Filtering of second order generalized stochastic processes corrupted by additive noise

We treat the optimal linear filtering problem for a sum of two second order uncorrelated generalized stochastic processes.

This is an operator equation involving covariance operators.

We study both the wide-sense stationary case and the non-stationary case.

In the former case the equation simplifies into a convolution equation.

The solution is the Radon--Nikodym derivative between non-negative tempered Radon measures, for signal and signal plus noise respectively, in the frequency domain.

In the non-stationary case we work with pseudodifferential operators with symbols in Sjöstrand modulation spaces which admits the use of its spectral invariance properties.