Blind Deconvolution and Ica with a Banded Mixing Matrix

Convolution is a linear operation, and, consequently, can be formulated as a linear system of equations. If only the output of the system (the convolved signal) is known, then the problem is blind so that given one equation, two unknowns are sought. Here, the blind deconvolution problem is solved using independent component analysis (ICA). To facilitate this, several time lagged versions of the convolved signal are extracted and used to construct realizations of a random vector. For ICA, this random vector is the, so called, mixture vector, created by the matrix-vector multiplication of the two unknowns, the mixing matrix and the source vector. Due to the properties of convolution, the mixing matrix is banded with its nonzero elements containing the convolution’s filter. This banded property is incorporated into the ICA algorithm as prior information, giving rise to a banded ICA algorithm (B-ICA) which is, in turn, used in a new blind deconvolution method. B-ICA produces as many independent components as the dimension of the filter; whereas for blind deconvolution, only one signal is sought (the deconvolved signal). Fortunately, the convolutional model provides additional information which enables one best independent component to be extracted from the pool of candidate solutions. This, in turn, yields estimates of both the filter and the deconvolved signal.