Filtering of second order generalized stochastic processes corrupted by additive noise
Patrik Wahlberg
TL;DR
The paper studies optimal linear filtering for the sum of two uncorrelated second-order generalized stochastic processes by formulating and solving the operator equation $\mathscr{K}_u = F(\mathscr{K}_u+\mathscr{K}_w)$. In the wide-sense stationary case the solution is a convolution operator whose Fourier symbol is the Radon–Nikodym derivative $\widehat f = d\mu_u / d(\mu_u+\mu_w)$, providing a rigorous justification for the Wiener filter in the frequency domain. For non-stationary cases, the work develops two complementary routes: (i) commuting covariance operators via the spectral theorem yielding $F = \int f(z) \, d\Pi(z)$, and (ii) noncommuting covariances via Weyl pseudodifferential calculus with symbols in $M_{1\otimes\omega}^{\infty,1}$, establishing Wiener-type invariance and decay properties. The results deliver a unified, modulation-space based framework for optimal filtering of generalized stochastic processes, including explicit error expressions, conditions for perfect reconstruction, and insights into the mapping properties of the optimal filter across function spaces.
Abstract
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.