Filtering of second order generalized stochastic processes corrupted by additive noise

2025-04-25

AI Summary

This paper addresses the optimal linear filtering of the sum of two second-order uncorrelated generalized stochastic processes corrupted by additive noise. For the wide-sense stationary case, we introduce the Radon–Nikodym derivative to characterize frequency-domain measure relationships and derive an explicit closed-form filter solution in the spectral domain. For the nonstationary case, we propose a novel modeling framework based on pseudodifferential operators acting on Sjöstrand modulation spaces, leveraging their spectral invariance to ensure well-posedness and stability of the filter. The methodology integrates generalized stochastic process theory, covariance operator analysis, tempered Radon measures, Fourier analysis, and pseudodifferential operator theory. Our main contributions are: (i) removing the conventional stationarity assumption; (ii) establishing an analytic spectral-domain solution for the wide-sense stationary setting; and (iii) constructing a robust time-frequency filtering framework for nonstationary generalized processes—thereby providing both a rigorous mathematical foundation and a computationally feasible implementation pathway.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization
• Intelligent Robots: State Estimation

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Systems and Infrastructure for Web, Mobile and WoT: Data management and stream processing for Web, mobile and wireless applications

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 Sj""ostrand modulation spaces which admits the use of its spectral invariance properties.

Problem

• Optimal linear filtering for second-order generalized stochastic processes
• Solving operator equations involving covariance operators
• Analyzing stationary and non-stationary cases with pseudodifferential operators

Innovation

• Optimal linear filtering for uncorrelated generalized processes
• Radon-Nikodym derivative in frequency domain solution
• Pseudodifferential operators with Sjöstrand modulation spaces