In many applications of Independent Component Analysis (ICA) and Blind Source Separation (BSS) estimated sources signals and the mixing or separating matrices have some special structure or some constraints are imposed for the matrices such as symmetries, orthogonality, non-negativity, sparseness and specified invariant norm of the separating matrix. In this paper we present several algorithms and overview some known transformations which allows us to preserve several important constraints.

In this paper we consider blind source separation (BSS) problem of signals which are spatially uncorrelated of order four, but temporally correlated of order four (for instance speech or biomedical signals). For such type of signals we propose a new sufficient condition for separation using fourth order statistics, stating that the separation is possible, if the source signals have distinct normalized cumulant functions (depending on time delay). Using this condition we show that the BSS problem can be converted to a symmetric eigenvalue problem of a generalized cumulant matrix Z(4)(b) depending on L-dimensional parameter b, if this matrix has distinct eigenvalues. We prove that the set of parameters b which produce Z(4)(b) with distinct eigenvalues form an open subset of RL, whose complement has a measure zero. We propose a new separating algorithm which uses Jacobi's method for joint diagonalization of cumulant matrices depending on time delay. We empasize the following two features of this algorithm: 1) The optimal number of matrices for joint diago- nalization is 100-150 (established experimentally), which for large dimensional problems is much smaller than those of JADE; 2) It works well even if the signals from the above class are, additionally, white (of order two) with zero kurtosis (as shown by an example).