<s>[INST] Simplifying transfer functions in Z domain I have difficulties to check whether the below transfer function is recursive or non-recursive:
$$H(z)=\frac{1-z^{-1}+z^{-2}-3z^{-3}}{z^{-2}(1-z^{-1})}$$
I know that I have to multiply the num and denum by ${z^3}$ but the problem here is the denum. I think it must be first simplified even more, or maybe it should be flatted like:
$$z^{-2}(1-z^{-1}) = z^{-2} - 2z^{-3}$$
I cant go further from here...can you please tell me what is the correct procedure?
 [/INST] Starting from
$$H(z)=\frac{1-z^{-1}+z^{-2}-3z^{-3}}{z^{-2}(1-z^{-1})}$$
then
\begin{align}
H(z) &= \frac{z^{3}}{z^{3}} \cdot \frac{1-z^{-1}+z^{-2}-3z^{-3}}{z^{-2}(1-z^{-1})} \\
&= \frac{z^{3} - z^{2} + z - 3}{z-1} = \frac{z^{2} (z-1) + (z-1) -2}{ z-1} \\
&= 1 + z^{2} + \frac{2}{1-z} = 1 + z^{2} + 2 \, \sum_{n=0}^{\infty} z^{n} \\
&= 3 + 2 z + 3 z^{2} + 2 z^{3} + 2 z^{4} + \cdots.
\end{align}
</s>