Question

16.3 (a) Use the impulse invariance method to show that the analog transfer function given by 2.6702 H(s)=3+2.7747s2 +3.8494s +2.6702 B. Friedlander and B. Porat, the modified Yule-Walker method of ARMA spectral estimation, IEEE Transactions on Aerospace Electronic Systems (1984), AES-20(2), 158-173. (2] L. B. Jackson, Digital Filters and Signal Processing. 3rd edn. Kluwer Academic Publishers (1996), Chap. 10, pp. 345-355 16 lIR filter design results in the following z-transfer function: 0.4695220.1907z H(z)=3-0.610622 +0.3398z- 0.0624 as stated in Example 16.3 for the third-order Butterworth filter. (b) Use the impulse invariance method to show that the analog transfer function given by 6.2902 H(s) +4.1 383s3+8.5630s2 + 10.3791s +6.2902 results in the following z-transfer function: 0.3298z30.4274z20.0427z H(z) z4-0.4978z3 0.3958z2 - 0.1197z 0.0159 = as stated in Example 16.3 for the fourth-order Butterworth filter.

Please solve these questions. Thank you.

Answer 1

H (S)= 2 6702 3 2.77412 3.8494s+ 2.6102 using partial fraction 1383 34 H(S)= -69367-.400512 (t.69 367-1.20145) - -69367+4u0512 (St.69367t1.2o | 45) Let T Sempling perod 1 using impulse invariance H(2) 1.38734 -.69367--40051 - -69367t 20 lA5 -1-38736 -1 1- E - 69369 + .46051 a 69367 -1 20145 2 IN

H(2) 1.38734 69367--4005) 1 - 24972 1-18 04 +.466D1) 69367+.4u05) 2 (2(0997.-7 81) - H(2) 1387 Z -1-387 2t6236 Z Z- 2497 z2.36082+ 249 469522 +-1907 2 23 -610622 +0.3398-0624 11