# Two meanings for "innovation" in Wiener filter are the same?

This is related question to A question about Wiener filter based on Linear Estimation by Kailath, based on the textbook Linear Estimation by Kailath. In that link I talk about how I first learned what innovation is, but in this next class we now learned a different "innovation" and I wonder how these two seemingly different concepts are related. Basically, I learned that whitening filter of y(s) is transfer function from y(s) to white-noise v(s), whereas the inverse transfer function from v(s) to y(s) is innovation filter.

However, in this next class, we learned relationship among estimates $$\hat{x}$$ based on different measurements y1, y2... Knowing only $$y_1$$, we can make estimate $$\hat{x}_{y_1} = E[x y_1^T](E[y_1 y_1^T])^{-1} y_1$$ which is analogous to projecting x to the space of $$y_1$$ Then knowing $$y_2$$, as well, we can get rid of redundant info by projecting to the space of $$y_1$$ first to get $$\hat{y_2}_{y_1} = E[y_2 y_1^T](E[y_1 y_1^T])^{-1} y_1$$ and using $$e_2 = \tilde{y_2}_{y_1} = y_2 - \hat{y_2}_{y_1}$$ to find the second estimate component $$\hat{x}_{e_2}$$ to add onto the previous estimate $$\hat{x}_{y_1}$$ to get $$\hat{x}=\hat{x}_{\tilde{y_2}_{y_1}}+\hat{x}_{y_1}$$

But why is this process of basically doing Gram-Schmidt to create orthogonal bases $$e_1 = y_1$$ , $$e_2 = \tilde{y_2}_{y_1}$$ etc. and projecting to them also called an innovation process? He said innovation is like adding new information. But I thought that had to do with going back and forth with the white noise in the s-domain? I don't see the connection between the two concepts.