While Blind Source Separation can use sparsity in order to decompose signals in high dimensions, one wonders how this relates to Compressed Sensing /Compressive Sampling. Compressed Sensing is enabled by Sparsity and one then wonders how sampling could take advantage of that and provide the ability to solve the blind source separation problem.
A first step in that direction is taken by Thomas Blumensath and Mike Davies in Compressed Sensing and Source Separation
Separation of underdetermined mixtures is an important problem in signal processing that has attracted a great deal of attention over the years. Prior knowledge is required to solve such problems and one of the most common forms of structure exploited is sparsity. Another central problem in signal processing is sampling. Recently, it has been shown that it is possible to sample well below the Nyquist limit whenever the signal has additional structure. This theory is known as compressed sensing or compressive sampling and a wealth of theoretical insight has been gained for signals that permit a sparse representation. In this paper we point out several similarities between compressed sensing and source separation. We here mainly assume that the mixing system is known, i.e. we do not study blind source separation. With a particular view towards source separation, we extend some of the results in compressed sensing to more general overcomplete sparse representations and study the sensitivity of the solution to errors in the mixing system.
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