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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+0.5−j0.866
A=(s+1)1(s+1)(s2+s+1)|s=−1=1(−1)2−1+1=1B=(s+0.5+j0.866)1(s+1)(s+0.5+j0.866)|s=−0.5−j0.866=1(−0.5−j0.866+1)(−j0.866−j0.866)=1−j1.732(0.5−j0.866)=1−j0.866−1.5=−1.5+j0.8663=−0.5+j0.288C=B∗=−0.5−j0.288
Hence,
H(s)=1s+1+−0.5+0.288js+0.5+j0.866+−0.5−0.288js+0.5−j0.866=1s−(−1)+−0.5+0.288js−(−0.5−j0.866)+−0.5−0.288js−(−0.5+j0.866)
In the impulse invariant technique,
if H(s)=∑Nk=1cks−pk, then H(z)=∑Nk=1ck1−epkTz−1
Therefore,
H(z)=11−e−1z−1+−0.5+j0.2881−e−0.5e−j0.866z−1+−0.5−j0.2881−e−0.5ej0.866z−1=11−0.368z−1+−1+0.66z−11−0.786z−1+0.368z−2
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