Question

HI, PLEASE ANSWER ALL PARTS AND PLEASE SHOW ALL WORKINGS STEP BY STEP. THANK YOU. a) A discrete-time filter is defined by the difference equation; Vk - 0.5yk-1= xk Describe the filter response to a unit step input applied at k = 0, for the range 0 Sk <5. (8 Marks) b) Show from first principles the z-transform of the following; i. Z{e-k), where k = 0,1,2... c) Draw a block diagram to represent the system modelled by the difference equation and give the corresponding z transfer function; Yk+2 - 3yk+1 +2yk = UK

Answer 1

One a 88 - 0:59k- = XK and к 1 r u(k) -4(K-5) >] A ahhling 2 z toonsform to equatim 412) -005zt 412) = x(2) 412) X(2) 3 (1-0.5 -52') applying z bonsform to equation ② X(z) 25 1-27 1-24 - x12) 4 (i=27) (1-25) Som equation ③ and ④ Y(z) r a (1-25) (1-0.57") (1-24) ( 21h (1-0-52 1-0-521) (1.2") (1-25) 4/2) + 1(ien ( ' -0-52") 1-27

Ylz) 2 -(1-25) + 1-0.52" in (1-25) IZY 1 anhlying inners z transtorn to both Side Yk y (0.5)* (UCK-UCK-5) )+ 2 (uk) - ulk-5)) answer b cck2 e-kuck) we know x 12) = z-K Deck) Σ K=-00 X(2) = Ezt* erka) K= xl2)= I + (ze) +(ze)? + x (2) 1-eve I-etzt l-lez) zle*). 1 1-lez)

given equation, 39 KPI +8Yk UK K+2 applying a z doansform to both sides z² ulz) - 3412) z 2 2! & g 4/2) U(z) Y(z) U12) 2² - 3 2 + 2 - Y(z) ?=2 11 U12) 1-327 +22² -2 412) 2 u(z) -( 32 ㅔ -22-2 bansfer femetim ULZ) 412) 3 21 - 2 modelling of froms fes function.