there is o uique retioship etwee sequece d its -trsform Hece, the -trsform must wys e specified with its ROC 5 6 ROC of Rtio ROC of

5 Rtio s ROC of Rtio Oserve tht the mgitude pot exhiits very rge peks roud the poits. 4 ± j. 698 which re the poes of ) It so exhiits very rrow d deep wes roud the octio of the eros t. ± j. ROC of -trsform is importt cocept Without the kowedge of the ROC, there is o uique retioship etwee sequece d its -trsform Hece, the -trsform must wys e specified with its ROC 5 6 ROC of Rtio ROC of Rtio oreover, if the ROC of -trsform icudes the uit circe, the DTFT of the sequece is otied y simpy evutig the -trsform o the uit circe There is retioship etwee the ROC of the -trsform of the impuse respose of cus LTI discrete-time system d its BIBO stiity The ROC of rtio -trsform is ouded y the octios of its poes To uderstd the retioship etwee the poes d the ROC, it is istructive to exmie the poe-ero pot of -trsform Cosider gi the poe-ero pot of the - trsform µ() ROC of Rtio I this pot, the ROC, show s the shded re, is the regio of the -pe just outside the circe cetered t the origi d goig through the poe t 3 ROC of Rtio Exmpe-The -trsform H() of the sequece h[ (.6) µ [ is give y H( ), +. 6 >.6 Here the ROC is just outside the circe goig through the poit. 6 5

6 ROC of Rtio ROC of Rtio A sequece c e oe of the foowig types: fiite-egth, right-sided, eft-sided d two-sided I geer, the ROC depeds o the type of the sequece of iterest Exmpe-Cosider fiite-egth sequece g[ defied for, where d re o-egtive itegers d g[ < Its -trsform is give y ) + + g g [ ] [ ] 3 3 ROC of Rtio ROC of Rtio ote:) hs poes t d poes t As c e see from the expressio for ), the -trsform of fiite-egth ouded sequece coverges everywhere i the -pe except possiy t d/or t Exmpe-A right-sided sequece with oero smpe vues for is sometimes ced cus sequece Cosider cus sequece u [ Its -trsform is give y ( ) u[ U ROC of Rtio It c e show tht U ( ) coverges exterior to circe R, icudig the poit O the other hd, right-sided sequece u [ with oero smpe vues oy for with oegtive hs -trsform U ( ) with poes t The ROC of U ( ) is exterior to circe R, excudig the poit 36 ROC of Rtio Exmpe-A eft-sided sequece with oero smpe vues for is sometimes ced ticus sequece Cosider ticus sequece v [ Its -trsform is give y ( ) v[ V 6

7 37 ROC of Rtio It c e show tht V ( ) coverges iterior to circe R 3, icudig the poit O the other hd, eft-sided sequece with oero smpe vues oy for with oegtive hs -trsform V ( ) with poes t The ROC of V ( ) is iterior to circe R 4, excudig the poit ROC of Rtio Exmpe-The -trsform of two-sided sequece w[ c e expressed s W ( ) w[ w[ + [ w The first term o the RHS, w[, c e iterpreted s the -trsform of right-sided sequece d it thus coverges exterior to the circe R 5 38 ROC of Rtio The secod term o the RHS, w[, c e iterpreted s the -trsform of eftsided sequece d it thus coverges iterior to the circe R 6 If R 5 < R 6, there is overppig ROC give y R 5 < < R 6 If R 5 > R 6, there is o overp d the -trsform does ot exist 39 ROC of Rtio Exmpe-Cosider the two-sided sequece u[ α where α c e either re or compex Its -trsform is give y U ( ) α The first term o the RHS coverges for > α, wheres the secod term coverges for < α 4 α + α 4 ROC of Rtio There is o overp etwee these two regios Hece, the -trsform of u[ α does ot exist 4 ROC of Rtio The ROC of rtio -trsform cot coti y poes d is ouded y the poes To show tht the -trsform is ouded y the poes, ssume tht the -trsform X() hs simpe poes t α d β Assume tht the correspodig sequece x[ is right-sided sequece 7

43 ROC of Rtio The x[ hs the form x[ ( r α + r β ) µ [ o], α < β where o is positive or egtive iteger ow, the -trsform of the right-sided sequece γ µ [ o] exists if for some o γ < The coditio ROC of

