The z-transform. LTI System description. Prof. Siripong Potisuk

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1 The -Trsform Prof. Srpog Potsuk LTI System descrpto Prevous bss fucto: ut smple or DT mpulse The put sequece s represeted s ler combto of shfted DT mpulses. The respose s gve by covoluto sum of the put d the reflected d shfted mpulse respose ow, use egefuctos of ll LTI systems s bss fucto

2 Egefuctos of LTI Systems A egefucto of system s put sgl tht, whe ppled to system, results the output beg the scled verso of tself. The sclg fctor s kow s the system s egevlue. Comple epoetls re egefuctos of LTI systems,.e., the respose of LTI system to comple epoetl put s the sme comple epoetl wth oly chge mpltude. s = s + jw = re jw

3 If the put s ler combto of comple epoetls, the the output s lso ler combto of the sme set of comple epoetls both CT d DT. = re jw Z-trsform of DT Sgls The blterl -trsform of geerl DT sgl s defed s where s comple vrble,.e., 3

4 Covergece Issue If the Fourer trsform of -trsform coverges coverges, the the The -trsform of sequece hs ssocted wth t rego of covergece ROC o the comple -ple defed s rge of vlues of for whch coverges. E. For r =, the -trsform reduces to the DTFT o the ut crcle cotour the comple -ple. Rego of Covergece 4

5 5 E. Fd the -trsform of the followg rght-sded sequece u u 0 Ths form to fd verse ZTusg PFE E. Fd the -trsform of the followg left-sded sequece

6 Propertes of the ROCs of the -trsform Propertes of the ROCs of the -trsform 6

7 Propertes of the ROCs of the -trsform The Ulterl Z-trsform - The ulterl -trsform of cusl DT sgl s defed s 0 - Equvlet to the blterl -trsform of u - Sce u s lwys rght-sded sequece, ROC of s lwys the eteror of crcle. - Useful for solvg dfferece equtos wth tl codtos 7

8 E. Fd the -trsform of the followg sequece = {, -3, 7, 4, 0, 0,..} , 0 3 The ROC s the etre comple - ple ecept the org. E. Fd the -trsform of d 3 d wth ROC cosstg of the etre - ple. E. 3 Fd the -trsform of d - d wth ROC cosstg of the etre - ple ecept 0. E. 4 Fd the -trsform of d + d wth ROC cosstg of the etre - ple ecept.e., there s pole t fty., 8

9 E. 5 Fd the -trsform of u wth ROC cosstg of the eteror of the ut crcle,.e., u s u 0 cusl or rght - sded sequece., E. 6 Fd the -trsform of Rewrtg s sum of left-sded d rght-sded sequeces d fdg the correspodg -trsforms, where otce from the ROC tht the -trsform does t est for b > 9

10 Propertes of -Trsform 3 Lerty : by by Propertes of -Trsform 4 Z- scle Property: 5 Itl Vlue : 0 lm 6 Fl Vlue : Applcbl e oly f lm the ROC of cludes the ut crcle,.e., ll the poles re sde the ut crcle 7 Covoluto : h H 0

11 Rtol -Trsform For most prctcl sgls, the -trsform c be epressed s rto of two polyomls where, d of p, p of b0 D p p p,, the umertor polyoml M,, p 0 re the eroes of,.e., the roots re the poles of,.e., the roots the deomtor polyoml. M Rtol -Trsform It s customry to ormle the deomtor polyoml to mke ts ledg coeffcets oe,.e., b0 D p p p b M 0 b M b Also, t s cusl sgl, the wll be proper rtol polyoml wth M,.e., # of eroes # of poles. M M

12 where where, for fed r, IDTFT DTFT A cotour tegrl Iverse -Trsform Sythetc Dvso Method Perform log dvso of the umertor polyoml by the deomtor polyoml to produce the quotet polyoml 0 q q q r r r r q r q r 0, 0, 0, 0 Wrte s ormled rtol polyoml by multplyg the umertor d deomtor by M M r b b b 0 Idetfy coeffcets the power seres defto of where

13 3 E. Fd the verse -trsform of Equtg coeffcets, },0,0,0,0,,0,,0,3,0, { Remrks: Ths method does t produce closed-form epresso for E. Fd the verse -trsform of 3 } 5, 3, {0,,3,3,

14 4 Prtl Frcto Method Produce closed-form epresso for Wrte s sum of terms, ech of whch c be verse -trsformed by usg -trsform tble Sutble for rtol whose deomtor polyoml hs dstct rel poles multplcty of oe or smple rel poles Prtl Frcto Epso, p p p b p R p R p,, Gve wth dstct poles fctored form s epress / terms of prtl frcto epso: The verse -trsform of c be wrtte s u R p

15 5 E. Fd the verse -trsform of u E. Fd the verse -trsform of b u b b

16 6 E. Fd the verse -trsform of u E. Fd the verse -trsform of s u

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