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We examine the recovery of block sparse signals and extend the recovery framework in two important directions; one by exploiting the signals' intra-block correlation and the other by generalizing the signals' block structure. We propose two families of algorithms based on the framework of block sparse Bayesian learning (BSBL). One family, directly derived from the BSBL framework, require knowledge of the block structure. Another family, derived from an expanded BSBL framework, are based on a weaker assumption on the block structure, and can be used when the block structure is completely unknown. Using these algorithms, we show that exploiting intra-block correlation is very helpful in improving recovery performance. These algorithms also shed light on how to modify existing algorithms or design new ones to exploit such correlation and improve performance.