The z Transform. The Discrete LTI System Response to a Complex Exponential

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1 The Trsform The trsform geerlies the Discrete-time Forier Trsform for the etire complex ple. For the complex vrible is sed the ottio: jω x+ j y r e ; x, y Ω rg r x + y {} The Discrete LTI System Respose to Complex Expoetil ipt j [ ] x re Ω System h[] otpt [ ] [ ] y h H [ ] [ ] [ ] [ ] ( ) y h h h H ottio: eige fctio of y discrete-time LTI system : complex expoetil with o itry bsolte vle. ssocited eige vle. [ ] H h H ( )

2 [ ] y H fst compttio of the respose of y discrete-time LTI system to lier combitios of complex expoetils of the form: ipt [ ] c x System h[] otpt [ ] y ch trsfer fctio, trsform of the implse respose h[]. [ ] H h 3 Bilterl Trsform The bilterl trsform of the sigl x[] : jω { [ ]} [ ],, Z x X x r e r geerlitio of the discrete-time Forier trsform jω Z{ x[ ] }( ) ( x[ ] r ) e F { r x[ ] }( Ω); r jω { [ ]} F { [ ]} r Z x e x Ω Discrete-time Forier trsform prticlr cse of trsform o the it circle 4

3 Exmples [ ] [ ]. x σ, < X regio of covergece (ROC) set of vles of for which X() is coverget coverget: X ( ), > rtiol fctio, eros roots of mertor ; poles roots of deomitor. Pole/ero plot or costelltio (PZC). < 5 [ ] [ ]. x σ, < X coverget if: < X ( ) ; < oe pole i. 6 3

sme trsform ; differet ROCs X First exmple Secod exmple Regio of covergece ROC : set of vles of for which the series X() is coverget All the complex vles hvig the bsolte vle r stisfyig the coditio:

4 sme trsform ; differet ROCs X First exmple Secod exmple Regio of covergece ROC : set of vles of for which the series X() is coverget All the complex vles hvig the bsolte vle r stisfyig the coditio: belog to the ROC of X(). 7 Properties of the ROC of the Bilterl Trsform ROC of bilterl trsform c ot coti y pole for right sided sigls (icldig csl sigls), the ROC exteds otwrd from the otermost pole, for left sided sigls (icldig ticsl sigls), the ROC exteds iwrd from the iermost pole for ifiite drtio sigls, ROC is rig tht does t iclde poles boded o the iterior d exterior by pole. for fiite drtio sigls, the ROC is the etire - ple, except possibly or. stble system s trsfer fctio hs the poles iside the it circle. ROC is otside the it circle. 8 4

The ROC of X() is the itersectio of the ROCs of its compoets (rig) 9 x [ ] [ ] [ ] + [ ] 3.

5 [ ] [ ] + [ ] x x x first compoet of x[] is right sided; ROC is otside the circle with rdis R -. secod compoet of x[] is left sided, ROC is iside the circle with rdis R +. The ROC of X() is the itersectio of the ROCs of its compoets (rig) 9 x [ ] [ ] [ ] + [ ] 3., < < x x x Exmple x [ ] σ[ ], > x [ ] σ[ ] σ[ ], < Right sided (csl) sigl Left sided (ticsl) sigl ; < < ( ) 5

6 Iverse Trsform Z { X ( ) }[ ] X ( ) d x[ ] ; Γ ROC πj Γ Γ - coterclocwise closed pth (cotor) iclded i the regio of covergece. Γ - ecircles the origi d mst ecircle ll poles of X(). itegrl log cotor. trsforms i DSP : rtiol fctios. prtil frctio decompositio d tbles sigl trsform pirs (owig the pole/ero plot). The Bilterl Trsform Compttio Usig its pole/ero plot X() - rtiol fctio M ( ) N X ; ROC ( p ) poles d eros Z trsform withot costt X( ) lso ow Z trsform d costt 6

