Question

HI, PLEASE ANSWER ALL PARTS AND PLEASE SHOW ALL WORKINGS STEP BY STEP. THANK YOU. a) Draw the block diagram representation of the discrete-time system Yk+2+ Yk+1 +0.5yk = Ux (8 Marks) b) Find the impulse response of the system with the z transfer function; Z G(z) = 22 + 3z + 2 c) Use z-transforms to solve the following difference equation; 6Yk+2 + Yk+1 - Yx = 3 subject to yo = y1 = 0

Answer 1

simply applied the basic z transform rules and formulas.
Dues Q Clueu , discrete time system, Yk+2+ Jeti + 0.5 = UK be written as, So our 0.5 Yk given difference equation can Y CK+2) + yekto) yk +1) + 6.5 luck) Now, ) here, UK is the input and yck) is the oufpect . As we or cence see in difference equation their is a differen advanced delay is given. diagram as, forward So, the block draw we can yCk) u(K) E E 105 Z 22 8 where, the advanced delay of signal 2 & 2² are the ges).

b Now, given, system as a(z) = Z z²+ 32 +2 G(z) z (2+2)(2+1) Now, we Know that, x(2) 412) on 412) din) gin) yin) a(z) Yez) where, xin) x(2) no s tolp ill yin gln) system. on here, we n) 1 as So the green y (n) i.e. have to take xl2) the impulse input and determine the impulse response of sipten, Y(2) - Alz). %(2) know that the impulse signal is to decin) esin) z (xing z transform of and the Z Soi YCZ) = a(z). the inverse z-transfrom Now, will determine we

Bo by using partial fraction method, (2) = Z (2+3)(2+2) (12) = A B 2+3 2+2 we get here, & B=-1/ A=t 3 - ot (2) = 1/3 (2+3) (*): 2+2 S By taking A(2) 12-1 2-1 2 3.2.1) (2) (1+22-1) 2 common 3 (1+ 32-1) from denominados z- transform. Now, applying interse z '[alz)] 러 1 3 (1432-1) 2-) 1+22-1 - (-2.)" uin) I (-3)" ucn) 11 -19 3 Si 2 Litaza= casums &o. The impeelse rexpouse of given system ů yini= gen?= Į (-3) "un) - 2 (-21" um) Aus

C Given difference equation, 6 Як 1 Які — Як = 3 Lince, all with initial conditious ds = = the initial conditions are zero we will tranform of giuen difference equation att4) - Хек» г[3] take the 2. as i 2 [64k + я — 3 6 | 22. (2) 3 Y(2) .2* - 112) = 3 уг (z) [622 + 2 - 17 2 ГА 3 = А where, A= coustant Үlz) - 6222- [{( к+n) - Aus zn ylk) 3 Х 2.