Uniqueness of Phase Retrieval From Offset Linear Canonical Transform

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) is a more general type of linear integral transform, including the Fourier transform (FT), the fractional Fourier transform (FRFT), and the linear canonical transform (LCT) as its special cases. Hence, in this paper, we focus on the uniqueness and reconstruction algorithm 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. In addition, we demonstrate that a nonseparable function can be uniquely recovered from its magnitudes of short‐time OLCT (STOLCT) up to a global phase. Furthermore, we show that signals which are bandlimited in the FT or OLCT domain can be reconstructed from their sampled STOLCT magnitude measurements, up to a global phase, providing the ambiguity function of the window function satisfies some mild conditions. Finally, we present two reconstruction algorithms for OLCT phase retrieval, namely Gerchberg‐Saxton (GS) algorithm and matrix completion (MC) algorithm, and evaluate their reconstruction performance via numerical simulations.