## Abstract

In this paper we consider the following real-valued and finite dimensional specific instance of the 1-D classical phase retrieval problem. Let F∈R^{N} 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 language | American English |
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Pages (from-to) | 463-485 |

Number of pages | 23 |

Journal | Applied and Computational Harmonic Analysis |

Volume | 45 |

Issue number | 3 |

DOIs | |

State | Published - Nov 2018 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2017 Elsevier Inc.

## Keywords

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