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Uncovering the limits of uniqueness in sampled Gabor phase retrieval: A dense set of counterexamples in L2

by R. Alaifari and F. Bartolucci and M. Wellershoff

(Report number 2025-05)

Abstract
Sampled Gabor phase retrieval — the problem of recovering a square-integrable signal from the magnitude of its Gabor transform sampled on a lattice — is a fundamental problem in signal processing, with important applications in areas such as imaging and audio processing. Recently, a classification of square-integrable signals which are not phase retrievable from Gabor measurements on parallel lines has been presented. This classification was used to exhibit a family of counterexamples to uniqueness in sampled Gabor phase retrieval. Here, we show that the set of counterexamples to uniqueness in sampled Gabor phase retrieval is dense in \(L^2(\mathbb{R})\), but is not equal to the whole of \(L^2(\mathbb{R})\) in general. Overall, our work contributes to a better understanding of the fundamental limits of sampled Gabor phase retrieval.

Keywords: Phase retrieval, Gabor transform, uniqueness, stability

@Techreport{ABW25_1126,
  author = {R. Alaifari and F. Bartolucci and M. Wellershoff},
  title = {Uncovering the limits of uniqueness in sampled Gabor phase retrieval: A dense set of counterexamples in L2},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2025-05},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2025/2025-05.pdf },
  year = {2025}
}

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