Beurling-type density criteria for system identification

Authors

Céline Aubel, Helmut Bölcskei, and Verner Vlačić

Reference

Journal of Fourier Analysis and Applications, Vol. 29, Article No. 45, July 2023.

DOI: https://doi.org/10.1007/s00041-023-10020-8

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, 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 find 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.

Keywords

Beurling density, Bargmann-Fock space, interpolation, modulation spaces, short-time Fourier transform, harmonic analysis, uniformly discrete sets, system identification, linear time-varying systems, time-frequency analysis

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