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Design a third-order Butterworth digital filter using the impulse invariant technique. Assume sampling period T=1sec.
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Solution:

For N=3, the transfer function of a normalized Butterworth filter is given by,

H(s)=1(s+1)(s2+s+1)=As+1+Bs+0.5+j0.866+Cs+0.5j0.866

$ \begin{aligned} A & =\left.(s+1) \frac{1}{(s+1)\left(s^2+s+1\right)}\right|_{s=-1}=\frac{1}{(-1)^2-1+1}=1 \\\\ B & =\left.(s+0.5+j 0.866) \frac{1}{(s+1)(s+0.5+j 0.866)}\right|_{s=-0.5-j 0.866} \\\\ & =\frac{1}{(-0.5-j 0.866+1)(-j 0.866-j 0.866)} \\\\ & =\frac{1}{-j 1.732(0.5-j …

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