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Find the direct form II structure of, $ H(S)=\frac{5 S^3-4 S^2+11 S-2}{\left(S-\frac{1}{4}\right)\left(S^2-S+\frac{1}{2}\right)} $
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Solution:

$ H(S)=\frac{5 S^3-4 S^2+11 S-2}{\left(S-\frac{1}{4}\right)\left(S^2-S+\frac{1}{2}\right)}\\ $

$ =\frac{5 S^3-4 S^2+11 S-2}{S^3-\frac{S^2}{4}-S^2+\frac{S}{4}+\frac{S}{2}-\frac{1}{8}}\\ $

$ H(S)=\frac{5 S^3-4 S^2+11 S-2}{S^3-\frac{5 S^2}{4}+\frac{3 S}{4}-\frac{1}{8}}\\ $

$ =\frac{5-\frac{4}{S}+\frac{11}{S^2}-\frac{2}{S^3}}{1-\frac{5}{4 S}+\frac{3}{4 S^2}-\frac{1}{8 S^3}}\\ $

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