Definitions

We give a set of definitions that will help us to better understand the properties of the noise and signal filters in equation (5).

Definition 1: an operator P is a projector if

$${\bf PP} = {\bf P}.$$ (22)

Definition 2: two operators P and Q are complementary operators if

$${\bf P+Q} = {\bf I}.$$ (23)

Definition 3: two operators P and Q are mutually orthogonal if

$${\bf PQ} = {\bf QP} = {\bf 0}.$$ (24)

Definition 4: the $\ell^2$ norm of a vector v is

$$\Vert{\bf v}\Vert^2 = \sum_i v_i^2$$ (25)

or

$$\Vert{\bf v}\Vert^2 = {\bf v'v},$$ (26)

where (') is the adjoint.
Definition 5: two vectors u and v are orthogonal if

$${\bf u'v}={\bf v'u}=0.$$ (27)