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

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1 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 Respose to a Complex Expoetial ipt j [ ] x re Ω System h[] otpt [ ] [ ] y h H [ ] [ ] [ ] [ ] ( ) y h h h H otatio: For ay discrete-time LTI system: -the eige fctio: -the eige vale: [ ] H h H ( )

2 [ ] y H fast comptatio of the respose of ay discrete-time LTI system to liear combiatios of complex expoetials of the form: ipt [ ] c x System h[] otpt [ ] y ch trasfer fctio, trasform of the implse respose h[]. [ ] H h 3 Bilateral Trasform The bilateral trasform of the sigal x[] : jω { [ ]} [ ],, Z x X x r e r geeraliatio of the discrete-time Forier trasform jω Z{ x[ ] }( ) ( x[ ] r ) e F { r x[ ] }( Ω); r jω { [ ]} F { [ ]} r Z x e x Ω Discrete-time Forier trasform particlar case of trasform o the it circle 4

3 Examples [ ] [ ]. x aσ, a< X a a regio of covergece (ROC) set of vales of for which X() is coverget coverget: X ( ), a a > ratioal fctio, eros roots of merator ; poles roots of deomiator. Pole/ero plot or costellatio (PZC). a < 5 [ ] [ ]. x aσ, a< X a a a a a coverget if: a < X ( ) ; a < a a a oe pole i a. 6 3

4 same trasform ; differet ROCs a X First example Secod example Regio of covergece ROC : set of vales of for which the series X() is coverget. The Z trasform exists where the Forier trasform of x[]r - exists, r 7 Properties of the ROC of the Bilateral Trasform ROC of a bilateral trasform ca ot cotai ay pole for right sided sigals (icldig casal sigals), the ROC exteds otward from the otermost pole, for left sided sigals (icldig aticasal sigals), the ROC exteds iward from the iermost pole for ifiite dratio sigals, ROC is a rig that does t iclde poles boded o the iterior ad exterior by a pole. for fiite dratio sigals, the ROC is the etire - plae, except possibly or. a casal ad stable system s trasfer fctio has the poles iside the it circle. ROC is otside the it 8 circle. 4

5 [ ] [ ] + [ ] x x x first compoet of x[] is right sided; ROC is otside the circle with radis R -. secod compoet of x[] is left sided, ROC is iside the circle with radis R +. The ROC of X() is the itersectio of the ROCs of its compoets (rig) 9 x [ ] a a [ ] [ ] + [ ] 3., < < x x x Example x [ ] a σ[ ], > a a x a [ ] σ[ ] σ[ ], < a a a Right sided (casal) sigal Left sided (aticasal) sigal a a ; a< < a a ( a) a 5

6 Iverse Trasform Z { X ( ) }[ ] X ( ) d x[ ] ; Γ ROC πj Γ Γ - coterclocwise closed path (cotor) iclded i the regio of covergece. Γ - ecircles the origi ad mst ecircle all poles of X(). itegral alog a cotor. trasforms i DSP : ratioal fctios. partial fractio expasio ad tables sigal trasform pairs (owig the pole/ero plot). The Bilateral Trasform comptatio sig its pole/ero plot X() - ratioal fctio M ( ) N X ; ROC ( p ) poles ad eros Z trasform withot costat X( ) also ow Z trasform ad costat 6

7 pole/ero plot ; p.5 Example X for.5.5 casal sigal, ROC: > p.5 it circle ROC discrete-time Forier trasform exists: jω jω e X ( Ω ) X ( e ) jω e.5.5e A it circle; (OA, Ox) Ψ the spectrm of sigal x[] OA X ( Ω ) ; Φ( Ω ) ψ ϕ AP 3 jω Geeral case: M ( ) N X ; ROC ( p ) vectors A p, A o : (AZ, Ox) Ψ ; (AZ p, Ox) ϕ Magitde ad phase spectrm: M A M N X ( Ω ) ; Φ Arg+ N ψ ϕ A p The freqecy Ω legth of the arc (of circle) i radias o the it circle, betwee its itersectio with the positive real axis ad the poit A, i trigoometric sese. 4 7

