The Inverse z-transform

Transcription

1 The Iverse -Trasfor I sciece oe tries to tell people, i such a way as to be uderstood by everyoe, soethig that o oe ever ew before. But i poetry, it's the eact opposite. Paul Dirac Cotet ad Figures are fro Discrete-Tie Sigal Processig, e by Oppehei, Shafer, ad Buc, Pretice Hall Ic.

2 The Iverse Z-Trasfor Foral iverse -trasfor is based o a Cauchy itegral Less foral ways sufficiet ost of the tie Ispectio ethod Partial fractio epasio Power series epasio Ispectio Method Mae use of ow -trasfor pairs such as a u a [ ] Z > a Eaple: The iverse -trasfor of > [ ] u[ ] X

3 Iverse Z-Trasfor by Partial Fractio Epasio ssue that a give -trasfor ca be epressed as X pply partial fractioal epasio X M r 0 B r r First ter eist oly if M> B r is obtaied by log divisio Secod ter represets all first order poles Third ter represets a order s pole There will be a siilar ter for every high-order pole Each ter ca be iverse trasfored by ispectio M 0 0, i b a d s C ( d ) i 3

4 Partial Fractioal Epressio Coefficiets are give as Easier to uderstad with eaples s i i, M 0 r r r d C d B X d X d [ ] d i w s i s s s i w X dw dw d d! s C

5 5 Eaple: d Order Z-Trasfor Order of oiator is saller tha deoiator (i ters of - ) o higher order pole ROC : X > X X X

6 6 Eaple Cotiued ROC eteds to ifiity Idicates right sided sequece X > [ ] [ ] [ ] u - u

7 7 Eaple # Log divisio to obtai B o 3 X > X X 9 X 8 X

8 Eaple # Cotiued ROC eteds to ifiity Idicates right-sides sequece 8 > X 9 [ ] δ[ ] 9 u[ ] - 8u[ ] 8

9 9 Iverse Z-Trasfor by Power Series Epasio The -trasfor is power series I epaded for Z-trasfors of this for ca geerally be iversed easily Especially useful for fiite-legth series Eaple [ ] X [ ] [ ] [ ] [ ] [ ] 0 X X [ ] [ ] [ ] [ ] [ ] δ δ δ δ [ ] 0 0

10 Copyright (C) 005 Güer rsla 35M Digital Sigal Processig 0 [ ] otherwise 0 0 a a a a a a a X 0 0

11 otatio Liearity Z-Trasfor Properties: Liearity Z [ ] X ROC R a Z [ ] b[ ] ax bx ROC R R ote that the ROC of cobied sequece ay be larger tha either ROC This would happe if soe pole/ero cacellatio occurs Eaple: [ ] a u[ ] - a u[ - ] Both sequeces are right-sided Both sequeces have a pole a Both have a ROC defied as > a I the cobied sequece the pole at a cacels with a ero at a The cobied ROC is the etire plae ecept 0 We did ae use of this property already, where?

12 Z-Trasfor Properties: Tie Shiftig Z [ o ] X ROC R o Here o is a iteger If positive the sequece is shifted right If egative the sequece is shifted left The ROC ca chage the ew ter ay dd or reove poles at 0 or Eaple X > [ ] u[ - ] -

13 Z-Trasfor Properties: Multiplicatio by Epoetial o Z [ ] X( / o ) ROC o R ROC is scaled by o ll pole/ero locatios are scaled If o is a positive real uber: -plae shris or epads If o is a cople uber with uit agitude it rotates Eaple: We ow the -trasfor pair Z u - - Let s fid the -trasfor of r cos ωo u [ ] ROC : > [ ] [ ] [ ] jω jω re u re u[ ] / > r re o / X jω jω o re o o 3

14 Z-Trasfor Properties: Differetiatio [ ] Eaple: We wat the iverse -trasfor of Let s differetiate to obtai ratioal epressio Maig use of -trasfor properties ad ROC dx d Z ROC dx d log( a ) > a X a dx a d [ ] a( a) u[ ] a [ ] ( ) u[ ] a R a

15 Z-Trasfor Properties: Cojugatio Z * * [ ] X ( ) ROC R * Eaple X X [ ] X [ ] [ ] ( ) [ ] ( ) [ ] Z{ [ ] } 5

16 ROC is iverted Eaple: Z-Trasfor Properties: Tie Reversal Z [ ] X( / ) Tie reversed versio of ROC [ ] a u[ ] a a u[ ] - - a - - a R < a X 6

17 Z-Trasfor Properties: Covolutio Z [ ] [ ] X X ROC : R R Covolutio i tie doai is ultiplicatio i -doai Eaple:Let s calculate the covolutio of [ ] [ ] [ ] u[ ] a u ad X ROC : > a a Multiplicatios of -trasfors is ROC : Y X X X > ( a )( ) ROC: if a < ROC is > if a > ROC is > a Partial fractioal epasio of Y() a a u a asue ROC : Y > [ ] [ ] a u[ ] y 7