The discrete sign problem: uniqueness, recovery algorithms and phase retrieval applications

Ben Leshem, Oren Raz, Ariel Jaffe, Boaz Nadler

Research output: Contribution to journalArticlepeer-review

9 Scopus citations


In this paper we consider the following real-valued and finite dimensional specific instance of the 1-D classical phase retrieval problem. Let F∈RN be an N-dimensional vector, whose discrete Fourier transform has a compact support. The sign problem is to recover F from its magnitude |F|. First, in contrast to the classical 1-D phase problem which in general has multiple solutions, we prove that with sufficient over-sampling, the sign problem admits a unique solution. Next, we show that the sign problem can be viewed as a special case of a more general piecewise constant phase problem. Relying on this result, we derive a computationally efficient and robust to noise sign recovery algorithm. In the noise-free case and with a sufficiently high sampling rate, our algorithm is guaranteed to recover the true sign pattern. Finally, we present two phase retrieval applications of the sign problem: (i) vectorial phase retrieval with three measurement vectors; and (ii) recovery of two well separated 1-D objects.

Original languageAmerican English
Pages (from-to)463-485
Number of pages23
JournalApplied and Computational Harmonic Analysis
Issue number3
StatePublished - 2018

Bibliographical note

Funding Information:
We thank Dan Oron and Nirit Dudovich for useful discussions. We thank Gaurav Thakur for interesting discussions and for providing us with the code for his algorithm. We also thank Uri Feige for pointing us to Ref. [22] . The research of B.N. was supported in part by grant 892/14 from the Israel Science Foundation . O.R. acknowledges financial support from the James S. McDonnell Foundation .

Publisher Copyright:
© 2017 Elsevier Inc.


  • Compact support
  • Phase retrieval
  • Sampling theory
  • Signal reconstruction


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