Exact Distribution and High-dimensional Asymptotics for Improperness Test of Complex Signals

Improperness testing for complex-valued vectors and processes has been of interest lately due to the potential applications of complex-valued time series analysis in several research areas. This paper provides exact distribution characterization of the GLRT (Generalized Likelihood Ratio Test) statistics for Gaussian complex-valued signals under the null hypothesis of properness. This distribution is a special case of the Wilks’s lambda distribution, as are the distributions of the GLRT statistics in multivariate analysis of variance (MANOVA) procedures. In the high dimensional setting, i.e. when the size of the vectors grows at the same rate as the number of samples, a closed form expression is obtained for the asymptotic distribution of the GLRT statistics. This is, to our knowledge, the first exact characterization for the GLRT-based improperness testing.