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 f 1 , . . . , f k from the Fourier samples. Following [5], for each j = 1, . . . , k we choose sampling set S j to be a subset of the common set of zeroes of the Fourier transforms F(f ), = 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 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 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.

Norming Sets and Related Remez-Type Inequalities

“…This is an important nonlinear version of the Remez inequality having numerous applications in Analysis, Random Functions theory, Sampling theory etc. (see [36,30,31,4] and references therein), formulated as follows:…”