Question

2. (50 marks) Consider using the impulse invariance method to design a prototype IIR digital filter corresponding to the analogue prototype filter: He(s) = 52 +58 +6 a) Write the correct sequence of basic steps involved in this method. b) Determine the transfer function H(z) of the resulting digital filter. Simplify H(2) as much as possible. Assume a sampling frequency of fs = 100 HZ.

Answer 1

@)(a) Given Hals) 4 s²75516 Impulse Invariant_transformation procedure ů) Express the given transfer function Hals) in partial expansion form Hals) Fraction N. E Ai - i=1 S-li 려 Aints (") Find the impulse response by applying Inverse Laplace transform of Hals) halt) = [[tag] = ² A, hit walt) By applying sampling with its hon) hachts) - 2 Age (ATS) (1) Apply z transform HCZ) = CS mot C.CAT)]" H (2) = EARRANTES A (15)] E ial n=o 려 . huo H(2) = Ai PATI I-ê ỉ =1 2 fransform to Pits -1 S-P; l-ez

b) Given Hals) 4 $75sto Hals) = 4 (5+2) Cs+3) AL sta St3 4 t Az (S +a)Cs+3) = A (5+3) +A, (s+2) = 4 s=-3 Az(-1)=4 A t2 =-4 Se-2 ) Ap (1) -4 4 i. Hals) - 4 - 4 St3 sta Ai l-ě z' 털 We know that S-P Guren fs = 100Hz =oool :. Here Pi -2, P=-3 :. H(z) Pits-I l-e z l-eetsal 4 H(2) = -0.02 I 4 _0103 -1 l-e zi l-e z

니 HC2). = 4 10118 (-047 - 4-3882 - 4 +3.7/ (i-otga 1-4) HM 195년 409522 0004일 H2 = 042 2 2 11952 to095 Transfer function of digital I IR filter.