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 (45) with filter definitions:

$$Q(\bold f\T, \bold f) \eq (\bold X\bold f - \bold y)\T\, (\bold X\bold f - \bold y)$$ (47)

The solution $\bold f$ is found just as we found (46), and it is the set of simultaneous equations $\bold 0 = \bold X\T\,(\bold X\bold f - \bold y)$.