Definition of z-transform.

Transcription

1 - Trasforms Frequecy domai represetatios of discretetime sigals ad LTI discrete-time systems are made possible with the use of DTFT. However ot all discrete-time sigals e.g. uit step sequece are guarateed to have DTFT because of covergece coditio. As a result for these cases we are ot able to use such frequecy domai characteriatio.

2 Defiitio of -Trasform. DTFT is give by - Ge ω Certai fuctio of g[] will ot allow the above summatio to coverge. Therefore we eed to fid a way of overcomig this o - covergece. Let us modified the sigal g[] to g[]r DTFT of g[]r g[ ] e - ω is Ge. ω g[ ] r - e ω g[ ] re ω. Lettig re ω, therefore G g[ ]... Eq.6.

3 where re Regio of Covergece. Aother way lookig at the trasform G - trasform is the DTFT of modified sigal ω. g[]r g[ ] By choosig the right value of r, we ca make the summatio above to coverge. The values of r that make this covergece possible is kow as Regio of Covergece ROC. ROC are circles or aular rigs i the - plae. -..-

4 Poit i a Complex - plae Imagiary Axis re ω r ω Real Axis Uit Circle

5 Aular rig i a Complex - plae Imagiary Axis r ω re ω ROC Real Axis Uit Circle

6 Example 6. -Trasform of Causal What is X X Expoetial Sequece. Usig eq. 6., the - trasform of x[ ] G The above summatio is a sum of - α oly if The - trasform here is a algebraic expressio but must be accompay with the covergece or else it is -, x[] g[ ] α µ [ ] α ot valid. <. α geometric series coditio of ROC here is outside a circleaular regio > α - α µ [ ]? 0.

7 Amplitude r Time idex α Amplitude x[ ] α µ [ ]. µ [ ] Time idex

8 Imagiary Axis ROC α < Real Axis Uit Circle

9 If α, x[] DTFT of 0 X e oly if Uit step sequece The uit step sequece is hece it DTFT does But - trasform of - α µ [ ] ω - DTFT does - µ [ ] µ [ ]: - µ [ ] e ot coverge. - < i.e. >.ROC., µ [ ]. ω ot exit ot absolutely summable,

10 Imagiary Axis ROC Real Axis Uit Circle Sice uit circle is ot i ROC, DTFT of uit step does ot coverge.

11 Example 6. -Trasform of Aticausal Expoetial Sequece. What is the - trasform of x[] α µ [ ]?{left had sequece} Usig eq. 6., X α - α, m α x[ ] m m 0 α provided that α α µ [ ] m <. m 0 α m α. - α X - α α α The ed algebraic expressio is exactly the same as i example 6., the oly that is differet here is ROC, α <. ROC is iside a circleaular regio < α - provided that α <.

12 Imagiary Axis α < ROC Real Axis Uit Circle

13 -Trasform of Fiite Sequeces G If g[]is fiite e.g.g[] [,-,3,-,0,3,7]. G ROC for this G is the etire - plae except for If g[]is fiite e.g.g[] [-,0,3,7]. G ROC for this G is the etire - plae except for If g[]is fiite e.g.g[] [,-,3,-] G g[ ] g[ ] g[ ] g[ ] ROC for this G is the etire - plae except for ad 0. 3.

14 Ratioal -trasforms - plae. 0origi of poles at N M - there are additiol M, N If - plae. 0origi of eros at N - M there are additiol M, N If. i.e. obtaied by takig H H o - plae. are N poles of 0. i.e. obtaied by takig H o - plae. H are M eros of Factorig above equatios : polyomials i : ratio of Alterate Represetatio as : ad D i P two polyomials Ratio of < > Π Π Π Π l l l N l l M l M N l N l l M l N N N N N M M M M M M N N N N N M M M M Eq d p d p H Eq d d d d d p p p p p H Eq d d d d d p p p p p D P H λ ξ λ ξ λ ξ

15 Examples of Ratioal -Trasforms - trasform of µ [ ]: µ - havig a ero at 0, ad a pole at. H -, for >.ROC Cougate poles at 0.4 ± 0.698, - Cougate eros at. ±.

16

17

18

19

20

21

22

23

24 ROC for right-sided sequece

25

26

27

28 Methods of computig the Iverse -trasform Usig Cauchy s Residue Theorem Table look-up Partial Fractio Expasio Power Series Expasio or Log Divisio.

