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Design a digital Butterworthlow pass filter that satisfies the following constraint using IIM. Assume T=1 sec

0.707≤|H(w) |≤1 ; for 0<ω<0.3π

|H(w) |≤0.2 ; for 0.75π<ω<π

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Step-1: Identify the filters specification

Ap=0.707;As=0.2;ωp=0.3π;ωs=0.75π;T=1sec

Now,

Ωp=ωpT=0.942rad/sec

Ωs=ωsT=39.25rad/sec

Step-2: Calculation of order of filter

The order of filter is given by

N12log[(1/(As2)1)(1/(Ap2)1)]log(ΩsΩp

N1.732

Step-3: Calculation of cut off frequency

Ωc=Ωp(1(Ap21))12N

Ωc=0.941rad/sec

Step-4: Calculation of poles

Pk=Ωc=ej(N+2k+1π2N

Now,

when k=0;

∴P_o=-0.665+j0.665

when k=1;

∴P_1=-0.665-j0.665

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

H(s)=\frac{(Ω_c)^N}{((s-P_o )(s-P_1))}

=\frac{0.885}{((s+0.665-j0.665)(s+0.665+j0.665))}

∴H(s)=\frac{0.885}{((s+0.665)^2+(0.665)^2 )}

Step-6: Conversion of analog Transfer function to digital Transfer function

We know that,

\frac{b}{((s+a^2 )+b^2 )}=\frac{(e^{-aTs} [sin⁡bTs ] z^{-1})}{(1-2e^{-aTs} [cos⁡bTs ] z^{-1}+e^{-2aTs} z^{-2} )}

∴H(z)=\frac{(e^{-0.665Ts} [sin⁡0.665Ts ] z^{-1})}{(1-2e^{-0.665Ts} [cos⁡0.665Ts ] z^{-1}+e^{-1.33Ts} z^{-2})}

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