In this thesis, we focus on the application of sparsity-encouraging regularization techniques to the problem of Direction-Of-Arrival (DOA) estimation using sensor arrays. By developing the sparse representation models for the spatial covariance matrix of correlated or uncorrelated sources respectively, the DOA estimation problem is reformulated under the framework of Sparse Signal Reconstruction (SSR). The L1-Norm regularization technique is employed to formulate sparsity-exploiting DOA estimation algorithms, which can estimate DOAs and signal powers simultaneously. The algorithm specialized for uncorrelated sources, Sparse Spectrum Fitting (SpSF), is attractive for its computational complexity, resolution capability, low large error threshold and robustness with respect to source correlation (when combined with spatial smoothing and forward-backward smoothing). Despite the similarity between the formulation of SpSF and ordinary SSR algorithms, such as Lasso , SpSF can perform much better than predicted by the existing theoretical results in SSR literature, because of the extra positivity
constraints in its formulation (which are included since the optimization variables of SpSF represent the signal powers). Thus, we begin this thesis by developing and justifying its formulations. This is followed by a discussion of the existence and uniqueness of the solutions of SpSF, which provides an explicit formula for the maximum number of sources whose DOAs can be reliably estimated by SpSF. Although it is hard to directly relate this maximum number of sources to the number of sensors M, we recognize that this maximum number is actually the degree of freedom of the co-array and, as an exception, we prove that by using Uniform Linear Arrays (ULA), SpSF can work with M-by-1 sources. Further, by combining with the uniqueness result, the estimation consistency of SpSF with respect to infinitely many snapshots and sensors is obtained.
Another focus of this thesis is the selection of the regularization parameter (RP) of SpSF. Inspired by the theoretical analysis in the first part of this thesis, Diagonal Loading technique is used to significantly reduce the sensitivity of SpSF with respect to its RP, which further enables the control of such sensitivity. Based on the Discrepancy Principle, an iterative version of SpSF, Iterative-SpSF (I-SpSF), is formulated as a parameter-free optimization algorithm and shown to achieve similar or even better performance than SpSF using exhaustively searched but fixed RPs. Following that, by analyzing SpSF's probability of perfect support recovery, an upper bound on such probability is proposed. This upper bound can accurately approximate the probability of perfect DOA estimates of SpSF (perfect DOA estimates is only possible in the context of DOAs belonging to the grid of candidate directions). Based on this upper bound and a Monte Carlo evaluation process, an automatic and data-adaptive RP selector is proposed for the purpose of DOA estimation when the number of snapshots is finite. This technique is robust with respect to its parameters and can help SpSF to achieve almost the same or even better performance than exhaustively searched and fixed RPs for uncorrelated and correlated sources respectively.
The extensions of SpSF to the cases of o -grid DOAs and moving sources are considered and two algorithms SpSF with Modeling Uncertainty (SpSFMU) and Persistently Active Block Sparsity (PABS) are proposed for the two cases, respectively. By using extra variables to parameterize the modeling errors caused by o -grid DOAs, SpSFMU is formulated as a convex optimization problem that can provide continuous DOA estimates. Under the presence of o -grid DOAs, SpSFMU can be used either to improve the estimation performance of SpSF or to reduce its computational complexity. In contrast, PABS is introduced as a general estimator for inconsistent but persistently active sparse models. PABS is formulated by using a novel objective function promoting Block-Level Sparsity and Element-Level Sparsity simultaneously. Then, on the basis of the sparse representation model of array snapshots, PABS is applied to the DOA estimation of moving sources and shown to exhibit significant performance improvement over C-HiLasso and SPICE. Intensive simulation results are included in each chapter to illustrate the effectiveness
and performance of the presented algorithms and theoretical analyses.