Sbaiz, Luciano and Vandewalle, Patrick and Vetterli, Martin (2008) Groebner Basis Methods for Multichannel Sampling with Unknown Offsets. Applied and Computational Harmonic Analysis, 25 (3). pp. 277-294.
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Abstract:
In multichannel sampling, several sets of sub-Nyquist sampled signal values are acquired. The offsets between the sets are unknown, and have to be resolved, just like the parameters of the signal itself. This problem is nonlinear in the offsets, which once found, make a linear problem for the signal parameters. We show that when the basis functions for the signal space are related to polynomials, we can express the joint offset and signal parameter estimation as a set of polynomial equations. This is the case for example with polynomial signals or Fourier series. The unknown offsets and signal parameters can be computed exactly from such a set of polynomials using Groebner bases and Buchberger's algorithm. This solution method is developed in detail after a short and tutorial overview of Groebner basis methods. We then address the case of noisy samples, and consider the computational complexity, exploring simplifications due to the special structure of the problem.
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