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Recoverability of band-passed signals from non-zero threshold crossings

In this paper we consider the necessary condition for the recoverability with probability close to one (3 sigma criterion) of a band-limited band-passed signal from event-based samples. Specifically, we consider sampling events derived from non-zero level crossings of a stochastic differentiable signal on a finite time interval. Using a Fourier approach we write the system in matrix form allowing a sufficient condition of the recoverability of the signal to be specified. A necessary condition establishes a link between the number of Fourier amplitudes of the signal and the number of level crossings; a sufficient condition establishes a link between the number of Fourier amplitudes and the rank of a rectangular matrix.