A striking feature of many sensory processing problems is that there appear to be many more neurons engaged in the internal representations of the signal than in its transduction. For example, humans have similar to 30,000 cochlear neurons, but at least 1000 times as many neurons in the auditory cortex. Such apparently redundant internal representations have sometimes been proposed as necessary to overcome neuronal noise. We instead posit that they directly subserve computations of interest. Here we provide an example of how sparse overcomplete linear representations can directly solve difficult acoustic signal processing problems, using as an example monaural source separation using solely the cues provided by the differential filtering imposed on a source by its path from its origin to the cochlea [the head-related transfer function (HRTF)]. In contrast to much previous work, the HRTF is used here to separate auditory streams rather than to localize them in space. The experimentally testable predictions that arise from this model, including a novel method for estimating the optimal stimulus of a neuron using data from a multineuron recording experiment, are generic and apply to a wide range of sensory computations.