I have a fraction and I want to break down stairs.
$$ F(z) = \frac{1 - 1.37 z^{-1} + 37 z^{-2}}{1 - z^{-1} + 0.6 z^{-2}} $$
I want it in form of $$ F(z) = A_{0} + \frac{1}{B_{1} z + \frac{1}{A_{1} + B_{2}(z)}} $$
where $A_{0}$, $A_{1}$, $B_{1}$ and $B_{2}$ are constants and $i$ is the order of stairs which would be bigger as much as it could be.
Is it possible to obtain this form from $F(z)$?
You can do the usual continued fraction development: $$\begin {align} \frac{1 - 1.37 z^{-1} + 37 z^{-2}}{1 - z^{-1} + 0.6 z^{-2}}&=1+\frac {-.37z^{-1}+36.4z^{-2}}{1 - z^{-1} + 0.6 z^{-2}}\\ &=1+\frac 1{\frac {1 - z^{-1} + 0.6 z^{-2}}{-.37z^{-1}+36.4z^{-2}}} \\&=1+\frac1{z\frac {1 - z^{-1} + 0.6 z^{-2}}{-.37+36.4z^{-1}}}\\&= 1+\frac1{\frac {-1}{0.37}z(1+\frac{12.468z^{-1}+0.6z^{-2}}{-.37+36.4z^{-1}})} \end {align}$$ and so on.