High resolution sparse estimation of exponentially decaying signals

We consider the problem of sparse modeling of a signal consisting of an unknown number of exponentially decaying sinusoids. Since such signals are not sparse in an oversampled Fourier matrix, earlier approaches typically exploit large dictionary matrices that include not only a finely spaced frequency grid but also a grid over the considered damping factors. The resulting dictionary is often very large, resulting in a computationally cumbersome optimization problem. Here, we instead introduce a novel dictionary learning approach that iteratively refines the estimate of the candidate damping factor for each sinusoid, thus allowing for both a quite small dictionary and for arbitrary damping factors, not being restricted to a grid. The performance of the proposed method is illustrated using simulated data, clearly showing the improved performance as compared to previous techniques.