On the asymptotic distribution of GLR for impropriety of complex signals

In this paper, the problem of testing impropriety (i.e., second-order noncircularity) of a sequence of complex-valued random variables (RVs) based on the generalized likelihood ratio test (GLRT) for Gaussian distributions is considered. Asymptotic (w.r.t. the data length) distributions of the GLR are given under the hypothesis that RVs are proper or improper, and under the true, not necessarily Gaussian distribution of the RVs. The considered RVs are independent but not necessarily identically distributed: assumption which has never been considered until now. This enables us to deal with the practical important situations of noncircular RVs disturbed by residual frequency offsets and additive circular noise. The receiver operating characteristic (ROC) of this test is derived as byproduct, an issue previously overlooked. Finally illustrative examples are presented in order to strengthen the obtained theoretical results.

► Some new enlightening results about the asymptotic distribution of the GLR for impropriety of complex signals have been investigated.
► The associated GLRT derived under the usual assumption of independent identically distributed Gaussian RVs is studied under non necessarily identical Gaussian distributions of the RVs.
► The asymptotic distribution of the circularity coefficient has been given under H_0 and H_1 for independent identical or independent non identical arbitrary distributions of the RVs.
► This allow us to deal with the important practical situations where discrete RVs are disturbed by residual frequency offsets and additive Gaussian circular noise.