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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)
\end{displaymath} (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)$.