8 43 ROC of Rtio The x[ hs the form x[ ( r α + r β ) µ [ o], α < β where o is positive or egtive iteger ow, the -trsform of the right-sided sequece γ µ [ o] exists if for some o γ < The coditio ROC of Rtio o < hods for > γ ut ot for γ Therefore, the -trsform of x r α + r β µ [ ], γ ( ) α < β [ o hs ROC defied y β < ROC of Rtio Likewise, the -trsform of eft-sided sequece x[ ( r α + r β ) µ [ o], α < β hs ROC defied y < α Fiy, for two-sided sequece, some of the poes cotriute to terms i the pret sequece for < d the other poes cotriute to terms 46 ROC of Rtio The ROC is thus ouded o the outside y the poe with the smest mgitude tht cotriutes for < d o the iside y the poe with the rgest mgitude tht cotriutes for There re three possie ROCs of rtio -trsform with poes t α d β ( α < β ) 47 ROC of Rtio 48 ROC of Rtio I geer, if the rtio -trsform hs poes with R distict mgitudes, the it hs R + ROCs Thus, there re R +distict sequeces with the sme -trsform Hece, rtio -trsform with specified ROC hs uique sequece s its iverse -trsform 8

9 49 ROC of Rtio The ROC of rtio -trsform c e esiy determied usig ATLAB [,p,k] tfp(um,de) determies the eros, poes, d the gi costt of rtio -trsform with the umertor coefficiets specified y the vector um d the deomitor coefficiets specified y the vector de 5 ROC of Rtio [um,de ptf(,p,k) impemets the reverse process The fctored form of the -trsform c e otied usig sos psos(,p,k) The ove sttemet computes the coefficiets of ech secod-order fctor give s L 6 mtrix sos 5 sos where ROC of Rtio ) L L k L k k + + L k k L + + L k k L 5 ROC of Rtio The poe-ero pot is determied usig the fuctio pe The -trsform c e either descried i terms of its eros d poes: pe(eros,poes) or, it c e descried i terms of its umertor d deomitor coefficiets: pe(um,de) 53 ROC of Rtio Exmpe- The poe-ero pot of ) otied usig ATLAB is show eow Imgiry Prt Re Prt poe o ero 54 Iverse Geer Expressio: Rec tht, for r e, the -trsform ) give y G ( ) g[ g[ r e is merey the DTFT of the modified sequece g[ r Accordigy, the iverse DTFT is thus give y π g[ r π re ) e dω π 9

10 55 Iverse By mkig chge of vrie r e, the previous equtio c e coverted ito cotour itegr give y g[ ) d πj C where C is coutercockwise cotour of itegrtio defied y r 56 Iverse But the itegr remis uchged whe is repced with y cotour C ecircig the poit i the ROC of ) The cotour itegr c e evuted usig the Cuchy s residue theorem resutig i g[ residuesof ) t the poes iside C The ove equtio eeds to e evuted t vues of d is ot pursued here 57 Iverse Trsform y Prti-Frctio Expsio A rtio -trsform ) with cus iverse trsform g[ hs ROC tht is exterior to circe Here it is more coveiet to express ) i prti-frctio expsio form d the determie g[ y summig the iverse trsform of the idividu simper terms i the expsio 58 Iverse Trsform y Prti-Frctio Expsio A rtio ) c e expressed s ) P( ) D( ) pi i di i i i If the ) c e re-expressed s P ( ) ) η + D( ) where the degree of P () is ess th 59 Iverse Trsform y Prti-Frctio Expsio The rtio fuctio proper frctio Exmpe-Cosider ( )/ D( ) is ced ) By og divisio we rrive t ) P 3 6 Iverse Trsform y Prti-Frctio Expsio Simpe Poes: I most prctic cses, the rtio -trsform of iterest ) is proper frctio with simpe poes Let the poes of ) e t λ k, k A prti-frctio expsio of ) is the of the form ρ ) λ

11 Iverse Trsform y Prti-Frctio Expsio ρ The costts i the prti-frctio expsio re ced the residues d re give y ρ ( λ ) ) λ Ech term of the sum i prti-frctio expsio hs ROC give y > λ d, thus hs iverse trsform of the form ρ ( λ ) µ [ 6 6 Iverse Trsform y Prti-Frctio Expsio Therefore, the iverse trsform g[ of ) is give y g[ ρ ( λ ) µ [ ote:the ove pproch with sight modifictio c so e used to determie the iverse of rtio -trsform of ocus sequece Iverse Trsform y Prti-Frctio Expsio Exmpe-Let the -trsform H() of cus sequece h[ e give y ( + ) + H( ) (. )( +. 6) (. )( +. 6 A prti-frctio expsio of H() is the of the form ρ ρ H ( ) ) Iverse Trsform y Prti-Frctio Expsio ow + ρ (. ) H ( ) d. + ρ ( +.6 ) H ( ) Iverse Trsform y Prti-Frctio Expsio Hece. 75 H( ) The iverse trsform of the ove is therefore give y h[.75(.) µ [.75(.6) µ [ Iverse Trsform y Prti-Frctio Expsio utipe Poes: If ) hs mutipe poes, the prti-frctio expsio is of sighty differet form Let the poe t ν e of mutipicity L d the remiig L poes e simpe d t λ, L 65 66

