Long k-step division method

The steady state solution of an $ARMA$ process can be obtained with a long k-step division of $C(z)$ and $A(z)$ seen as polynomial of $z$ performing only k-steps.

$$\frac{C(z)}{A(z)}=E(z)+\frac{z^{-k}F(z)}{A(z)}$$ Where:

• $E(z)$ is the quotient
• $z^{-k}F(z)$ is the reminder

Then: $$\hat{y}(t+k|t)=\frac{z^{-k}F(z)}{A(z)}e(t+k)=\frac{F(z)}{A(z)}e(t)$$

IDEA OF PROOF (complete in notes)

If we sobstitute $\frac{C(z)}{A(z)}=E(z)+\frac{z^{-k}F(z)}{A(z)}$ inside: $$y(t+k)=\frac{C(z)}{A(z)}e(t+k)$$ and then considering $E(z)=w_0+w_1z^{-1}+...+w_{k-1}z^{-k+1}$,w e obtain: $$y(t+k)=w_0e(t+k)+...+w_{k-1}e(t+1)+\frac{z^{-k}F(z)}{A(z)}e(t+k)$$ Dropping the unpredictable part: $$\hat{y}(t+k|t)=\frac{z^{-k}F(z)}{A(z)}e(t+k)=\frac{F(z)}{A(z)}e(t)$$