Identification of stochastic operators

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.

Sampling of Stochastic Operators

“…Analogous to the deterministic criterion µ(S) < 1, the requirement µ(U ) < 1 is also necessary for stochastic identifiability of a stochastic operator, as we show in an sibling paper [25]. In this paper, we demonstrate patterns that correspond to regions of 4D volume less than one but are nonetheless unidentifiable by our methods.…”

“…In the sibling paper [25] we develop and use the theory of stochastic modulation spaces to rigorously define and prove identification results for operators with spreading functions belonging to a class of generalized random processes -including delta functions and white noise. The norm inequalities that are proven there are essential to justify use of delta trains as sounding signals.…”

“…Two common types of methods to identify the scattering function are deconvolution and direct measurement methods [1,9,21,13]. In [32] we apply the methodology developed here and in [25] to study WSSUS channels in depth. Theorem 16 below guarantees identifiability of a WSSUS channel whenever its scattering function is merely compactly supported.…”

Cornerstones of Sampling of Operator Theory

“…In such models, the spread-ing function η(t, ν) (and therefore the operator's kernel and Kohn-Nirenberg symbol) are random processes, and the operator is split into the sum of its deterministic portion, representing the mean behavior of the channel, and its zero-mean stochastic portion that represents the noise and the environment. The detailed analysis of the stochastic case was carried out in [52,51]. One of the foci of these works lies in the goal of determining the second-order statistics of the (zero mean) stochastic process η(τ, ν), that is, its so called covariance function R(τ, ν, τ , ν ) = E{η(τ, ν) η(τ , ν )}.…”

“…For details, formal definitions of identifiability and detailed statements of results we refer to the papers [37,52,51,53].…”