Exact Inversion of MIMO Nonlinear Polynomial Mixtures

This paper deals with the inversion of MIMO mixing systems, which are instantaneous and nonlinear but polynomial. An exact inverse is searched in the class of polynomial systems. It is shown that Groebner bases techniques offer an attractive solution for testing the existence of such an inverse and computing it. If such an inverse does not exist we propose to test the existence of a polynomial relationship between the sources and the observations and to compute simple polynomial functions, which each depend on one source only. Relying on the fact that for finite alphabet sources, polynomials span the whole set of nonlinear mappings, we tackle the general nonlinear case. We generalize the first results to give a condition for the existence of an exact nonlinear inverse. The proposed method allows to compute this inverse in polynomial form.