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Sparse constraints on signal decompositions are justified by typical sensor data used in a variety of signal processing fields such as acoustics, medical imaging, or wireless, but moreover can lead to more effective algorithms. The specific sparseness assumption used in this work is that the maximum number of statistically independent sources active at any time and frequency point in a mixture of signals is small. This is shown to result from an assumption of sparseness of the sources themselves, and allows us to solve the maximum likelihood formulation of the noninstantaneous acoustic mixing source estimation problem. We consider an additive noise mixing model with an arbitrary number of sensors and possibly more sources than sensors, when sources satisfy the sparseness assumption above. The solution obtained is applicable to an arbitrary number of microphones and sources, but works best when the number of sources simultaneously active at any time frequency point is a small fraction of the total number of sources.