Sparse signal recovery algorithms have significant impact on many fields. The core of these algorithms is to find a solution to an underdetermined inverse system of equations, where the solution is expected to be sparse or approximately sparse. However, most algorithms ignored correlation among nonzero entries of a solution, which is often encountered in a practical problem. Thus, it is unclear what role the correlation plays in signal recovery.;This work aims to design algorithms which can exploit a variety of correlation structures in solutions and reveal the impact of these correlation structures on algorithms' recovery performance.;First, a block sparse Bayesian learning (BSBL) framework is proposed. Based on it, a number of sparse Bayesian learning (SBL) algorithms are derived to exploit intra-block correlation in a block sparse model, temporal correlation in a multiple measurement vector model, spatiotemporal correlation in a spatiotemporal sparse model, and local temporal correlation in a time-varying sparse model. Several optimization approaches are employed in the algorithm development, such as the expectation-maximization method, the bound-optimization method, and a fixed-point method. Experimental results show that these algorithms have superior performance.;With these algorithms, we find that different correlation structures affect the quality of estimated solutions to different degrees. However, if these correlation structures are present and exploited, algorithms' performance can be largely improved. Inspired by this, we connect these algorithms to Group-Lasso type algorithms and iterative reweighted ℓ1 and ℓ 2 algorithms, and suggest strategies to modify them to exploit the correlation structures for better performance.;The derived algorithms have been used with considerable success in various challenging applications such as wireless telemonitoring of raw physiological signals and prediction of patients' cognitive levels from their neuroimaging measures. In the former application, where raw physiological signals are neither sparse in the time domain nor sparse enough in transformed domains, the derived algorithms are the only algorithms so far that achieved satisfactory results. In the latter application, the derived algorithms achieved the highest prediction accuracy on common datasets, compared to published results around 2011.
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