In this letter, the problem of blind separation of n independent sources from their m linear instantaneous mixtures is considered. First, a generalized contrast function is defined as a valuable extension of the existing classical and nonsymmetrical contrast functions. It is applicable to the overdetermined blind separation ( m > n ) with an unknown number of sources, because not only independent components but also redundant ones are allowed in the outputs of a separation system. Second, a natural gradient learning algorithm developed primarily for the complete case ( m = n ) is shown to work as well with an n × m or m × m separating matrix, for each optimizes a certain mutual information contrast function. Finally, we present stability analysis for a newly proposed generalized orthogonal natural gradient algorithm (which can perform the overdetermined blind separation when n is unknown), obtaining an expectable result that its local stability conditions are slightly stricter than those of the conventional natural gradient algorithm using an invertible mixing matrix ( m = n ).
The blind source separation (BSS) problem with an unknown number of sources is an important practical issue that is usually skipped by assuming that the source number n is known and equal to the number m of sensors. This letter studies the general BSS problem satisfying m ≥ n . First, it is shown that the mutual information of outputs of the separation network is a cost function for BSS, provided that the mixing matrix is of full column rank and the m × m separating matrix is nonsingular. The mutual information reaches its local minima at the separation points, where the m outputs consist of n desired source signals and m − n redundant signals. Second, it is proved that the natural gradient algorithm proposed primarily for complete BSS ( m n ) can be generalized to deal with the overdetermined BSS problem ( m > n ), but it would diverge inevitably due to lack of a stationary point. To overcome this shortcoming, we present a modified algorithm, which can perform BSS steadily and provide the desired source signals at specified channels if some matrix is designed properly. Finally, the validity of the proposed algorithm is confirmed by computer simulations on artificially synthesized data.