


 Model fitting by least squares  

Next: KRYLOV SUBSPACE ITERATIVE METHODS
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For solving the unknowninput problem,
we put the known filter in a matrix of downshifted columns .
Our statement of wishes is now to find so that
.
We can expect to have trouble finding unknown inputs
when we are dealing with certain kinds of filters,
such as bandpass filters.
If the output is zero in a frequency band,
we are never able to find the input in that band
and need to prevent from diverging there.
We prevent divergence by the statement that we wish
,
where is a parameter that is small
with exact size chosen by experimentation.
Putting both wishes into a single, partitioned matrix equation gives:

(48) 
To minimize the residuals and ,
we can minimize the scalar
.
Expanding:
We solved this minimization
in the frequency domain
(beginning from equation (4)).
Formally the solution is found just as with equation (46),
but this solution looks unappealing in practice
because there are so many unknowns and
the problem can be solved much more quickly
in the Fourier domain.
To motivate ourselves to solve this problem in the time domain,
we need either to find an approximate solution method that is
much faster, or find ourselves with an application
that needs boundaries,
or needs timevariable weighting functions.



 Model fitting by least squares  

Next: KRYLOV SUBSPACE ITERATIVE METHODS
Up: From the frequency domain
Previous: Unknown filter
20141201