A sparse Bayesian learning algorithm with dictionary parameter estimation

This paper concerns sparse decomposition of a noisy signal into atoms which are specified by unknown continuous-valued parameters. An example could be estimation of the model order, frequencies and amplitudes of a superposition of complex sinusoids. The common approach is to reduce the continuous parameter space to a fixed grid of points, thus restricting the solution space. In this work, we avoid discretization by working directly with the signal model containing parameterized atoms. Inspired by the “fast inference scheme” by Tipping and Faul we develop a novel sparse Bayesian learning (SBL) algorithm, which estimates the atom parameters along with the model order and weighting coefficients. Numerical experiments for spectral estimation with closely-spaced frequency components, show that the proposed SBL algorithm outperforms state-of-the-art subspace and compressed sensing methods.