Sparse spatial spectral estimation: A covariance fitting algorithm, performance and regularization

In this paper, the sparse spectrum fitting (SpSF) algorithm for the estimation of directions-of-arrival (DOAs) of multiple sources is introduced, and its asymptotic consistency and effective regularization under both asymptotic and finite sample cases are studied. Specifically, through the analysis of the optimality conditions of the method, we prove the asymptotic, in the number of snapshots, consistency of SpSF estimators of the DOAs and the received powers of uncorrelated sources in a sparse spatial spectra model. Along with this result, an explicit formula of the best regularization parameter of SpSF estimator with infinitely many snapshots is obtained. We then build on these results to investigate the problem of selecting an appropriate regularization parameter for SpSF with finite snapshots. An automatic selector of such regularization parameter is presented based on the formulation of an upper bound on the probability of correct support recovery of SpSF, which can be efficiently evaluated by Monte Carlo simulations. Simulation results illustrating the effectiveness and performance of this selector are provided, and the application of SpSF to direction-finding for correlated sources is discussed.