General properties of the noise and signal filters

Following the preceding definitions, we can define the noise and signal filters more precisely. But first, recall that

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

with

$$\begin{eqnarray} {\bf \overline{R_s}}&=&{\bf I}-{\bf L_s}({\bf L_s'L_s})^{-1}{\b... ...line{{\bf R_n}}&=&{\bf I}-{\bf L_n}({\bf L_n'L_n})^{-1}{\bf L_n'}.\end{eqnarray}$$ (109)

${\bf \overline{R_s}}$ and ${\bf \overline{R_n}}$ are signal and noise filtering operators respectively. If we define

$$\begin{eqnarray} {\bf \overline{R_s}}&=&{\bf I}-{\bf R_s}, \nonumber \\ \overline{{\bf R_n}}&=&{\bf I}-{\bf R_n},\end{eqnarray}$$ (110)

with ${\bf R_s}={\bf L_s}({\bf L_s'L_s})^{-1}{\bf L_s'}$ and ${\bf R_n}={\bf L_n}({\bf L_n'L_n})^{-1}{\bf L_n'}$ the signal and noise resolution operators, we deduce that ${\bf R_s}$ and ${\bf \overline{R_s}}$, ${\bf R_s}$ and ${\bf \overline{R_s}}$ are complementary operators (definition 2).

It can be shown that ${\bf \overline{R_s}}$, ${\bf \overline{R_n}}$,${\bf R_s}$ and ${\bf R_n}$ are projectors. Indeed, for ${\bf R_s}$ and ${\bf \overline{R_s}}$, we have

$$\begin{eqnarray} {\bf R_sR_s}&=&{\bf L_s}({\bf L_s'L_s})^{-1}{\bf L_s'}{\bf L_s... ... L_s'L_s})^{-1}{\bf L_s'}, \nonumber \\ {\bf R_sR_s}&=&{\bf R_s},\end{eqnarray}$$ (111)

and

$$\begin{eqnarray} {\bf \overline{R_s}}{\bf \overline{R_s}}&=&({\bf I}-{\bf R_s})... ... {\bf \overline{R_s}}{\bf \overline{R_s}}&=&{\bf \overline{R_s}}.\end{eqnarray}$$ (112)

Thus, ${\bf R_s}$ and ${\bf \overline{R_s}}$ are projectors as defined in definition 1. The same proofs work for ${\bf R_n}$ and ${\bf \overline{R_n}}$.

We can prove also that ${\bf \overline{R_s}}$ and ${\bf R_s}$, ${\bf \overline{R_n}}$ and ${\bf R_n}$ are mutually orthogonal. For ${\bf R_s}$ and ${\bf \overline{R_s}}$, we have

$$\begin{eqnarray} {\bf \overline{R_s}}{\bf R_s}&=&({\bf I}-{\bf R_s}){\bf R_s}, ... ...\bf R_s}), \nonumber \\ {\bf \overline{R_s}}{\bf R_s}&=&{\bf 0}.\end{eqnarray}$$ (113)

Hence, ${\bf \overline{R_s}}$ and ${\bf R_s}$, ${\bf \overline{R_n}}$ and ${\bf R_n}$ are complementary, mutually orthogonal projectors.