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Phase Retrieval from Gabor Measurements

Abstract

Compressed sensing investigates the recovery of sparse signals from linear measurements. But often, in a wide range of applications, one is given only the absolute values (squared) of the linear measurements. Recovering such signals (not necessarily sparse) is known as the phase retrieval problem. We consider this problem in the case when the measurements are time-frequency shifts of a suitably chosen generator, i.e. coming from a Gabor frame. We prove an easily checkable injectivity condition for recovery of any signal from all \(N^2\) time-frequency shifts, and for recovery of sparse signals, when only some of those measurements are given.

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Notes

  1. We in fact always obtain a frame in this way if \(g\ne 0\). The frame is even \(N \left\| g \right\| ^2\)-tight [23].

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Acknowledgments

The authors would like to thank Gitta Kutyniok and Peter Jung for fruitful discussions and remarks, and Martin Schäfer, who assisted in the proof-reading of the manuscript. I. B. acknowledges support by the Berlin Mathematical School. A. F. acknowledges support by Deutsche Forschungsgemeinschaft (DFG) Grant KU 1446/18-1 and by the Deutscher Akademischer Austausch Dienst (DAAD).

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Correspondence to Axel Flinth.

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Communicated by Peter G. Casazza.

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Bojarovska, I., Flinth, A. Phase Retrieval from Gabor Measurements. J Fourier Anal Appl 22, 542–567 (2016). https://doi.org/10.1007/s00041-015-9431-0

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  • DOI: https://doi.org/10.1007/s00041-015-9431-0

Keywords

  • Phase retrieval
  • PhaseLift
  • Gabor frames
  • Time-frequency analysis
  • Sparse signals
  • Difference sets

Mathematics Subject Classification

  • 42C15
  • 42A38
  • 94A12
  • 65T50