Question

digital butter worth filter that satisfies the following constract using bilinear tranformation assume t=1s 0.9

Answer 1

Answer:

Step-1: Identification of filters specification

Ap=0.6;Ap=0.1;ωp=0.35π;ωs=0.7π;T=0.1sec

Now,

Ωp=2Ttan(ωp2)=12.25rad/sec

Ωs=2Ttan(ωs2)=39.25rad/sec

Step-2: Calculation of order of filter

The order of filter is given by

N>12log[1As2−11Ap2−1]log(ΩsΩp)

N≥1.72≅2

Step-3: Calculation of cut off frequency

Ωc=Ωp(1Ap2−1)12N

Ωc=10.60rad/sec

Step-4: Calculation of poles

Pk=Ωcej(N+2k+1)π2N

when k=0;

∴Po=−7.49+j7.49

when k=1;

∴P1=−7.49−j7.49

Step-5: Calculation of Transfer function H(s)

H(s)=(Ωc)N((s−Po)(s−P1))

=(10.60)2((s+749−j7.49)(s+7.49+j7.49))

∴H(s)=112.36((s+7.49)2+(7.49)2)

Conversion of analog Transfer function to digital Transfer function:

H(z)=H(s)(s=2T(z−1)(z+1)

H(z)=112.36(20((z−1)(z+1)+(7.49)2+(7.49)2