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

Ben Leshem, Oren Raz, Ariel Jaffe, Boaz Nadler*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 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 - Nov 2018
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2017 Elsevier Inc.


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


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