Sparse Component Analysis for Blind Source Separation with Less Sensors than Sources

A sparse decomposition approach of observed data matrix is presented in this paper and the approach is then used in blind source separation with less sensors than sources. First, sparse representation (factorization) of a data matrix is discussed. For a given basis matrix, there exist infinite coefficient matrices (solutions) generally such that the data matrix can be represented by the product of the basis matrix and coefficient matrices. However, the sparse solution with minimum 1-norm is unique with probability one, and can be obtained by using linear programming algorithm. The basis matrix can be estimated using gradient type algorithm or Kmeans clustering algorithm. Next, blind source separation is discussed based on sparse factorization approach. The blind separation technique includes two steps, one is to estimate a mixing matrix (basis matrix in the sparse representation), the second is to estimate sources (coefficient matrix). If the sources are sufficiently sparse, blind separation can be carried out directly in the time domain. Otherwise, blind separation can be implemented in time-frequency domain after applying wavelet packet transformation preprocessing to the observed mixtures. Three simulation examples are presented to illustrate the proposed algorithms and reveal algorithms performance. Finally, concluding remarks review the developed approach and state the open problems for further studying.