II The Z Transform. Topics to be covered. 1. Introduction. 2. The Z transform. 3. Z transforms of elementary functions

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1 II The Z Trnsfor Tocs o e covered. Inroducon. The Z rnsfor 3. Z rnsfors of eleenry funcons 4. Proeres nd Theory of rnsfor 5. The nverse rnsfor 6. Z rnsfor for solvng dfference equons II. Inroducon The role of he rnsfor n dscree-e conrol syse s slr o h of he Llce rnsfor n connuous-e syse. Dscree e sgnls: The sled sgnl s, T, T where T s he slng erod. The sgnl cn e wren s,, cn e consdered s sled sgnl of T re nerchngele f doesn e confuson. where T s sec. nd II. The Z rnsfor In consderng he rnsfor of e funcon, we consder he sled vlue of, h s, T, T The rnsfor of e funcon, where s nonnegve, s defned s follows

2 T T. T Z Defnon: Z-rnsfor of generl dscree-e sgnl s defned s Z. Z Noe: eq.. nd. s referred o s he one-sded rnsfor or unlerl rnsfor. Z s cole vrle. If, or,,, hen he rnsfor wll e defned s T T.3 T Z Defnon: Z-rnsfor of generl dscree-e sgnl s defned s.4 Z Z Noe: eq..3 nd.4 s referred o s he wo-sded rnsfor or lerl rnsfor. Z s cole vrle. We re only focused on one-sded rnsfor n hs course.

3 3 II.3 Z Trnsfors of eleenry funcons Un se funcon Un r funcon Polynol funcon,,

4 Eonenl funcon e Ele. On he rnsfor of e sn w Ele. On he rnsfor of s s 4

5 Tle of rnsfors 5

6 Tle of rnsfors 6

7 II.4 Proeres nd Theory of rnsfor Lnery, ROC R, ROC R Sclng n he -Don, ROC R, ROC R, ROC R R Z Ele.3 Deerne he rnsfor nd he ssoced regon of convergence for followng u funcon of e: u nd Te Shfng nt n, n n nt T for nuer sequence. we hve n n, n n n Ele.4. Deerne he rnsfor for u u v Cole rnslon heore T e e 7

8 v Dfferenon n he -Don d d, ROC R, ROC R v The nl vlue heore If,, hen l Z Ele.5 Deerne he rnsfor for followng funcon of e: u v Fnl Vlue heore If,, hen l l Ele.6 Deerne he fnl vlue of e T y usng he fnl vlue heore. 8

9 Proery le: 9

10 II.5 The nverse rnsfor Noe: he nverse rnsfor yelds he corresondng e sequence, u doesn yeld unque If he rnsfor s gven s ro of wo olynols n, hen he nverse rnsfor y e oned y severl dfferen ehods, such s drec dvson ehod, he couonl ehod, he rl-frcon-enson ehod, nd he nverson negrl ehod. drec dvson ehod T T T T Z or Z Ele.7 Deerne he nverse rnsfor of

11 The couonl ehod MATLAB roch Ele.8 fnd he nverse rnsfor of Le G,, Y nu s he rnsfor of he Kronecer del nu. In MATLAB, he Kronecer del nu s gven y N eros,, where N corresonds o he end of he dscree e duron of he rocess consdered. Dfference equon roch y y y Y Y G

12 3 The rl-frcon-enson ehod n n n n n Cse ll ole re dsngushed: n n, where Cse doule ole c c Then c, nd d d c Ele.9 fnd he nverse rnsfor of 4 The nverson negrl ehod. C d j T Where C s crcle wh s cener he orgn of he lne such h ll oles of re nsde. Usng cole vrle heory, we hve of ole of resdue K K K T Cse : conns sle ole, l K Cse conns ulle ole of of order q. l! q q q d d q K

13 Ele. Fnd he nverse rnsfor of Noe h 3

14 Ele. Fnd he nverse rnsfor of II.6 Z rnsfor for solvng dfference equons Noe: dfference equons cn e solved usng dgl couer. However, closed for eressons cnno e oned fro he couer soluon. Mle cn do soe sle ones. Tle: rnsfor of nd 4

15 Ele.. For followng dfference equon nd ssoced nu nd nl condons, deerne he ero-nu nd ero-se resonses y usng he rnsfor. y 3y, y u 5

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