Accuracy of algebraic fourier reconstruction for shifts of several signals

Dmitry Batenkov, Niv Sarig, Yosef Yomdin

Research output: Contribution to journalArticlepeer-review

Abstract

We consider the problem of “algebraic reconstruction” of linear combinations of shifts of several known signals f1,…, fk from the Fourier samples. Following [5], for each j = 1,…, k we choose sampling set Sj to be a subset of the common set of zeroes of the Fourier transforms F (f), ℓ ≠ j, on which F (fj) ≠ 0. It was shown in [5] that in this way the reconstruction system is “decoupled” into k separate systems, each including only one of the signals fj. The resulting systems are of a “generalized Prony” form.

However, the sampling sets as above may be non-uniform/not “dense enough” to allow for a unique reconstruction of the shifts and amplitudes. In the present paper we study uniqueness and robustness of non-uniform Fourier sampling of signals as above, investigating sampling of exponential polynomials with purely imaginary exponents. As the main tool we apply a well-known result in Harmonic Analysis: the Turán-Nazarov inequality ([18]), and its generalization to discrete sets, obtained in [12]. We illustrate our general approach with examples, and provide some simulation results.

Original languageEnglish
Pages (from-to)151-173
Number of pages23
JournalSampling Theory in Signal and Image Processing
Volume13
Issue number2
StatePublished - 2014
Externally publishedYes

Keywords

  • Exponential fitting
  • Non-uniform sampling
  • Prony systems
  • Turán-Nazarov inequality

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