Enforcing uniqueness in one-dimensional phase retrieval by additional signal information in time domain

TL;DR

This paper investigates how additional time domain information, such as magnitudes or phase, can ensure unique recovery of signals in one-dimensional phase retrieval problems, reducing ambiguities significantly.

Contribution

It demonstrates that incorporating extra time domain magnitude or phase data guarantees unique signal recovery for most finite-support signals.

Findings

Almost all finite-support signals can be uniquely recovered with additional time domain magnitudes.

Additional phase information also ensures uniqueness in signal recovery.

The approach reduces the solution ambiguities inherent in phase retrieval.

Abstract

Considering the ambiguousness of the discrete-time phase retrieval problem to recover a signal from its Fourier intensities, one can ask the question: what additional information about the unknown signal do we need to select the correct solution within the large solution set? Based on a characterization of the occurring ambiguities, we investigate different a priori conditions in order to reduce the number of ambiguities or even to receive a unique solution. Particularly, if we have access to additional magnitudes of the unknown signal in the time domain, we can show that almost all signals with finite support can be uniquely recovered. Moreover, we prove that an analogous result can be obtained by exploiting additional phase information.