Question

4. (5 pts) Consider a discrete-time LTI system T that generates an output y[n] according to a2 y[n] bx[n] – ay[n – 1] - *[n – 2] where a, b are non-negative real constants. (a) (2 pts) Find the poles of the z-transform of the impulse response h[n] of T. (b) (3 pts) Let H(ejl be the frequency response of T. Find a, b so that the system is causal and stable, |H(1)| = |H(ejº)] = 0.04, and |H(-1)] = \H (237)= 1.

Answer 1

at by (0-2) On • Griven that, Y(n) = be(n)-ay(n-1) taking z-transform, we get Y(z) = b X(z)-az-'Y(z) - a?z?Y(2) Y(z) +az"Y(z)+ a zky(2) = bxcz) Y(z) [lt az't crne z] = bxcz) Y(z) b X(2) I+az az H(Z) = bz? 2²+ azt H(Z) = bz? (2+ 9/2² sh Poles of Z- Transform = (2=-9/2) Double pole Poles are at (%) 6 Put z=ejz eiertaeintalu 0.04 THEO | eotae tale tata Hrein) ? نعba beo b 25 2 a 4 tat1 = 255 - bejet 14 Cesty (ti Jb 1-a tahu ejen "taein +azy a²-a+1=+b 4 from eq" O $ 3.-4*+a+1) = 256-4+a-1)

-a-1 -250++ + 259-25 249²_260 +24=0 Đ 62²-269 +24 प 3a-132 2= 0 a= 13 I√ 169-144 6 6 1315 18 8 5 a= 3,43 the is Now, ROC of system 121 > for a=3 12/> 3) => 121>1-5 This ROC does not include unit circle. It means this system is not stable. {To be stable, ROC must include unit circle} For a=4 → 1217/ 2 x 4 ) » 121 > Now, this ROC is right sided and include unit circle Hence, system is a causal and stable. Đa=413 brak-a +1 = $^$*1-4+1 (5-1)+1 Ib= Yg = 1 -= So, a=43, b=1/g are the values for asb for the causal and stable system. Il