The paper presents a two-stage blind source separation method that leverages sparse decomposition in a signal dictionary. The first stage involves selecting an overcomplete signal dictionary where the sources are assumed to be sparsely representable. The second stage uses this dictionary to unmix the sources by exploiting their sparse representations. The authors consider both the general case of more sources than mixtures and a more efficient algorithm for the case of a non-overcomplete dictionary with an equal number of sources and mixtures. Experiments with artificial signals and musical sounds demonstrate superior separation compared to other known techniques. The paper also discusses the probabilistic framework for the maximum a posteriori approach and provides experimental results to validate the effectiveness of the proposed methods.The paper presents a two-stage blind source separation method that leverages sparse decomposition in a signal dictionary. The first stage involves selecting an overcomplete signal dictionary where the sources are assumed to be sparsely representable. The second stage uses this dictionary to unmix the sources by exploiting their sparse representations. The authors consider both the general case of more sources than mixtures and a more efficient algorithm for the case of a non-overcomplete dictionary with an equal number of sources and mixtures. Experiments with artificial signals and musical sounds demonstrate superior separation compared to other known techniques. The paper also discusses the probabilistic framework for the maximum a posteriori approach and provides experimental results to validate the effectiveness of the proposed methods.