A Sparse Signal Reconstruction Perspective for Source Localization with Sensor Arrays
Dmitry M. Malioutov, Müjdat Çetin, and Alan S. Willsky
We present a source localization method based upon a sparse representation of sensor measurements with an overcomplete basis composed of samples from the array manifold. We enforce sparsity by imposing penalties based on the l1-norm. A number of recent theoretical results on sparsifying properties of l1 penalties justify this choice. Explicitly enforcing the sparsity of the representation is motivated by a desire to obtain a sharp estimate of the spatial spectrum which exhibits superresolution. We propose to use the singular value decomposition (SVD) of the data matrix to summarize multiple time or frequency samples. Our formulation leads to an optimization problem, which we solve efficiently in a second-order cone (SOC) programming framework by an interior point implementation. We propose a grid refinement method to mitigate the effects of limiting estimates to a grid of spatial locations, and also introduce an automatic selection criterion for the regularization parameter involved in our approach. We demonstrate the effectiveness of the method on simulated data by plots of spatial spectra and by comparing the estimator variance to the Cramer-Rao bound (CRB). We observe that our approach has a number of advantages over other source localization techniques including increased resolution; improved robustness to noise, limitations in data quantity, and correlation of the sources; as well as not requiring an accurate initialization.