Question

Problem 5: (20=10+5+5 points) Consider a causal system with the following transfer function (the z-transform) 2 +42-2 H2) = 1 - 52-1 +6 1-52-1 +62- 2 2 < 121 <3. (a) Find the inverse z-transform of H(2). (b) Find the difference equation with input-output relation for this system. (c) Draw the diagram of realization (in form II) for this system.

Answer 1

:.yloj=apla) + upla-2) (2) Zht z]dt = (zazn+3) (Z]d = (7)CH (zid = r27 Numerator . ..plus=sclo) + spla-o) - 6 ploeas Inverse 2-transform Taking (zid,29+1770 ,-ZS-(37d = [z] .. c . : 32297,-ZS-1 iii (71* 317]d (?-29+1633-1) Już 1x = (Žit (E1% -1z7d 707 Denominator Implementation FORM II rz).A2.29 - (zlk,st (27x zhtr21x=rzih 2069+1-25-1 : zichts = 137H orby om

Direct Form IT - - Realization. - ocen) pln) 2ply yla) 14pl 4-2) 1 splny 21] La pisos placuje La

H(2) = 2tu202 2121C3 1-524622 | 22 +y (24422 리s246, 크 H2)= 2를 ty E223) H2 오라니 크리22) 모 +드 223 eY)' A2서드를 12-0= -320놓를 | A-2 12를 80년 09쁜122년 | 늘 | (27) CZtt) | , 화전기기타로는= = G -G 2 23 여3 3) HCZ) || Lz) | - (-3) | | 2-31 최근2 로병 . . . 2 +) + Hts) - 12 + 로 LG ) 2232 23 ) 23 t - . HC) (2-20 Selled with

:, H1Z) = 2 meer - 67 t 223Z' so Z-2 iz-3) S ROX 121<3, the terr s, the term obz 2-2 causal and 22/32 is non causar Apply Inverse z traustorm h(a) = 2 8(a) = 6.121" ulaj +1 -22, 3 ul-noos :h(a)= a scn) = 6 (2j" (la) - 22 23 ulanad) H(2) - 2+142?.. .. 1-52'+62-2 is : we have Hcz)= Y(Z) = 3+uz?? X12) 1-st 1+62:2 YLE) € 1-52 +62-2) = X(2) (27.4.20?) Y(z)-52-ly(22+ 6zo? . Y(E)=2X(z) tuzx(2) - transform Taking Inverse Laplace .ylns=as(n) tuscla-2) + Sylni) - byla-2)