Beurling-type density criteria for system identification
AuthorsCéline Aubel, Helmut Bölcskei, and Verner Vlačić
ReferenceJournal of Fourier Analysis and Applications, Volume 29, Article No. 45, July 2023.
AbstractThis 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.
KeywordsBeurling 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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Copyright Notice: © 2023 C. Aubel, H. Bölcskei, and V. Vlačić.
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