Definitions

First I introduce a set of important variables that will help us build the desired filters.

• ${\bf d}$: the data vector; input to the problem.
• ${\bf n}$: the noise vector; assumed to be known.
• ${\bf s}$: the signal vector; output of the problem.
• ${\bf D}$: annihilation filter for the data; a Prediction Error Filter (PEF).
• ${\bf N}$: annihilation filter for the noise; a PEF.
• ${\bf S}$: annihilation filter for the signal; a PEF.

The leading assumption is that the data vector is the sum of the signal and noise vectors, i.e,

$${\bf d}={\bf s}+{\bf n}.$$ (1)