Preconditioning the filtering method

There is a simple trick that modifies the fitting goals in equation (2). We can pose the following preconditioning transformations:

$$\begin{eqnarray} {\bf n} &=& {\bf N^{-1}m_n}, \nonumber \\ {\bf s} &=& {\bf S^{-1}m_s},\end{eqnarray}$$ (4)

where ${\bf m_n}$ and ${\bf m_s}$ are new variables. Now we can derive a new system of fitting goals as follows:

$$\begin{eqnarray} \bf m_n &\approx& \bf 0 \nonumber \\ \bf \epsilon m_s &\approx... ...ect to} &\leftrightarrow& \bf d = S^{-1}m_s + N^{-1}m_n. \nonumber\end{eqnarray}$$ (5)

This system is almost equivalent to what I introduced in Guitton (2001), except for the regularization that I omitted. With ${\bf L_n}={\bf N^{-1}}$ the noise-modeling operator and ${\bf L_s}={\bf S^{-1}}$ the signal-modeling operator, the least-squares inverse of equations (5) without the regularization terms is then given by  

$$\left( \begin{array} {c} {\bf \hat{m}_n} \\ {\bf \hat{m}_s... ..._n}L_s})^{-1}{\bf L_s'\overline{R_n}}\end{array}\right){\bf d},$$

with

$$\begin{array} {lcl} {\bf \overline{R_s}}&=&{\bf I}-{\bf L_s}... ...=&{\bf I}-{\bf L_n}({\bf L_n'L_n})^{-1}{\bf L_n'}. \end{array}$$ (6)

I showed in Guitton et al. (2001) that ${\bf \overline{R_s}}$ and $\overline{{\bf R_n}}$ can also be interpreted in term of projection filters.

The estimated noise and signal are then computed as follows

$$\begin{array} {lcl} \hat{\bf{n}} & = & \bf{L_n}{\bf \hat{m}_n}, \\ \hat{\bf{s}} & = & \bf{L_s}{\bf \hat{m}_s}. \end{array}$$ (7)

Because of the relationship that exists between the filtering and subtraction methods, the estimated noise or signal should be equivalent for both. This has been observed in a multiple attenuation problem by Guitton et al. (2001).