The generic crystallographic phase retrieval problem
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In this paper we consider the problem of recovering a signal x ∈ℝ^N from its power spectrum assuming that the signal is sparse with respect to a generic basis for ℝ^N. Our main result is that if the sparsity level is at most ∼ N/2 in this basis then the generic sparse vector is uniquely determined up to sign from its power spectrum. We also prove that if the sparsity level is ∼ N/4 then every sparse vector is determined up to sign from its power spectrum. Analogous results are also obtained for the power spectrum of a vector in ℂ^N which extend earlier results of Wang and Xu <cit.>.
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