Stable Phaseless Sampling and Reconstruction of Real-Valued Signals with Finite Rate of Innovation

Abstract

A spatial signal is defined by its evaluations on the whole domain. In this paper, we consider stable reconstruction of real-valued signals with finite rate of innovation (FRI), up to a sign, from their magnitude measurements on the whole domain or their phaseless samples on a discrete subset. FRI signals appear in many engineering applications such as magnetic resonance spectrum, ultra wide-band communication and electrocardiogram. For an FRI signal, we introduce an undirected graph to describe its topological structure, establish the equivalence between its graph connectivity and its phase retrievability by point evaluation measurements on the whole domain, apply the graph connected component decomposition to find its unique landscape decomposition and the set of FRI signals that have the same magnitude measurements. We construct discrete sets with finite density so that magnitude measurements of an FRI signal on the whole domain are determined by its phaseless samples taken on those discrete subsets, and we show that the corresponding phaseless sampling procedure has bi-Lipschitz property with respect to a new induced metric on the signal space and the standard $\ell ^{p}$ -metric on the sampling data set. In this paper, we also propose an algorithm with linear complexity to reconstruct an FRI signal from its (un)corrupted phaseless samples on the above sampling set without restriction on the noise level and apriori information whether the original FRI signal is phase retrievable. The algorithm is theoretically guaranteed to be stable, and numerically demonstrated to approximate the original FRI signal in magnitude measurements.