7 pole/ero plot ; p.5 Exmple X for.5.5 csl sigl, ROC: > p.5 it circle ROC discrete-time Forier trsform exists: jω jω e X ( Ω ) X ( e ) jω e.5.5e A it circle; (OA, Ox) Ψ the spectrm of sigl x[] OA X ( Ω ) ; Φ( Ω ) ψ ϕ AP 3 jω Geerl cse: M ( ) N X ; ROC ( p ) vectors A p, A o : (AZ, Ox) Ψ ; (AZ p, Ox) ϕ Mgitde d phse spectrm: M A M N X ( Ω ) ; Φ Arg+ N ψ ϕ A p The freqecy Ω legth of the rc (of circle) i rdis o the it circle, betwee its itersectio with the positive rel xis d the poit A, i trigoometric sese. 4 7

8 The Uilterl Trsform Z x X x { [ ]} [ ] ilterl trsform bilterl trsform of csl sigl ROC of ilterl trsform : etire complex ple or the otside regio of disc cetered i. csl systems described by differece eqtios, o ero iitil coditios. 5 ottios: Trsform Properties Z Z [ ], ; x [ ] Z Z [ ], ; [ ] x X ROC x X y Y ROC y Y y. Lierity [ ] [ ] [ ] + [ ] + x + by X + by, ROC ROC t lest x by X by Proof. Directly, sig the defiitios. Homewor - Prove it. x y 6 8

9 . Time shiftig Proof. If >, ROC. If <, ROC. ilterl trsform: x X, ROC x X ( ) x[ ], > bilterl trsform: m ( m + ) Z { x } x x[ m] x[ m] m m m m { } [ ] m m Z x x x m m m x[ m] + x[ m], > m m 7 3. Modltio i time Proof. { [ ]} [ ] [ ] ( jω jω jω jω Z e x e x x e X e Ω [ ] jω j, e x X e ROC More geerlly: x[ ] X, ROC x[ ] X Homewor - Prove it. 8 9

10 4. Time reversl x[ ] X ( ), ROC Proof. m m { [ ]} [ ] [ ] Z x x x m X m 9 5. Differetitio i time [ ] [ ] x x X ; ROC [ ] [ ] ( ) [ ] x x X x Proof. direct pplictio of the defiitios d previos properties. Homewor. Prove these properties.

11 6. Additio i time X x[ ] X ( ) ; ROC ROC + [ ] X x x [ ] ; > Proof. [ ] [ ] [ ] [ ] [ ] ( ) y x x y y X Y [ ] [ ] [ ] ( ) [ ]; [ ] [ ] x y y X Y y y x 7. Differetitio i domi dx x[ ] ; ROC d dx ( ) x[ ] d Proof. By direct pplictio of defiitios. Homewor. Prove it.

12 8. Complex cojgtio i time domi [ ] [ ] x X ; ROC x X Proof. By direct pplictio of defiitios. Homewor. Prove it Time covoltio (covoltio theorem) [ ], y [ ] [ ] [ ] [ ] [ ] x x y X Y ; ROC ROC t lest x y X Y Proof. x y { [ ] [ ]} ( [ ] [ ]) [ ] [ ] Z x y x y xy ( ) x y m [ ] [ ] m [ ] [ ] ym x Y X 4

13 . Prodct theorem or covoltio theorem i the complex domi d xy [ ] [ ] X( Y ), ROC j Γ Γ π d x[ ] y[ ] X( ) Y, ROC j Γ Γ π Proof. x x x x ROCx : R < < R ; ROCy : R < < R R < < R d R < < R y y y y x y x ROC : R R < < R R y + + Z{ x[ ] y[ ] } x[ ] y[ ] X ( ) d y[ ] j Γ π d d X ( ) y[ ] X ( ) Y j Γ j Γ π π 5 Prticlr Cses 6 3

14 . The Iitil Vle Theorem Proof. At the limit : [ ] lim lim x X X trsform of csl sigl X ( ) X ( ) x[ ] x[ ] + x[] + X X x x x [ ] + [] + [ ] 7. The Fil Vle Theorem [ ] lim [ ] lim ( ) lim ( ) x x X X Proof. + m ( m ) ( x[ + ] x[ ] ) x[ + ] X( ) x[ m] X( ) m m x m x X X x m [ ] [ ] [ ] ( x [ + ] x [ ]) ( ) X ( ) x[ ] lim lim ( x [ + ] x [ ]) lim ( x [ + ] x [ ]) lim ( x [ + ] x[ ] ) [ ] lim [ ] lim lim x x X X 8 4

