Topics in combinatorial phase retrieval
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In this dissertation, we study and prove results concerning the finite alphabet phase retrieval problem, and the problem of sampling real frames that preserves the phase retrieval property. The finite alphabet phase retrieval problem considers the following: under what conditions can one recover a signal whose entries lie in a small alphabet of possible values from its Fourier magnitudes. We extend the definition of homometric sets to homometric partitions of sets and prove that for generic values of the alphabet, two signals have the same Fourier magnitude if and only if their support partitions are homometric. The real frame sampling phase retrieval problem studies when a subset of a phaseretrievable real frame is phase retrievable. We call a frame for which no subset is phase-retrievable a vital frame. We construct a family of real vital frames of size 2M in dimension M and show that in dimension 3, there are no real vital frames of size 7.
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Ph. D.