8 The Uilateral Trasform Z x X x { [ ]} [ ] For casal sigals, ilateral trasform bilateral trasform ROC: etire complex plae or the otside regio of a disc cetered i. Usefl for casal systems described by differece eqatios, with o ero iitial coditios. 5 otatios: Trasform Properties Z Z [ ], ; x [ ] Z Z [ ], ; [ ] x X ROC x X y Y ROC y Y y. Liearity [ ] [ ] [ ] + [ ] + ax + by ax + by, ROC ROC at least ax by ax by Proof. Directly, sig the defiitios. Homewor - Prove it. x y 6 8

9 . Time shiftig Proof. If >, ROC. If <, ROC. ilateral trasform: x X, ROC x X ( ) x[ ], > bilateral trasform: 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. Modlatio 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 geerally: x[ ] X, ROC x[ ] X Homewor - Prove it. 8 9

10 4. Time reversal x[ ] X ( ), ROC Proof. m m { [ ]} [ ] [ ] Z x x x m X m 9 5. Differetiatio i time [ ] [ ] x x X ; ROC [ ] [ ] ( ) [ ] x x X x Proof. direct applicatio of the defiitios ad 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. Differetiatio i domai dx x[ ] ; ROC d dx ( ) x[ ] d Proof. By direct applicatio of defiitios. Homewor. Prove it.

12 8. Complex cojgatio i time domai [ ] [ ] x X ; ROC x X Proof. By direct applicatio of defiitios. Homewor. Prove it Time covoltio (covoltio theorem) [ ], y [ ] [ ] [ ] [ ] [ ] x x y X Y ; ROC ROC at least 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 domai 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 ad 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 Particlar Cases 6 3

14 . The Iitial Vale Theorem Proof. At the limit : [ ] lim lim x X X trasform of a casal sigal X ( ) X ( ) x[ ] x[ ] + x[] + X X x x x [ ] + [] + [ ] 7. The Fial Vale 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 Relatio betwee the Trasform ad the Laplace Trasform Ideal samplig: a x t X s () δ( ) xˆ t xa Ts t Ts { ˆ ()} a( s) L{ δ( s) } a( s) Discrete-time sigal: xa( Ts) xd [ ] Z{ xd [ ] } xd [ ] xa ( Ts ) a L x t x T t T x T e sts Laplace of the sampled aalog sigal { () ()} st [ ] { }; [ ] L x t δ t e Z x x x T st e s a Ts s d d a s 9 Usig the Trasform for the Stdy of the Discrete LTI Systems The trasform is sefl for stdyig discrete-time LTI systems (theorem of covoltio of discrete-time sigals) For a casal system, with o ero iitial coditios, the ilateral trasform is sed. 3 5

16 The Trasfer Fctio for a Discrete LTI System trasfer fctio H() - trasform of the implse respose of discrete-lti system. describes completely the system i the complex domai. 3 Stable system: It has a freqecy respose. Discrete-time Forier trasform of the implse respose is coverget. The it circle belogs to the ROC of its trasfer fctio ROC Casal system: H () H() ROC is otside of a disc. Casal ad stable system: Uit circle belogs to the ROC: ROC. All poles of H(): iside the it disc p < 3 6

17 Comptatio of the respose of a discrete LTI system sig the trasform If we ow the ipt sigal, x[] ad the system, h[] H (), Compte the trasform of the ipt, X (). Compte the prodct Y()H() X (). Apply iverse trasform the respose y[]. For a casal ipt sigal ad a casal system with o ero iitial coditios, we se the ilateral trasform 33 The Comptatio of the Iverse Trasform There are three methods that ca be sed:. Direct comptatio of the itegral,. Partial fractio expasio of the fctio Y(), 3. Power series expasio of fctio Y (). 34 7

18 . Partial fractio expasio of the fctio Y () Y() ratioal fctio, ratio of polyomials i - or. We se -, deote - x ( trasforms expressed i fctio of - i tables) 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 a s m i + m m i ( ) D x x x x x b i a 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 Example: d order Z-Trasform Casal sigal y[] Y Y ; ROC: >.5 Y Y (.5)(.5) 8 8x 8 6 x 4 x x 4 x ( )( 4 ) [ ] [ ] For >.5: y σ [ ] [ ] For <.5 : y σ [ ] [ ] [ ] For.5 < <.5: y 4.5 σ 4.5 σ Power series expasio of fctio Y () Expasio of fctio Y() ito a power series Example #a Y e, ROC: > e !!!! y [ ]! 38 9

