Conjugate Phase Retrieval on ${\mathbb C}^M$ by real vectors

TL;DR

This paper introduces conjugate phase retrieval, a relaxed version of phase retrieval allowing recovery up to conjugacy, and characterizes real frames that achieve this in complex spaces, with results for low dimensions and generic frames.

Contribution

It defines conjugate phase retrieval, develops its theory for real frames, and provides complete characterizations for low dimensions and conditions for generic frames.

Findings

Real frames are not phase retrievable but can be conjugate phase retrievable.

Complete characterization of conjugate phase retrievable real frames in ${f C}^2$ and ${f C}^3$.

Generic real frames with at least $4M - 6$ measurements are conjugate phase retrievable for $M \\ge 4$.

Abstract

In this paper, we will introduce the notion of {\it conjugate phase retrieval}, which is a relaxed definition of phase retrieval allowing recovery of signals up to conjugacy as well as a global phase factor. It is known that frames of real vectors are never phase retrievable on ${\mathbb C}^M$ in the ordinary sense, but we show that they can be conjugate phase retrievable in complex vector spaces. We continue to develop the theory on conjugate phase retrievable real frames. In particular, a complete characterization of conjugate phase retrievable real frames on ${\mathbb C}^2$ and ${\mathbb C}^3$ is given. Furthermore, we show that a generic real frame with at least $4M - 6$ measurements is conjugate phase retrievable in ${\mathbb C}^M$ for $ M \ge 4.$