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We consider the problem of "algebraic reconstruction" of linear combinationsof shifts of several known signals $f_1,\ldots,f_k$ from the Fourier samples.Following \cite{Bat.Sar.Yom2}, for each $j=1,\ldots,k$ we choose sampling set$S_j$ to be a subset of the common set of zeroes of the Fourier transforms${\cal F}(f_\ell), \ \ell \ne j$, on which ${\cal F}(f_j)\ne 0$. It was shownin \cite{Bat.Sar.Yom2} that in this way the reconstruction system is"decoupled" into $k$ separate systems, each including only one of the signals$f_j$. The resulting systems are of a "generalized Prony" form.  However, the sampling sets as above may be non-uniform/not "dense enough" toallow for a unique reconstruction of the shifts and amplitudes. In the presentpaper we study uniqueness and robustness of non-uniform Fourier sampling ofsignals as above, investigating sampling of exponential polynomials with purelyimaginary exponents. As the main tool we apply a well-known result in HarmonicAnalysis: the Tur\'an-Nazarov inequality (\cite{Naz}), and its generalizationto discrete sets, obtained in \cite{Fri.Yom}. We illustrate our generalapproach with examples, and provide some simulation results.

Accuracy of Algebraic Fourier Reconstruction for Shifts of Several Signals