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Q1: You should ask yourself if $$\frac{z}{3z^2 - 4z + 1}\stackrel{?}{=}\frac{\frac 3 2}{z-1} - \frac{\frac 1 2 }{z-\frac 1 3}$$ really holds. You'll find out that you forgot to scale correctly. Q2: There is no reason to consider $X(z)/z$ instead of $X(z)$ in this case. And concerning your question about the anti-causal sequence, a multiplication by $z^{-... 3 Your two transfer functions are in parallel, i.e they simply add up. So your feedback transfer function is simply$G(z) = H_1(z)+H_2(z)$. You want to makes sure that the magnitude of$G(z)$is smaller than one. and the overall closed loop transfer function is $$H(z) = \frac{Y(z)}{X(z)} = \frac{1}{1+H_1(z)+H_2(z)}$$ 1 It does matter that$a=b=2$, because this gives a relatively nice looking solution. As mentioned in a comment, you should split up the given transfer function$H(s)$. But first, let's introduce a normalized variable$p\$: $$p=\frac{s}{\omega_n}\tag{1}$$ Now we can write the given transfer function as $$\hat{H}(p)=\frac{2p^2+2p+1}{p^3+2p^2+2p+1}\tag{2}$$ ...