Beurling-type density criteria for system identification

TL;DR

This paper establishes a density criterion for identifying linear time-varying systems with delay-Doppler shifts, linking system identification to complex analysis and ensuring robust recovery under certain density conditions.

Contribution

It introduces a Beurling density condition for system identifiability without geometric discretization constraints, connecting it to Bargmann-Fock space interpolation.

Findings

Identifiability when density is less than 1/2

Density condition is necessary for certain classes of systems

Robust support and weight recovery with vanishing error

Abstract

This paper addresses the problem of identifying a linear time-varying (LTV) system characterized by a (possibly infinite) discrete set of delay-Doppler shifts without a lattice (or other geometry-discretizing) constraint on the support set. Concretely, we show that a class of such LTV systems is identifiable whenever the upper uniform Beurling density of the delay-Doppler support sets, measured uniformly over the class, is strictly less than 1/2. The proof of this result reveals an interesting relation between LTV system identification and interpolation in the Bargmann-Fock space. Moreover, we show that this density condition is also necessary for classes of systems invariant under time-frequency shifts and closed under a natural topology on the support sets. We furthermore show that identifiability guarantees robust recovery of the delay-Doppler support set, as well as the weights of the individual delay-Doppler shifts, both in the sense of asymptotically vanishing reconstruction error for vanishing measurement error.