One problem with noise removal by filtering is that the response of the noise to the filter is left in the signal. As an example, consider a model of the recorded data as the sum of signal and noise, or , where is the recorded data, is signal, and is noise. If a signal-prediction filter is calculated from the data such that ,and the noise is assumed to be what is left after the filter is applied, ,is inconsistent with the original model of .Instead the output of the filter is , since ,assuming that .
An alternative to accepting this filtering result is to set up the problem as a matrix inversion. If the signal is predicted by a filter , such that ,the noise is predicted by a filter , such that ,and the recorded data is a sum of the signal and noise, ,then the signal may be predicted with a system of regressions such as
If is difficult to estimate, or if the noise is unpredictable, an alternative to the previous system of regressions is