One-dimensional phase retrieval with additional interference measurements

TL;DR

This paper demonstrates that in one-dimensional phase retrieval, the original complex signal can be uniquely reconstructed using Fourier intensity data combined with two interference measurements, even in continuous-time scenarios.

Contribution

It introduces a novel approach showing unique recovery of signals from Fourier intensity and interference measurements, extending to continuous-time signals and unknown references.

Findings

Unique recovery with two interference measurements in discrete-time

Extension of results to continuous-time signals

Recovery possible with known or unknown reference signals

Abstract

The one-dimensional phase retrieval problem consists in the recovery of a complex-valued signal from its Fourier intensity. Due to the well-known ambiguousness of this problem, the determination of the original signal within the extensive solution set is challenging and can only be done under suitable a priori assumption or additional information about the unknown signal. Depending on the application, one has sometimes access to further interference measurements between the unknown signal and a reference signal. Beginning with the reconstruction in the discrete-time setting, we show that each signal can be uniquely recovered from its Fourier intensity and two further interference measurements between the unknown signal and a modulation of the signal itself. Afterwards, we consider the continuous-time problem, where we obtain an equivalent result. Moreover, the unique recovery of a continuous-time signal can also be ensured by using interference measurements with a known or an unknown reference which is unrelated to the unknown signal.