Accuracy of Algebraic Fourier Reconstruction for Shifts of Several Signals

We consider the problem of “algebraic reconstruction” of linear combinations of shifts of several known signals $f_1,\dots ,f_k$ from the Fourier samples. Following \cite{Bat.Sar.Yom2}, for each $j=1,\dots ,k$ we choose sampling set $S_j$ to be a subset of the common set of zeroes of the Fourier transforms $\mathcal{F}(f_{\mathrm{\ell }}),\mathrm{\ell }\mathrm{\ne }j$, on which $\mathcal{F}(f_j)\mathrm{\ne }0$. It was shown in \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” 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 Tur'an-Nazarov inequality (\cite{Naz}), and its generalization to discrete sets, obtained in \cite{Fri.Yom}. We illustrate our general approach with examples, and provide some simulation results.