Accuracy of algebraic fourier reconstruction for shifts of several signals

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 (fℓ), 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.