Blind separation of convolved mixtures in the frequency domain

In this paper we employ information theoretic algorithms, previously used for separating instantaneous mixtures of sources, for separating convolved mixtures in the frequency domain. It is observed that convolved mixing in the time domain corresponds to instantaneous mixing in the frequency domain. Such mixing can be inverted using simpler and more robust algorithms than the ones recently developed. Advantages of this approach are improved efficiency and better convergence features. The problem of blind source separation was traditionally approached by observing instantaneous mixtures of sources. Assume that N signals s i are ordered in a vector where t is a time index. Upon transmission through a medium these signals are collected from N sensors from which we obtain. Assuming linear superposi-tion the vector can be expressed as: (1) where is an unknown matrix called the mixing matrix. The objective is to recover the original signals given only the vectors. These signals can be recovered using. If is invertible then separation is feasible and the inverse of this mixing matrix will be called the unmixing matrix. and Sejnowski (1995) and Amari et al. (1996) who introduced fast and robust solutions. Both approaches employed information theoretic principles to find an unmixing matrix that would maximize the statistical independence of the estimated original sources. The computational structure that was used was a matrix multiplication between the estimated unmixing matrix and the mixed inputs. After every sample presentation the estimate of the unmixing matrix was updated using the following learning rules, where : (2) which was derived by Bell and Sejnowski (1995), and: (3) derived by Amari et al. (1996) who also performed adaptation accounting for the Riemannian structure of the problem. The matrix is our estimate of the inverse of the mixing matrix and the function f(⋅) is a non-linear sigmoid function. It has been shown that for f(⋅) = tanh(⋅) the algorithm performs very well for zero mean super-Gaussian input data. The unmixing matrix that is obtained this way will recover the original sources, but arbitrarily scaled. In addition the rows of the unmixing matrix might have a different ordering than the true inverse of the mixing matrix, so that: (4) Where is a scaling and permutation matrix. s T t () s 1 t () … s N t () [ ] = x T t () x 1 t () … x N t () [ ] = x x t …