Phase retrieval from sampled Gabor transform magnitudes: Counterexamples

TL;DR

This paper demonstrates that, in general, signals cannot be uniquely recovered from sampled Gabor transform magnitudes, providing counterexamples that challenge the feasibility of phase retrieval from such samples.

Contribution

It constructs explicit counterexamples showing non-uniqueness in phase retrieval from sampled Gabor magnitudes for any lattice, highlighting fundamental limitations.

Findings

Existence of non-unique signals with identical sampled Gabor magnitudes

Counterexamples can be constructed with good time-frequency concentration

Real-valued counterexamples are possible on rectangular lattices

Abstract

We consider the recovery of square-integrable signals from discrete, equidistant samples of their Gabor transform magnitude and show that, in general, signals can not be recovered from such samples. In particular, we show that for any lattice, one can construct functions in $L^2(\mathbb{R})$ which do not agree up to global phase but whose Gabor transform magnitudes sampled on the lattice agree. These functions have good concentration in both time and frequency and can be constructed to be real-valued for rectangular lattices.