Sparse representations & compressed sensing with application to the problem of direction-of-arrival estimation

The significance of sparse representations has been highlighted in numerous signal processing applications ranging from denoising to source separation and the emerging field of compressed sensing has provided new theoretical insights into the problem of inverse systems with sparsity constraints. In this thesis, these advances are exploited in order to tackle the problem of directionof-arrival (DOA) estimation in sensor arrays. Assuming spatial sparsity e.g. few sources impinging on the array, the problem of DOA estimation is formulated as a sparse representation problem in an overcomplete basis. The resulting inverse problem can be solved using typical sparse recovery methods based on convex optimization i.e. `1 minimization. However, in this work a suite of novel sparse recovery algorithms is initially developed, which reduce the computational cost and yield approximate solutions. Moreover, the proposed algorithms of Polytope Faces Pursuits (PFP) allow for the induction of structured sparsity models on the signal of interest, which can be quite beneficial when dealing with multi-channel data acquired by sensor arrays, as it further reduces the complexity and provides performance gain under certain conditions. Regarding the DOA estimation problem, experimental results demonstrate that the proposed methods outperform popular subspace based methods such as the multiple signal classification (MUSIC) algorithm in the case of rank-deficient data (e.g. presence of highly correlated sources or limited amount of data) for both narrowband and wideband sources. In the wideband scenario, they can also suppress the undesirable effects of spatial aliasing. However, DOA estimation with sparsity constraints has its limitations. The compressed sensing requirement of incoherent dictionaries for robust recovery sets limits to the resolution capabilities of the proposed method. On the other hand, the unknown parameters are continuous and therefore if the true DOAs do not belong to the predefined discrete set of potential locations the algorithms’ performance will degrade due to errors caused by mismatches. To overcome this limitation, an iterative alternating descent algorithm for the problem of off-grid DOA estimation is proposed that alternates between sparse recovery and dictionary update estimates. Simulations clearly illustrate the performance gain of the algorithm over the conventional sparsity approach and other existing off-grid DOA estimation algorithms.