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

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 L2(ℝ), but is not equal to the whole of L2(ℝ) in general. Overall, our work contributes to a better understanding of the fundamental limits of sampled Gabor phase retrieval.