  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 innovations (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. We
establish the equivalence between the graph connectivity and phase
retrievability of FRI signals, and we apply the graph connected component
decomposition to find all FRI signals that have the same magnitude measurements
as the original FRI signal has. We construct discrete sets with finite density
explicitly so that magnitude measurements of FRI signals on the whole domain
are determined by their samples taken on those discrete subsets. In this paper,
we also propose a stable algorithm with linear complexity to reconstruct FRI
signals from their phaseless samples on the above phaseless sampling set. The
proposed algorithm is demonstrated theoretically and numerically to provide a
suboptimal approximation to the original FRI signal in magnitude measurements.
