Noise removal by inversion

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 $\sv d=\sv s+\sv n$, where $\sv d$ is the recorded data, $\sv s$ is signal, and $\sv n$ is noise. If a signal-prediction filter $\st S$ is calculated from the data $\sv d$ such that $\st S \sv d \approx \sv 0$,and the noise is assumed to be what is left after the filter is applied, $\sv n = \st S \sv d$,is inconsistent with the original model of $\sv d=\sv s+\sv n$.Instead the output of the filter is $\st S \sv n$, since $\st S \sv d = \st S \sv s + \st S \sv n$,assuming that $\st S \sv s =\sv 0$.

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 $\st S$, such that $\st S \sv s \approx \sv 0$,the noise is predicted by a filter $\st N$, such that $\st N \sv n \approx \sv 0$,and the recorded data is a sum of the signal and noise, $\sv d=\sv s+\sv n$,then the signal may be predicted with a system of regressions such as  

$$\left( \begin{array} {c} \sv 0 \\ \st N \sv d\end{array}\rig... ...ft( \begin{array} {c} \st S \\ \st N\end{array}\right) \sv s.$$ (1)

In this expression, the assumption has been made that $\st S$ and $\st N$ have been previously calculated. Since $\sv s$ and $\sv n$ are not available before system () is solved, $\st S$ and $\st N$ must be approximated somehow from the available data. In most of the cases in this thesis, $\st S$ can be calculated by assuming the noise is unpredictable laterally or that $\st S$ may be calculated from some subset of the full dataset. $\st N$ may be calculated from some subset of the full dataset where the noise is expected to dominate.

If $\st N$ is difficult to estimate, or if the noise is unpredictable, an alternative to the previous system of regressions is  

$$\left( \begin{array} {c} \st S \sv d \\ \epsilon \st S \sv ... ... \begin{array} {c} \st S \\ \epsilon\end{array}\right) \sv n.$$ (2)

This system will be examined in greater detail in chapter . The calculated signal in this case is $\sv s=\sv d-\sv n$.Here the assumption is made that the actual noise is close to the noise predicted by the prediction filtering process.