12 Iverse Trsform y Prti-Frctio Expsio The the prti-frctio expsio of ) is of the form L ρ L γ ) η + + i λ i ( ν i ) γ i where the costts re computed usig Li d L γi [( ν ) ) ], Li Li ( L i)!( ν) d( ) ν i L The residues re ccuted s efore ρ Prti-Frctio Expsio Usig ATLAB [r,p,k] residue(um,de) deveops the prti-frctio expsio of rtio -trsform with umertor d deomitor coefficiets give y vectors um d de Vector r cotis the residues Vector p cotis the poes Vector k cotis the costtsη 69 Prti-Frctio Expsio Usig ATLAB [um,deresidue(r,p,k) coverts -trsform expressed i prti-frctio expsio form to its rtio form Iverse vi Log Divisio The -trsform ) of cus sequece {g[} c e expded i power series i I the series expsio, the coefficiet mutipyig the term is the the -th smpe g[ For rtio -trsform expressed s rtio of poyomis i, the power series expsio c e otied y og divisio 7 7 Iverse vi Log Divisio Exmpe-Cosider + H ( ) +.4. Log divisio of the umertor y the deomitor yieds 3 4 H( ) As resut { h [ } { }, 7 Iverse Usig ATLAB The fuctio imp c e used to fid the iverse of rtio -trsform ) The fuctio computes the coefficiets of the power series expsio of ) The umer of coefficiets c either e user specified or determied utomticy

13 73 Te 6.: Properties 74 Properties Exmpe-Cosider the two-sided sequece v[ α µ [ β µ [ ] Let x[ α µ [ d y[ β µ [ ] with X() d Y() deotig, respectivey, their -trsforms ow X ( ), > α α d Y ( ), < β β 75 Properties Usig the ierity property we rrive t V ( ) X ( ) + Y( ) + α β The ROC of V() is give y the overp regios of > α d < β If α < β, the there is overp d the ROC is ur regio α < < β If α > β, the there is o overp d V() does ot exist 76 Properties Exmpe-Determie the -trsform d its ROC of the cus sequece x[ r (cosωo) µ [ We c express x[ v[ + v*[ where v[ r e [ ] [ ] o µ α µ The -trsform of v[ is give y V ( ) α r e j ωo, > α r 77 Properties Usig the cojugtio property we oti the -trsform of v*[ s V *( *), α* r e o > α Fiy, usig the ierity property we get X ( ) V ( ) + V *( *) + r e o r e o 78 Properties or, ( r cosω X o ) ( ), > r (r cosω ) + r o Exmpe-Determie the -trsform Y() d the ROC of the sequece y[ ( + ) α µ [ We c write y [ x[ + x[ where x[ α µ [ 3

14 79 Properties ow, the -trsform X() of x[ α µ [ is give y X ( ), > α α Usig the differetitio property, we rrive t the -trsform of x[ s dx ( ) α, d ( α ) > α 8 Properties Usig the ierity property we fiy oti Y ( ) α + α ( α ) ( α ), > α 8 LTI Discrete-Time Systems i the Trsform Domi A LTI discrete-time system is competey chrcteried i the time-domi y its impuse respose sequece {h[} Thus, the trsform-domi represettio of discrete-time sig c so e equy ppied to the trsform-domi represettio of LTI discrete-time system 8 LTI Discrete-Time Systems i the Trsform Domi Such trsform-domi represettios provide dditio isight ito the ehvior of such systems It is esier to desig d impemet these systems i the trsform-domi for certi ppictios We cosider ow the use of the DTFT d the -trsform i deveopig the trsformdomi represettios of LTI system 83 Fiite-Dimesio LTI Discrete-Time Systems I this course we sh e cocered with LTI discrete-time systems chrcteried y ier costt coefficiet differece equtios of the form: k k d y[ k] k k p x[ k] 84 Fiite-Dimesio LTI Discrete-Time Systems Appyig the -trsform to oth sides of the differece equtio d mkig use of the ierity d the time-ivrice properties of Te 6. we rrive t d k k k Y( ) p X ( ) where Y() d X() deote the -trsforms of y[ d x[ with ssocited ROCs, respectivey k k k 4

15 Fiite-Dimesio LTI Discrete-Time Systems A more coveiet form of the -domi represettio of the differece equtio is give y k k dk Y( ) pk X ( ) k k 85 5

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS. Assoc. Prof. Dr. Burak Kelleci. Spring 2018

DEPARTMENT OF ELECTRICAL &ELECTRONICS ENGINEERING SIGNALS AND SYSTEMS Assoc. Prof. Dr. Bur Kelleci Sprig 8 OUTLINE The Z-Trsform The Regio of covergece for the Z-trsform The Iverse Z-Trsform Geometric