Separation Of Stochastic Processes

Assume we observe an unknown linear combination of p stochastic processes. The problem we are addressing here is to recover original source processes by applying an inverse transform, F, based solely upon their statistical independence. The conditions under which this problem is solvable are first pointed out. Then, by imposing the cancellation of output crosscumulants, we obtain a polynomial system of equations that the entries of the linear transform F must cancel. In the ring of polynomials with real or complex coefficients, obtaining a greatest common divisor is not easy; for instance Euclid's algorithm is unstable. Our approach is rather based on a detailed analysis of properties of input cumulants, and provides when p=2 a direct solution, which can be also implemented in an adaptive fashion. The extension to p>2 sources raises several specific problems; the solution we propose in this more general case becomes iterative.