15 Reltio betwee the Trsform d the Lplce Trsform Idel smplig: x t X s () δ( ) xˆ t x Ts t Ts { ˆ ()} ( s) L{ δ( s) } ( s) Discrete-time sigl: x( Ts) xd [ ] Z{ xd [ ] } xd [ ] x ( Ts ) L x t x T t T x T e sts { () ()} st [ ] { }; [ ] L x t δ t Z x x x T e Ts s d d s e sts 9 Usig the Trsform for the Stdy of the Discrete LTI Systems theorem of covoltio of discrete-time sigls. csl system, o ero iitil coditios. 3 5

The Trsfer Fctio for Discrete LTI System trsfer fctio H() - trsform of the implse respose of discrete-lti system. describes completely the system i the complex domi.

16 The Trsfer Fctio for Discrete LTI System trsfer fctio H() - trsform of the implse respose of discrete-lti system. describes completely the system i the complex domi. 3 Stble system: It hs freqecy respose. Discrete-time Forier trsform of the implse respose is coverget. The it circle belogs to the ROC of its trsfer fctio ROC Csl system: H () H() ROC is otside of disc. Csl d stble system: Uit circle belogs to the ROC: ROC. All poles of H(): iside the it disc p < 3 6

17 The Compttio of the Respose of Discrete LTI System with the Aid of the Trsform Kow: x[] d h [] H () Compte the trsform of the ipt sigl, X (). Prodct Y()H() X (). Iverse trsform the respose y[]. csl ipt sigl + csl system with o ero iitil coditios ilterl trsform 33 The Compttio of the Iverse Trsform There re three methods tht c be sed:. Direct compttio of the itegrl,. Prtil frctio expsio of the fctio Y(), 3. Power series expsio of fctio Y (). 34 7

18 . Prtil frctio expsio of the fctio Y () Y() rtiol fctio, rtio of polyomils i - or. We se -, deote - x ( trsforms expressed i fctio of - i tbles) Y( ) Y x I( x) Y x N N x D D x R x + D x Z δ[ ] I x c x c c 35 R x s m i + m m i ( ) D x x x x x b i x x m R x m D x i x x m s i d s R x b s! i x x s i dx D x x x 36 8

19 Exmple: d Order Z-Trsform Csl sigl y[] Y Y ; ROC: >.5 Y Y (.5)(.5) 8 8x 8 6 x 4 x x 4 x ( )( 4 ) [ ] 4 (.5) (.5) [ ] y σ [ ] [ ] For <.5 : y σ < < y[ ] σ [ ] 4(.5) σ[ ] For.5.5: Power series expsio of fctio Y () Expsio of fctio Y() ito power series Exmple # Y e, ROC: > e !!!! y [ ]! 38 9

20 Exmple #b Y e, ROC:, m e !! m! m! +!! ( ) ( ) ( ) ( ) m y[ ] δ[ ] σ[ ] σ!! m [ ] 39 Exmple #c Y l +, ROC: > csl sigl ( ) [ ] ( ) [ ] ( ) [ ] ( ) + + Y( ) y[ ] ; iitil vle theorem: y lim l + y σ 4

21 Exmple #d Y y, > [ ] σ[ ] 4 Exmple #e, sme trsform s #d, differet ROC Y( ), < ticsl sigl, trsform cotis oly powers of m [ ] [ ] Y y σ m 4

22 Discrete LTI Systems Described by Lier Costt Coefficiet Differece Eqtios N M [ ] [ ] y b x ; N M Y( ) b X( ) H( ) N M b N D trsfer fctio Trsfer fctio of discrete LTI system : rtiol fctio i or -. Zeros : roots of mertor N(); Poles : roots of deomitor D() 43 Csl d stble system: poles iside it circle < The iitil vle theorem c be pplied: p [ ] lim lim lim N h H H - fiite D The degree of the mertor of the trsfer fctio of csl d stble system smller or eql with the degree of its deomitor. 44

