Uncovering the limits of uniqueness in sampled Gabor phase retrieval: A dense set of counterexamples in $L^2(\mathbb{R})$

TL;DR

This paper demonstrates that the set of signals in $L^2( eal)$ that cannot be uniquely recovered from sampled Gabor phase measurements is dense, revealing fundamental limits in phase retrieval with practical implications.

Contribution

It proves the density of non-uniqueness counterexamples in $L^2( eal)$ for sampled Gabor phase retrieval, extending previous classifications.

Findings

Counterexamples are dense in $L^2( eal)$.

Non-uniqueness does not encompass all signals.

Provides insights into fundamental limits of phase retrieval.

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.