Compressed sensing framework for blind source separation have been considered in recent years because of the inherent sparsity in the assumptions used for blind source separation. Consider a noise free instantaneous mixture signals described below:
where there are sources and observations. Under the assumption that only one source is active, which implies the signal space is sparse, for each time, we define the following vectors and matrix.define the vector as:
where denotes the transpose. Also denote the vector as
Finally, denote the matrix as
The problem can then be posed as solving the equation
With being sparse by assumption, this is the classical compressed sensing signal recovery problem using
The signal space can be further given a sparser representation by using a dictionary D such that with the minimization done on .
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