29 Iverse -Trasform via Table Example 6.. H H α µ [ ] h[] Look-Up 0.5, > From Table 6.: - ZT 0.5 α α µ [ ], > α..

30 Iverse -Trasform via Partial Fractio Expasio Example H This is a improper fuctio sice M 3 > N. Perform log divisio first by reversig order of both polyomial s - 0.3/ / 0. H to get at the proper fuctio. H

31 Cotiue Example Now from the proper fuctio ad b ± b 4ac quadratic formula, a - 4 ± 6 0 Roots of deomiator of proper fuctio ± Performiig the partial fractio expasio for the proper fuctio :

32 Cotiue Example 6.3 ] [ ] [ ] [.5 ] [ 3.5 h[]., ] [ ad ZT µ µ δ δ α α α µ α >

33

34 Iverse -Trasform Via Power Series Expasio/Log Divisio... 4] [ 0.4 3] [ 0.4 ] [ 0.5 ] [.6 ] [ ] [ H H Example h δ δ δ δ δ

35 Properties of -Trasform Liearity Time Shiftig Scalig i the -domai Time Reversal Time Expasio Cougatio Covolutio Differetiatio i the -domai The iitial-value Theorem.

36 -T Property associated with Liearity T x [ ] X with ROC deoted by R x T [ ] X with ROC deoted by R ax [ ] bx I is LT [ ] ax bx the symbol for itersect with. with ROC cotaiig R I R.

37 -T Properties cotiued T x[ ] X, with ROC R, Time Shiftig : - the x[ 0 T ] 0 X, with ROC R,except for possible additio or deletio of the origi or ifiity. Scalig i the - domai : - T the 0x[ ] X 0, with ROC 0 R. Time Reversal: - T the x[ ] X, with ROC. R Time Expasio : - x k [ ] x[ ] if is a multiple of k k, 0.if is ot a multiple of k. the x k T [ ] X k, with ROC R k Cougatio : - the x * T [ ] X * *, with ROC R.

38 -T Property associated with Covolutio x T [ ] X with ROC deoted by R x T [ ] X with ROC deoted by R x I [ ]* x is T [ ] X X the symbol for itersect with. with ROC cotaiig R I R.

39 T Properties cotiued T x[ ] X, with ROC R, Differetiatio i the - Domai : - T dx the x[ ], with ROC d R. Iitial - value theorem : If x[] 0 for < 0, the x[0] Lim X. -

40 Aalysis & Characteriatio of LTI systems usig -Trasforms. If x[] where, y [ ] is eigefuctio of H is eigevalue of H., LTI system LTI system, x[] X h[] H y[]x[]*h[] YX.H H is kow as System Fuctio or Trasfer Fuctio Frequecy respose H with e ω uit circle i ROC

41 Causality. A LTI system is causal if : - Its impulse respose h[] 0 for < 0 ie. right -sided. or i other words : - ROC of its system fuctio H is the exterior of the circle icludig ifiity. A discrete - time LTI system with ratioal system fuctio H is causal if ad oly if : - a the ROC is the exterior of a circle outside the outermost pole; ad b with H expressed as a ratio of polyomials i, the order of the umerator caot be greater tha the order of the deomiator.

42 Stability. A LTI system is stable if ad oly if:- Its Impulse respose h[] is absolutely summable. Or Fourier Trasform of h[] coverges. Or the ROC of its system fuctio H icludes the uit circle, For causal LTI system, all poles of H must be i the uit circle,

43 Example 6.4 Iverse -trasform, Causality & Stability. H Determie all the possible impulse resposes h[] for the give H above. Associate each oe of Usig Partial - Fractio Expasio : - H.0, Poles are 0. ad -0.6 the above impulse reposes with the ROC, stability, ad causality ROC: - Right had impulse repose h[] µ [ ] µ [ ]. Causal & Stable. ROC : - R.H & L.H impulse respose h[].750. ROC3 : - Left had impulse respose h[] µ [ ] µ [ ] µ [ ]. Not Causal & Not Stable. µ [ ]. Not Causal & Not Stable. ROC :- >0.6 ROC :- 0.< <0.6 ROC 3:- <0. ROC ROC 3 ROC

44 System Fuctio for Itercoectios of LTI Systems H H.H H H H H H H H

45 System Fuctio for Itercoectios of LTI Systems X H Y H Y X H H H H

46 Block Diagram of Causal LTI systems described by Differece Equatios ad Ratioal System Fuctios. Example0.8 A Causal LTIsystem with system fuctio : - H - x[] 4 Y X Y { } X 4 Takig the iverse - trasform of y[ ] 4 y[ ] x[ ] or y[ ] the above equatio, x[ ] 4 4 y[ ] - y[-] y[]