Sampling in shift-invariant spaces is a realistic model for signals withsmooth spectrum. In this paper, we consider phaseless sampling andreconstruction of real-valued signals in a shift-invariant space from theirmagnitude measurements on the whole Euclidean space and from their phaselesssamples taken on a discrete set with finite sampling density. We introduce anundirected graph to a signal and use connectivity of the graph to characterizewhether the signal can be determined, up to a sign, from its magnitudemeasurements on the whole Euclidean space. Under the local complement propertyassumption on a shift-invariant space, we find a discrete set with finitesampling density such that signals in the shift-invariant space, that aredetermined from their magnitude measurements on the whole Euclidean space, canbe reconstructed in a stable way from their phaseless samples taken on thatdiscrete set. In this paper, we also propose a reconstruction algorithm whichprovides a suboptimal approximation to the original signal when its noisyphaseless samples are available only. Finally, numerical simulations areperformed to demonstrate the robust reconstruction of box spline signals fromtheir noisy phaseless samples.
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