Linear predictor from output

In the real world we cannot measur the white noise, the only aviable information is the values of the process up to time $t$.

We need to construct the white noise underlyning the generation of the process from the values of the process itself up to time $t$. This means that if we are able to define: $$e(t)=h_0y(t)$$ $$e(t)=h_0y(t)+h_1y(t-1)+...= \sum_{i=0}^{+\infty}{h_iy(t-i)}$$ Then the optimal predictor from noise can be written as a predictor from output: $$\hat{y}(t+k|t)=\sum_{i=0}^{+\infty}{w_{k+i}[\sum_{j=0}^{+\infty}{h_jy(t-i-j)}]}$$

Reconstructing WN from output

Since we imposed that $\frac{C(z)}{A(z)}$ is minimum phase then $\frac{A(z)}{C(z)}$ is asymptotically stable and: $$e(t)=\frac{C(z)}{A(z)}y(t)=\sum_{i=0}^{+\infty}{h_iy(t-i)}$$. So: $$\hat{y}(t+k|t)=\frac{F(z)}{C(z)}y(t)$$