Uniqueness of phase retrieval from offset linear canonical transform

TL;DR

This paper investigates the uniqueness of phase retrieval using the offset linear canonical transform (OLCT), proving conditions under which signals can be uniquely reconstructed from magnitude-only measurements in both continuous and discrete settings.

Contribution

It establishes the uniqueness conditions for phase retrieval in OLCT, including continuous, discrete, and short-time OLCT cases, extending phase retrieval theory beyond Fourier-based methods.

Findings

Nontrivial ambiguities are represented by convolution operators.

Unique recovery of compactly supported signals from multiple OLCT magnitudes.

Reconstruction of nonseparable functions from short-time OLCT magnitudes.

Abstract

The classical phase retrieval refers to the recovery of an unknown signal from its Fourier magnitudes, which is widely used in fields such as quantum mechanics, signal processing, optics, etc. The offset linear canonical transform (OLCT), which is a more general type of linear integral transform including Fourier transform (FT), fractional Fourier transform (FrFT), and linear canonical transform (LCT) as its special cases. Hence, in this paper, we focus on the uniqueness problem of phase retrieval in the framework of OLCT. First, we prove that all the nontrivial ambiguities in continuous OLCT phase retrieval can be represented by convolution operators, and demonstrate that a continuous compactly supported signal can be uniquely determined up to a global phase from its multiple magnitude-only OLCT measurements. Moreover, we investigate the nontrivial ambiguities in the discrete OLCT phase retrieval case. Furthermore, we demenstrate that a nonseparable function can be uniquely recovered from its magnitudes of short-time OLCT (STOLCT) up to a global phase. Finally, we show that signals which are bandlimited in FT or OLCT domain can be reconstructed from its sampled STOLCT magnitude measurements, up to a global phase, providing the ambiguity function of window function satisfies some mild conditions.