Finite rate of innovation sampling of Gaussian pulse streams with variable shape

Abstract

The existing Finite Rate of Innovation (FRI) sampling theory can realize the efficient measurement of Gaussian pulse streams with fixed modal functions. However, it is hard to sample that signal with variable pulse width factors by a sub-Nyquist rate. To get around this challenge, we propose an extended FRI sampling method based on Taylor series expansion in this paper, where the signal is first filtered by the exponential reproducing kernel, then each pulse can be reconstructed sequentially. For a single pulse, we first prove that the pulse width factor does not interfere with the estimation process of the time delay parameter, and we also present its Cramer-Rao lower Bound (CRB). The signal Fourier coefficient can then be reformulated by the Taylor series expansion of order N−1 to realize the decoupling of the pulse width factor before the final estimation. We derive the order limit of expansion from the perspective of numerical stability, and we also give the upper bound on the estimation error of the pulse width factor for N=2. Simulation results show that, compared with the classical methods, the proposed state-of-the-art algorithm can achieve a higher precision reconstruction of the Gaussian pulse stream signal.