In the following discussion, three assumptions are made to separate signal and noise from data. First, the data is defined to be a simple sum of the signal and noise; that is, , being the observed data, the signal, and the noise. Next, there exists a filter that predicts the signal, .Finally, there exists a filter that predicts the noise, .The methods of getting and will be covered later.
The assumed noise filter requires a change in the definition of the noise from the previous chapters, where unpredictable noise was separated from a predictable signal. Although it will be shown later that unpredictable noise may be removed with the techniques to be discussed here, more emphasis is given now to coherent noise.
Three conditions are expected to be met by the final solution for the signal and noise:
Using equations () to (), two systems of regressions may be generated, one to calculate the noise and one to calculate the signal. To calculate the signal, replaces in equation (), which is then combined with equation () to give a single system of regressions
A similar manipulation produces a calculation of the noise
Once the signal is calculated, the noise is simply .If the noise is calculated, the signal is .