Uniqueness of Short-Time Linear Canonical Transform Phase Retrieval for Bandlimited Signals

The short-time Fourier transform phase retrieval problem is reconstructing a signal from its short-time Fourier transform magnitude. This phase retrieval approach has a wide range of applications across various fields, such as ptychography and frequency-resolved optical gating. A recent contribution by Wellershof established that the complex-valued signals in the Paley-Wiener space can be uniquely recovered (up to global phase) by their short-time Fourier transform magnitudes sampled at 14 Omega Zx{omega 0,omega 1}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{1}{4\Omega }\mathbb {Z}\times \{\omega _{0},\omega _{1}\}$$\end{document}. In this paper, we generalize Wellershof's findings to the case of short-time linear canonical transform phase retrieval. Specifically, we demonstrate that complex-valued signals in the generalized Paley-Wiener space can be uniquely recovered (up to global phase) by their short-time linear canonical transform magnitudes sampled at b4 Omega Zx omega 0,omega 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{b}{4 \Omega } \mathbb {Z} \times \left\{ \omega _0, \omega _1\right\} $$\end{document}. Since the generalized Paley-Wiener space includes a broader class of signals, our results expand the application scope of the phase retrieval. Finally, we apply short-time linear canonical transform and short-time Fourier transform to the bandlimited signals and compare their standard deviation and energy concentration rate.