Identification of stochastic operators

TL;DR

This paper extends operator sampling theory to stochastic operators with spreading functions, establishing conditions under which such operators can be identified using delta train signals, highlighting the importance of support set geometry.

Contribution

It introduces a functional analytic framework for stochastic operator identification and characterizes the geometric conditions of support sets that enable identifiability.

Findings

Identification possible for stochastic operators with support sets of 4D volume less than one.

Support set geometry significantly affects operator identifiability.

Necessary condition: 4D volume of support set must be less than or equal to one.

Abstract

Based on the here developed functional analytic machinery we extend the theory of operator sampling and identification to apply to operators with stochastic spreading functions. We prove that identification with a delta train signal is possible for a large class of stochastic operators that have the property that the autocorrelation of the spreading function is supported on a set of 4D volume less than one and this support set does not have a defective structure. In fact, unlike in the case of deterministic operator identification, the geometry of the support set has a significant impact on the identifiability of the considered operator class. Also, we prove that, analogous to the deterministic case, the restriction of the 4D volume of a support set to be less or equal to one is necessary for identifiability of a stochastic operator class.