On the blind separation of non-circular sources

In this paper we address the blind separation of an instantaneous complex mixture of statistically independent non-circular signals. We show that, if the sources are non-circular at order 2, by exploiting all the second-order information, the mixing matrix can be estimated up to a real orthogonal factor. This is based on a link with the Takagi factorization, for the computation of which we derive a Jacobi-type algorithm. We prove that, if the sources are non-circular at order 4, after a classical prewhitening, the remaining unitary factor can be found via a simultaneous Takagi factorization / Hermitian Eigenvalue Decomposition (EVD). We also describe a variant in which no hard prewhitening is carried out. In addition, we pay some attention to the issue of dimensionality reduction, in the case where there are fewer sources than sensors.