# Phase Retrieval for Finitely-Supported Complex Measures via the Fourier Transform

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## Abstract

We study the recovery of a finitely-supported complex measure

Under some additional assumptions, we provide two recovery algorithms. The first algorithm is based on phase propagation and the Prony method. We show that the reconstruction problem can be reduced to applying linear inverses and finding roots of a polynomial in the case of exact measurements for almost every signal of the above form. In the second algorithm, at the cost of introducing a truncation error, we follow the technique presented by Cand`es and Fernandez-Granda in \cite{cand'es_fernandez-granda_2013} to show that the solution to a total-variation norm minimization problem defined by the given intensity measurements yields an approximation of