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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

A=(s+1)1(s+1)(s2+s+1)|s=1=1(1)21+1=1B=(s+0.5+j0.866)1(s+1)(s+0.5+j0.866)|s=0.5j0.866=1(0.5j0.866+1)(j0.866j0.866)=1j1.732(0.5j0.866)=1j0.8661.5=1.5+j0.8663=0.5+j0.288C=B=0.5j0.288

Hence,

H(s)=1s+1+0.5+0.288js+0.5+j0.866+0.50.288js+0.5j0.866=1s(1)+0.5+0.288js(0.5j0.866)+0.50.288js(0.5+j0.866)

In the impulse invariant technique,

if H(s)=Nk=1ckspk, then H(z)=Nk=1ck1epkTz1

Therefore,

H(z)=11e1z1+0.5+j0.2881e0.5ej0.866z1+0.5j0.2881e0.5ej0.866z1=110.368z1+1+0.66z110.786z1+0.368z2

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