23 The Cotribtio of the Poles of Csl Discrete LTI System t its Implse Respose Cosider two cses: simple poles d doble poles. # Simple complex cojgted poles jωp p p p p r e d r e jωp re re jωp jωp H p p Cotribtio of the two poles i the implse respose : r p e jω * p + e jω p σ [] 45 r < p r > p r p Prtil implse respose decreses i time No istbility from these poles. Prtil implse respose icreses i time Istbility. The pir of poles re o the it circle. Prtil implse respose oscilltio with fixed mplitde tht persists eve fter the ed of the ipt sigl: ( Ω +Φ ) Asi p p System oscilltor, criticlly stble. 46 3

24 #. A pir of doble complex cojgte poles jωp p p p p r e d r e jωp jωp jωp Ω Ω p H( ) re p re re re j p j p ( p ) ( p ) Cotribtio of the two poles i the implse respose : r < p r > p ( Ω +Φ ) + si ( Ω +Ψ ) σ[ ] Arp si p p Arp p p Prtil implse respose decreses i time No istbility from these poles. Prtil implse respose icreses i time Istbility. 47 Respose of Discrete LTI System Described by Differece Eqtio differece eqtio, o-ero iitil coditios ilterl Z trsform N M N M, y[ ] bx[ ] Z{ y[ ] } bz{ x[ ] } N M Y y b X x + [ ] + [ ] Csl ipt sigl, x[-] for >: N M Y( ) + y[ ] b X( ) 48 4

25 Exmple y Y Y jω [] [ ] [], [] y x x e σ[], y[ ] ( ) ( Y ( ) + y[ ] ) ( ) e j e y jω ( e )( ) jω Ω [] + y[ ], ; >. jω e y[ ] + [ ] y + +. jω jω e e + jω ( + e ) + + σ ; <, jω jω e e [] >. 49 pole/ero plot ; p.5 First Order Systems y [ ] y [ ] x [ ] H( ) ; ROC [ ] [ ] >, < h σ p OA j( Ψ ϕ) j( Ω ϕ) H( Ω ) e e PA PA < <, mx. freq. respose for Ω, < <, mx. for Ωπ, Positive low-pss filter Negtive high-pss filter 5 5

Secod Order Systems y + y + y x H [ ] [ ] [ ] [ ] + + + + + 4, p, < 4 complex cojgted poles p, ρe ± jθ 4 rel poles. Csl system. Coditios for stbility,?? + 4 ρ < mgitde.

26 Secod Order Systems y + y + y x H [ ] [ ] [ ] [ ] , p, < 4 complex cojgted poles p, ρe ± jθ 4 rel poles. Csl system. Coditios for stbility,?? + 4 ρ < mgitde. Complex cojgted poles <, > 4. Rel poles: 5 4 ; > ; >. The prbol correspods to the existece of sigle doble rel pole. The freqecy respose of the system: H j ( Ω ϕ ϕ Ω e ). P A P A Mgitde: eve; Phse: odd 5 6

27 7 53 Trsfer Fctio of Eqivlet System for Seril or Prllel Itercoectios of two discrete LTI Systems [ ] [ ] [ ]. H H H h h h e e ; + + [] [] [ ]. H H H h h h e e ; 54 Digitl Filters Implemettio Forms Obtied Usig the Trsform [ ] [ ] [ ] [ ] [ ].,, N M M N y x b y N M x b y

28 Direct form I (N+M dders). First form of implemettio sig trsform. Ech dder hs ipts. Oe dder with N ipts (MN, ). 55 Direct form II (N dders MN). Secod form of implemettio sig trsform. Ech dder hs ipts. Two dders with N ipts ech (MN, ). 56 8

The z Transform. The Discrete LTI System Response to a Complex Exponential

The z Transform. The Discrete LTI System Response to a Complex Exponential The Trasform The trasform geeralies the Discrete-time Forier Trasform for the etire complex plae. For the complex variable is sed the otatio: jω x+ j y r e ; x, y Ω arg r x + y {} The Discrete LTI System