Accuracy of Algebraic Fourier Reconstruction for Shifts of Several Signals

摘要

We consider the problem of 'algebraic reconstruction' of linear combi­nations of shifts of several known signals /1,…, 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), l ≠ j, on which F(f_j) ≠ 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-uniformot '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 Turan-Nazarov inequality ([18]).and its generalization to discrete sets, obtained in [12].We illustrate our general approach with examples, and provide some simulation results.