A Bernoulli-Gaussian Approach to the Reconstruction of Noisy Signals with Finite Rate of Innovation

Recently, it was shown that a large class of non-bandlimited signals that have a finite rate of innovation, such as streams of Diracs, non-uniform splines and piecewise polynomials, can be perfectly reconstructed from a uniform set of samples taken at the rate of innovation [1]. While this is true in the noiseless case, in the presence of noise the finite rate of innovation property is lost and exact reconstruction is no longer possible. In this paper we consider the problem of reconstructing such signals when noise is present. We focus on the case when a discrete-time signal is made up of a sum of weighted Diracs and we propose a stochastic reconstruction method based on the Bernoulli-Gauss model and on a maximum a posteriori optimization. Our approach is numerically stable and yields precise reconstruction by sampling the signal way below the Nyquist rate, significantly outperforming commonly used subspace methods [2, 3]. Applications of our method can be found in acquisition and processing of signals in wideband communication systems, such as ultra-wideband (UWB) systems.