An Impropriety Test Based on Block-Skew-Circulant Matrices

Since improper (noncircular) complex signals require adequate tools such as widely linear filtering, a generalized likelihood ratio test has been proposed in the literature to verify whether or not a given signal is improper. This test is based on the augmented complex formulation, which is sometimes regarded as the most convenient way of handling improper signals. In this paper, we show that a derivation of an impropriety test is also possible without making use of the augmented complex representation. Instead, we us a composite real formulation, and we apply the recently proposed framework of block-skewcirculant matrices. It turns out that this alternative derivation is of practical relevance since it reveals a computationally more efficient implementation of the impropriety test by avoiding redundant matrix structures. The paper is concluded by a discussion of redundancy in the description of second order statistical properties of complex random vectors.