Question

Given a system with pole zero plot shown in Figure 4-37 and the fact that H(0) = -5 Im{z} 2 1 -0.5 2 Re{z} 0 -1 -2 -2 -1 0 2 Figure 4-37 a) Find h[n] given that the system is causal b) Find h[n] given that the system is stable.

Answer 1

1 Page No. Date Dolls Givent H(o) = - 5 plu (2) and, Rece L . there are 2 - poles at e p = -0.5 P = 2 and No eroes H(z) is a given as :- H(Z) = K - (2 +0.5) (Z - a) System in Cansal - this means the ROC is. (Region of convergence) is given as - 171 > 0.5 & 12172 Impulse responce i Right sided Neme

Page No Date Now, Considering & ② » H( ak (2 to 5)(7-2) also H(o)= -s. put 2:0 in En ① s H(o)= k (0.5) (-2) H(Z) = 5 : (7 +0.5) (7-2) Now, - to Inverse find h(n) we will find transform of A(Z). f. We will use partial fraction method :-

3 Page No. Date H(Z) = 5x (Z+0.5)(2-2) = 5x 5) 1 (Z +0.5) - 1 (7-2) H(Z) = -25 - I 2 toisz Z-1 1-22- 11 Now h(n) = taking inverse : – 2 (-0.5) hüch-1) + 2 (2) "(n-1). 2 - 1 (2 - 4 (2)) = xcn-k) o i time shifting property :h(n) = 2 (2) - 1 u (n-1) - 2 (-0.5) - Lucuri) this the responce when Causal. beth System is

Page No. Date b) Now, System u stable. the dustem includes ROC u unit stable cirkle when - ROC.) 0. 5L12122 il 1st Causal & 2nd Non canal hon) is given as : Them) = -2 (2005) ucn-1) - 2 (2) "-1"(-n-2) this is the responce when System is stable.