Sampling FRI signals with the SOS kernel: Bounds and optimal kernel

Recently it has been shown that using appropriate sampling kernel, finite rate of innovation signals can be perfectly recon structed even tough they are non-bandlimited. In the presence of noise, reconstruction is achieved by an estimation procedure of all the parameters of the incoming signal. In this paper we consider the estimation of a finite stream of pulses using the Sum of Sincs (SoS) kernel. We derive the Cramér Rao Bound (BCRB) relative to the estimated parameters. The SoS kernel is used since it is configurable by a vector of weights: we propose a family of kernels which maximizes the Bayesian Fisher Information (BIM) i.e. the total amount of information about each of the parameter in the measures. The advantage of the proposed family is that it can be user-adjusted to favor one specific parameter. The variety of the resulting kernel goes from a perfect sinusoid to the Dirichlet kernel.