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### Unknown input: deconvolution with a known filter

deconvolution For solving the unknown-input problem, we put the known filter ft in a matrix of downshifted columns .Our statement of wishes is now to find xt so that .We can expect to have trouble finding unknown inputs xt when we are dealing with certain kinds of filters, such as bandpass filters. If the output is zero in a frequency band, we will never be able to find the input in that band and will need to prevent xt from diverging there. We do this by the statement that we wish ,where is a parameter that is small and whose exact size will be chosen by experimentation. Putting both wishes into a single, partitioned matrix equation gives
 (37)
To minimize the residuals and ,we can minimize the scalar .This is
 (38)
We solved this minimization in the frequency domain (beginning from equation (4)).

Formally the solution is found just as with equation (35), but this solution looks unappealing in practice because there are so many unknowns and because 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 to discover that constraints or time-variable weighting functions are required in some applications. This is an issue we must be continuously alert to, whether the cost of a method is justified by its need.

Next: KRYLOV SUBSPACE ITERATIVE METHODS Up: From the frequency domain Previous: Unknown filter
Stanford Exploration Project
4/27/2004