20 Example #b Y e, ROC:, m e !! m! m! +!! ( ) ( ) ( ) ( ) m y[ ] δ[ ] σ[ ] σ!! m [ ] 39 Example #c Y l + a, ROC: > a casal sigal ( ) [ ] ( a ) [ ] ( ) [ ] ( ) + + a a Y( ) y[ ] ; iitial vale theorem: y lim l + y aσ 4

21 Example #d Y a y a a, > [ ] σ[ ] 4 Example #e, same trasform as #d, differet ROC Y( ), < a a a aticasal sigal, trasform cotais oly powers of m [ ] [ ] Y a a y a σ m 4

22 Discrete LTI Systems Described by Liear Costat Coefficiet Differece Eqatios N M [ ] [ ] a y b x ; a N M a Y( ) b X( ) H( ) N M b N D a trasfer fctio Trasfer fctio of a discrete LTI system : ratioal fctio i or -. Zeros : roots of merator N(); Poles : roots of deomiator D() 43 Casal ad stable system: poles iside it circle < The iitial vale theorem ca be applied: p Degree M [ ] lim lim lim N h H H - fiite D Degree N The degree of the merator of the trasfer fctio of a casal ad stable system is smaller or eqal with the degree of its deomiator, M N 44

23 The Cotribtio of the Poles of a Casal Discrete LTI System at its Implse Respose Cosider two cases: simple poles ad doble poles. # Simple complex cojgated poles jωp p p p p r e ad r e jωp a a re re jωp jωp H p p Cotribtio of the two poles i the implse respose : r p a e jω * p + a e jω p σ [] 45 r < p r > p r p Partial implse respose decreases i time No istability from these poles. Partial implse respose icreases i time Istability. The pair of poles are o the it circle. Partial implse respose oscillatio with fixed amplitde that persists eve after the ed of the ipt sigal: ( Ω +Φ ) Asi p p System oscillator, critically stable. 46 3

24 #. A pair of doble complex cojgate poles jωp p p p p r e ad r e jωp jωp jωp Ω Ω p a a a a 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 Partial implse respose decreases i time No istability from these poles. Partial implse respose icreases i time Istability. 47 Respose of a Discrete LTI System Described by a Differece Eqatio differece eqatio, o-ero iitial coditios ilateral Z trasform N M N M, ay[ ] bx[ ] a az{ y[ ] } bz{ x[ ] } N M a Y y b X x + [ ] + [ ] Casal ipt sigal, x[-] for >: N M a Y( ) + y[ ] b X( ) 48 4

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

26 Secod Order Systems y + ay + ay x H [ ] [ ] [ ] [ ] + a + a + a + a a ± a 4a, p, a < 4 a complex cojgated poles p, ρe ± jθ a 4 a real poles. The system is cosidered casal ad stable system. a + 4a a ρ a < magitde The coditios imposed o a, a for stability are :. Complex cojgated poles a a <, a > 4. Real poles: a 4a ; a > a ; a > a. 5 The parabola correspods to the existece of a sigle doble real pole. The freqecy respose of the system: H j ( Ω ϕ ϕ Ω e ). P A P A Magitde: eve; Phase: odd 5 6

27 7 53 Trasfer Fctio of Eqivalet System for Serial or Parallel Itercoectios of two discrete LTI Systems [ ] [ ] [ ]. H H H h h h e e ; + + [] [] [ ]. H H H h h h e e ; 54 Digital Filters Implemetatio Forms Obtaied Usig the Trasform [ ] [ ] [ ] [ ] [ ].,, N M M N y a x b y N M a x b y a

28 Direct form I (N+M adders). First form of implemetatio sig trasform. Each adder has ipts. Oe adder with N ipts (MN, a ). 55 Direct form II (N adders MN). Secod form of implemetatio sig trasform. Each adder has ipts. Two adders with N ipts each (MN, a ). 56 8

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

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