A Time-Based Sampling Framework for Finite-Rate-of-Innovation Signals

Time-based sampling of continuous-time signals is an alternative to Shannon's sampling paradigm in which the signal is encoded using a sequence of nonuniform time instants. The standard methods for reconstructing signals in bandlimited and shift-invariant spaces from their nonuniform measurements employ alternating projections algorithms. In this paper, we consider the problem of sampling and perfect reconstruction of periodic finite-rate-of-innovation (FRI) signals using crossing-time-encoding machine (C-TEM) and integrate-and-fire TEM (IF-TEM). We formulate the reconstruction problem in the frequency domain and develop techniques to compute the Fourier coefficients, which contain the unknown parameters of the signal in the form of a sum of weighted complex exponentials. The parameters are then estimated using high-resolution spectral estimation techniques. Unlike state-of-the-art methods, the proposed method is generalized to incorporate reconstruction of periodic FRI signals consisting of weighted and shifted versions of an arbitrary pulse with arbitrarily close delays, and is compatible with a large class of sampling kernels. We provide sufficient conditions for sampling and perfect reconstruction using C-TEM and IF-TEM. We present simulation results to support our claims. We also discuss an extension to the sampling of aperiodic FRI signals.