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