Conditions on source signals for blind separation

In the blind source separation problem it is usually assumed that the source signals are mutually independent. It has been realized that this assumption is not necessary. In this paper, we provide some general conditions under which the source signals can be recovered from their linear mixtures. We first define two concepts, the mutually M-th order independence and the pairwise M-th order independence, on a set of random variables. Then we prove that if the source signals are mutually M-th order independent, then those source signals, each of which has at least one nonzero m-th order cumulant for 3⩽m⩽M, can be separated. This conclusion is in spirit similar to, but more general than, that proposed by Tong et al. (1993). The constructive proof suggests an algorithm for the blind separation of sources which are mutually M-th order independent. Simulation examples are presented to illustrate the results