Question

Suppose that the system function of an LTI system is 1+z H(z)=7 (1-12 '\1-22-X1 - 3z-") (a) Determine the ROC of H(z) if it is known that the system is stable. (b) Determine the ROC of H(z) if it is known that the system is causal. (c) Is it possible for the system to be both stable and causal?

Answer 1

on? Given LTI system is Hal H (2) = (32۔ ( 2) (-2- ۔ + (-(ا) (۱) ( 2 ح23 = z (22-1) (2-22-3) جلد - (2) ج (2) ( اج) 22 3ح2 H(2) 2 (22 -1) ( - 2) (2-3) 2=(2+) (2 - 1) (2) (2-3) Applying partial fractions - 9# ارچر 22 21 = لي-z) (2) + لوح ) -27) + لوح) (22) A لاحلت ) (1-2) (23 22 (2) ) A )6+ (+ 32 ج ح 2z - Z - ( 2 ) + c پہ 42 - 2 - 2) 22+22 ج وب 2 - 2 52 - = 2) + (3+12-22) 6+ لي+ 5 - ) A On Comparing A + 2 terms + 2 we = 2 get ) 22 - 2 = 5c ۔ 16 - 5A - GA + 38 + 2 =0 Scanned with CamScanner /

Te To solving equations ① , ① and ② on → Af 28+24= 2 XB2 = -10A – 148 - 10% = 4 GA + 3B A2C2O x 5 = 30 A +158 +100=0 SA - B= 2 6 20A + B = 4 > 0 - -5A-8=2 ③ > 20 ( 2 ) + B = 4 55 20A + B = 4 B: 4-8 15A=6 (2 - TA= 6 15 + 2(-4) +22=2 H(2) - G) - 4 + z (22-1) (2-2) et HB) 5103 foto polesa, 23 2= Ķ, 2, 3 (r<), teraz), (2«r<3), (033) are not including any poles. Therefore these are possible Roc's. Roc of H(²) if it is known that the system is stable a) o o is stable, it Roc include unitcircle & The system Scanned with CamScanner

includes unit circle stable < V 2 De 5(2-) (2-2) 52-3) HOH ROC o hoc of H(2) for the system to be stable is & T Roc ( 1 <r<2) for stable system a b) Roc of H(2) if it is known that the system is causalt The system is causal, if Roc is out ward directed . (rss) is outward directed. H(z) = - z z 4z + 24Z 5(2-) 5(2-3) causal VaROC for is H 5 8 ROC of H(2) to be causal TROG> (r>3) the system (r>3) for causal The system is stable and causal if Roe is out-ward including unit circe and all polee - inside the unit circle. No such RoC is present. CScannedd fith dømsdaater possible for the system to be both stable ce and causal.

- 24Z C * Finding hC) for the two cases: For stable system & H (2) = - 2 2 - 42 t 5(2-1) z-z 5(2-3) Roc V J (<r<3 roth rez raz. Applying inverse z-transform. B(-) :-) (n) - - - - -[-(-6%(-) Thm)- $ cyʻun) + BjMufth-1) - 2403,"ubon - → For causal system H(E)= 2 2 2 2 5(2-3) Roc 1 1 Cr>3) +242 - 42 r> 3 r > Applying inverse z-transform. Thane $(4) acn) – 4 (2)"u(m) + 24 (3)”uca) scanned with CamScanner