Model fitting by least squares

Unknown filter

When inputs and outputs are given, the problem of finding an unknown filter appears to be overdetermined, so we write $\bold y \approx \bold X \bold f$ where the matrix $\bold X$ is a matrix of downshifted columns like (). Thus the quadratic form to be minimized is a restatement of equation (2.37) with filter definitions:

$$Q(\bold f', \bold f) \eq (\bold X\bold f - \bold y)' (\bold X\bold f - \bold y)$$ (39)

The solution $\bold f$ is found just as we found (2.38), and it is the set of simultaneous equations $\bold 0 = \bold X'(\bold X\bold f - \bold y)$.