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We treat the optimal linear filtering problem for a sum of two second orderuncorrelated generalized stochastic processes. This is an operator equationinvolving covariance operators. We study both the wide-sense stationary caseand the non-stationary case. In the former case the equation simplifies into aconvolution equation. The solution is the Radon--Nikodym derivative betweennon-negative tempered Radon measures, for signal and signal plus noiserespectively, in the frequency domain. In the non-stationary case we work withpseudodifferential operators with symbols in Sj\"ostrand modulation spaceswhich admits the use of its spectral invariance properties.

Filtering of second order generalized stochastic processes corrupted by additive noise