Phase Retrieval from Gabor Measurements

TL;DR

This paper addresses the phase retrieval problem using Gabor measurements, providing conditions for signal recovery from time-frequency shifted measurements, applicable to both sparse and non-sparse signals.

Contribution

It introduces an injectivity condition for recovering signals from Gabor frame measurements, including sparse signals with partial data.

Findings

Injectivity condition for full Gabor measurements.

Recovery guarantees for sparse signals with partial measurements.

Applicable to a wide range of signals, not necessarily sparse.

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.