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A spatial signal is defined by its evaluations on the whole domain. In thispaper, we consider stable reconstruction of real-valued signals with finiterate of innovations (FRI), up to a sign, from their magnitude measurements onthe whole domain or their phaseless samples on a discrete subset. FRI signalsappear in many engineering applications such as magnetic resonance spectrum,ultra wide-band communication and electrocardiogram. For an FRI signal, weintroduce an undirected graph to describe its topological structure. Weestablish the equivalence between the graph connectivity and phaseretrievability of FRI signals, and we apply the graph connected componentdecomposition to find all FRI signals that have the same magnitude measurementsas the original FRI signal has. We construct discrete sets with finite densityexplicitly so that magnitude measurements of FRI signals on the whole domainare determined by their samples taken on those discrete subsets. In this paper,we also propose a stable algorithm with linear complexity to reconstruct FRIsignals from their phaseless samples on the above phaseless sampling set. Theproposed algorithm is demonstrated theoretically and numerically to provide asuboptimal approximation to the original FRI signal in magnitude measurements.

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