Uniqueness of phase retrieval from offset linear canonical transform

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.

SVD-Based Graph Fractional Fourier Transform on Directed Graphs and Its Application

Graph fractional Fourier transform (GFRFT) is an extension of graph Fourier transform (GFT) that provides an additional fractional analysis tool for graph signal processing (GSP) by generalizing temporal-vertex domain Fourier analysis to fractional orders. In recent years, a large number of studies on GFRFT based on undirected graphs have emerged, but there are very few studies on directed graphs. Therefore, in this paper, one of our main contributions is to introduce two novel GFRFTs defined on Cartesian product graph of two directed graphs, by performing singular value decomposition on graph fractional Laplacian matrices. We prove that two proposed GFRFTs can effectively express spatial-temporal data sets on directed graphs with strong correlation. Moreover, we extend the theoretical results to a generalized Cartesian product graph, which is constructed by $m$ directed graphs. Finally, the denoising performance of our proposed two GFRFTs are testified through simulation by processing hourly temperature data sets collected from 32 weather stations in the Brest region of France.

Robust Multidimentional Chinese Remainder Theorem for Integer Vector Reconstruction

The problem of robustly reconstructing an integer vector from its erroneous remainders appears in many applications in the field of multidimensional (MD) signal processing. To address this problem, a robust MD Chinese remainder theorem (CRT) was recently proposed for a special class of moduli, where the remaining integer matrices left-divided by a greatest common left divisor (gcld) of all the moduli are pairwise commutative and coprime. The strict constraint on the moduli limits the usefulness of the robust MD-CRT in practice. In this paper, we investigate the robust MD-CRT for a general set of moduli. We first introduce a necessary and sufficient condition on the difference between paired remainder errors, followed by a simple sufficient condition on the remainder error bound, for the robust MD-CRT for general moduli, where the conditions are associated with (the minimum distances of) these lattices generated by gcld's of paired moduli, and a closed-form reconstruction algorithm is presented. We then generalize the above results of the robust MD-CRT from integer vectors/matrices to real ones. Finally, we validate the robust MD-CRT for general moduli by employing numerical simulations, and apply it to MD sinusoidal frequency estimation based on multiple sub-Nyquist samplers.

Stable Recovery of Weighted Sparse Signals from Phaseless Measurements via Weighted l1 Minimization

The goal of phaseless compressed sensing is to recover an unknown sparse or approximately sparse signal from the magnitude of its measurements. However, it does not take advantage of any support information of the original signal. Therefore, our main contribution in this paper is to extend the theoretical framework for phaseless compressed sensing to incorporate with prior knowledge of the support structure of the signal. Specifically, we investigate two conditions that guarantee stable recovery of a weighted $k$-sparse signal via weighted l1 minimization without any phase information. We first prove that the weighted null space property (WNSP) is a sufficient and necessary condition for the success of weighted l1 minimization for weighted k-sparse phase retrievable. Moreover, we show that if a measurement matrix satisfies the strong weighted restricted isometry property (SWRIP), then the original signal can be stably recovered from the phaseless measurements.