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Stability estimates for phase retrieval from discrete Gabor measurements

by R. Alaifari and M. Wellershoff

(Report number 2019-67)

Abstract
Phase retrieval refers to the problem of recovering some signal (which is often modelled as an element of a Hilbert space) from phaseless measurements. It has been shown that in the deterministic setting phase retrieval from frame coefficients is always unstable in infinite-dimensional Hilbert spaces [7] and possibly severely ill-conditioned in finite-dimensional Hilbert spaces [7]. Recently, it has also been shown that phase retrieval from measurements induced by the Gabor transform with Gaussian window function is stable under a more relaxed semi-global phase recovery regime based on atoll functions [1]. In finite dimensions, we present first evidence that this semi-global reconstruction regime allows one to do phase retrieval from measurements of bandlimited signals induced by the discrete Gabor transform in such a way that the corresponding stability constant only scales like a low order polynomial in the space dimension. To this end, we utilise reconstruction formulae which have become common tools in recent years [6,13,19,21].

Keywords:

@Techreport{AW19_871,
  author = {R. Alaifari and M. Wellershoff},
  title = {Stability estimates for phase retrieval from discrete Gabor measurements},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2019-67},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2019/2019-67.pdf },
  year = {2019}
}

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