Estimating Signals With Finite Rate of Innovation From Noisy Samples: A Stochastic Algorithm

As an example of the recently introduced concept of rate of innovation, signals that are linear combinations of a finite number of Diracs per unit time can be acquired by linear filtering followed by uniform sampling. However, in reality, samples are rarely noiseless. In this paper, we introduce a novel
stochastic
algorithm to reconstruct a signal with finite rate of innovation from its
noisy
samples. Even though variants of this problem have been approached previously, satisfactory solutions are only available for certain classes of sampling kernels, for example, kernels that satisfy the Strang-Fix condition. In this paper, we consider the infinite-support Gaussian kernel, which does not satisfy the Strang-Fix condition. Other classes of kernels can be employed. Our algorithm is based on Gibbs sampling, a Markov chain Monte Carlo method. Extensive numerical simulations demonstrate the accuracy and robustness of our algorithm.