We consider the problem of “algebraic reconstruction” of linear combinations of shifts of several known signals f1,…, fk from the Fourier samples. Following , 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  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 (), and its generalization to discrete sets, obtained in . We illustrate our general approach with examples, and provide some simulation results.
|Number of pages||23|
|Journal||Sampling Theory in Signal and Image Processing|
|State||Published - 2014|
- Exponential fitting
- Non-uniform sampling
- Prony systems
- Turán-Nazarov inequality