Question

Bonus Question) A discrete-time LTI system with a sampling frequency of Ukm2 is shown in the following Figure. The rectangular boxes with the label z provide one sample period delay to their input signals. The circular components are adders or subtractors. The triangular components provide linear vain factors of ar or bi where i is 0,1 or 2. i) Derive the system transfer function H(2). ü) Find the difference equation relating the output y[n] and input x[n]. iii) Given that the gain values are: bo=1 ba =- Jab=1 Find the poles and zeros of this system.

Answer 1

"(2)= (bot be ??t be ž") x (2) = (bot myft ha) = (bot be Z² + by zt) x (2) (l+') (+225) Hazt Ifazl bot by z tbq z Citarz72224 X(z) to. Transfer function (2) X(2) b2 2 ² + b 2lt bo gzta, zlta I Difference equation relating the outent gen) and input na M(z) [azz +0,2441)= x(z) [1, 2 62746) az Y(2) { t an 4(2) ² + 4(2)= X(Z) by Z ²4 X(2) b 2 + b X/2) _ Taking inverse - Z-transform on both sides 1 az Y [n-2) Qq (n) ty (n)a by * {n-2} + bq8 [n+] + box(]] TOTAZZ

12 {id WS Given the gain values. bord, be = -2 62 214 A 2,-pace, on = .97 by z² + by z tho az z ² + 9, 2 tay obtained transfer function on b2t bi zot boz² 92 +92 +97 Poles i Ztaq 2+ Qx = 0 + 6432 +(0.9)720 Ź - 127.2 + 0.8920 = 1.274 kl. 278 (W(asi) 2 z = 1.27 6211 Z=1214(127) 273 0.635 - j (0.635) poles :) 240 0.125+; 10.635) 2cros62 by 2 + b2=0 2 - 22 +1:0 zire 1- € 7- 12 of Zeros: ziehen sie zu ñ DES