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Using the bilinear transformation, obtain H(z) from Ha(s) when T = 1s and Ha(s)=s3(s+1)(s2+2s+2)
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written 2.3 years ago by |
Solution:
Given this equation,
Ha(s)=s3(s+1)(s2+2s+2) and T=1 s.
To obtain H(z) using the bilinear transformation, put s=2T(1−z−11+z−1) in Ha(s).
H(z)=Ha(s)|s−2T(1−z−11+z−1)=s3(s+1)(s2+2s+2)|s=2(1−z−11+z−1)
=[2(1−z−1)(1+z−1)]3[2(1−z−1)1+z−1+1]{[2(1−z−1)1+z−1]2+2[2(1−z−1)1+z−1]+2}=8(1−z−1)3[2(1−z−1)+(1+z−1)][4(1−z−1)2+4(1−z−1)(1+z−1)+2(1+z−1)2]
=8(1−z−1)3(3−z−1)[10−4z−1+2z−2]=4(1−z−1)3(3−z−1)(5−2z−1+2z−2)=4(1−3z−1+3z−2−z−3)15−11z−1+8z−2−2z−3