Canonical coordinates for reduced-rank estimation of improper complex random vectors

We consider the problem of minimum mean squared error (MMSE) estimation of complex random vectors in the improper case. Accounting for the information present in the complementary covariance requires the use of widely linear transformations. Based on these, we present the eigenanalysis of improper complex random vectors. This paves the way for a study of two different rank-reduced implementations of the complex Wiener Filter that make use of canonical coordinates: one that is optimum with respect to maximizing mutual information and one that minimizes mean squared error.