Stochastic Wiener Filter in the White Noise Space

TL;DR

This paper introduces a stochastic Wiener filter framework using white noise space theory, enabling the analysis of systems with random parameters through operator theory and stochastic distributions.

Contribution

It generalizes the Wiener filter concept to stochastic systems by employing Hida's white noise space and Kondratiev's stochastic distributions, providing a new analytical approach.

Findings

Characterization of Wiener filter equations in stochastic settings

Application of operator theory to stochastic filtering

Use of nuclearity of white noise spaces for analysis

Abstract

In this paper we introduce a new approach to the study of filtering theory by allowing the system's parameters to have a random character. We use Hida's white noise space theory to give an alternative characterization and a proper generalization to the Wiener filter over a suitable space of stochastic distributions introduced by Kondratiev. The main idea throughout this paper is to use the nuclearity of this spaces in order to view the random variables as bounded multiplication operators (with respect to the Wick product) between Hilbert spaces of stochastic distributions. This allows us to use operator theory tools and properties of Wiener algebras over Banach spaces to proceed and characterize the Wiener filter equations under the underlying randomness assumptions.