Question

It is desired to design a digital BPF with center frequency/2 radians using the Bilinear Transformation technique discussed in class. The analog filter transfer function is given by Ha (s)-(s +0.1% + 16 a. b. c. Find the center frequency of the analog filter Find the transfer function of the digital filter and implement it Find and sketch (Matlab) the frequency response of both filters and comment on your answer on how good is the approximation.

Please Show All the Steps carefully with full explanation. Highest rating will be given to those who show all the parts with proper explanation. Thanks.

Answer 1

a. H(a) =   s + 0.1
-------------------
s^2 + 0.2 s + 16.01

from the characteristic equation w^2=16.01 and w= 4.0012

the center frequency f = 4.0012 /(2*pi ) = 0.6368 hertz

b. for digital filter replace s by

s = 2 / T (z-1)/(z+1) where T is the sampling time . let us assume T = 0.01

0.004996 z^2 + 4.993e-06 z - 0.004991
H(Z) = -------------------------------------
z^2 - 1.996 z + 0.998

c. Frequenct response of Analog and Digital Filters

s=tf('s');
g=(s+0.1)/((s+0.1)^2+16);
margin(g);
figure
h=c2d(g,0.01,'tustin')
z=tf('z');
margin(h)

Bode Diagram Gm Inf, Pm 100 deg (at 4.52 rad/s) 20 10 e10 E -20 2 30 40 .50 90 45 45 90 10 10 10 10 10 10- Frequency (rad/s)

Bode Diagram Gm Inf, Pm 100 deg (at 4.52 rad/s) 100 100 200 300 400 90 45 45 90 10 10 10 10 10 102 10 Frequency (rad/s)