For the unknown input problem we put the known filter f_{t} in a matrix of downshifted columns .Our statement of wishes is now to find x_{t} so that .We can expect to have trouble finding unknown filter inputs x_{t} 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 x_{t} 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
(28) |
(29) |
Formally the solution is found just as with equation (26), 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.
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