This paper presents a method for blind source separation using sparse decomposition in a signal dictionary. The approach involves two stages: first, selecting a possibly overcomplete signal dictionary in which the sources are assumed to be sparsely representable, and second, unmixing the sources by exploiting their sparse representability. The method is applicable to cases where there are more sources than mixtures, and a more efficient algorithm is derived for the case of an equal number of sources and mixtures. Experiments with artificial signals and musical sounds demonstrate significantly better separation than other known techniques. The blind source separation problem involves extracting underlying source signals from a set of linear mixtures, where the mixing matrix is unknown. The method assumes that natural signals can be sparsely represented in a proper signal dictionary. The paper introduces a probabilistic framework for the problem, where the goal is to maximize the posterior probability of the mixing matrix and source signals given the observed data. This is achieved by minimizing a cost function that includes a term for the fidelity of the reconstruction and a term for the sparsity of the source representation. The paper also presents an experiment where sources with very sparse representations can be separated almost perfectly, even when they are correlated and the number of samples is small. The method is tested on synthetic data and shows high-quality separation. The paper also discusses the computational challenges of the approach, including the presence of multiple local minima and the need for careful initialization. It also presents a more robust formulation for the case of an equal number of sources and sensors, and a faster method for nonovercomplete dictionaries. The paper concludes that the use of sparse decomposition in a proper signal dictionary provides high-quality blind source separation. The maximum a posteriori framework gives the most general approach, which includes the situation of more sources than sensors. Computationally more robust solutions can be found in the case of an equal number of sources and sensors. The paper also discusses the use of quadratic programming with nonconvex quadratic constraints for sequential source extraction and the use of nonovercomplete dictionaries for faster solutions. The experiments with artificial signals and digitally mixed musical sounds demonstrate a high quality of source separation compared to other known techniques.This paper presents a method for blind source separation using sparse decomposition in a signal dictionary. The approach involves two stages: first, selecting a possibly overcomplete signal dictionary in which the sources are assumed to be sparsely representable, and second, unmixing the sources by exploiting their sparse representability. The method is applicable to cases where there are more sources than mixtures, and a more efficient algorithm is derived for the case of an equal number of sources and mixtures. Experiments with artificial signals and musical sounds demonstrate significantly better separation than other known techniques. The blind source separation problem involves extracting underlying source signals from a set of linear mixtures, where the mixing matrix is unknown. The method assumes that natural signals can be sparsely represented in a proper signal dictionary. The paper introduces a probabilistic framework for the problem, where the goal is to maximize the posterior probability of the mixing matrix and source signals given the observed data. This is achieved by minimizing a cost function that includes a term for the fidelity of the reconstruction and a term for the sparsity of the source representation. The paper also presents an experiment where sources with very sparse representations can be separated almost perfectly, even when they are correlated and the number of samples is small. The method is tested on synthetic data and shows high-quality separation. The paper also discusses the computational challenges of the approach, including the presence of multiple local minima and the need for careful initialization. It also presents a more robust formulation for the case of an equal number of sources and sensors, and a faster method for nonovercomplete dictionaries. The paper concludes that the use of sparse decomposition in a proper signal dictionary provides high-quality blind source separation. The maximum a posteriori framework gives the most general approach, which includes the situation of more sources than sensors. Computationally more robust solutions can be found in the case of an equal number of sources and sensors. The paper also discusses the use of quadratic programming with nonconvex quadratic constraints for sequential source extraction and the use of nonovercomplete dictionaries for faster solutions. The experiments with artificial signals and digitally mixed musical sounds demonstrate a high quality of source separation compared to other known techniques.