Second-order analysis of improper complex random vectors and processes

We present a comprehensive treatment of the second-order theory of complex random vectors and wide-sense stationary (WSS) signals. The main focus is on the improper case, in which the complementary covariance does not vanish. Accounting for the information present in the complementary covariance requires the use of widely linear transformations. Based on these, we present the eigenanalysis of complex vectors and apply it to the problem of rank reduction through principal components. We also investigate joint properties of two complex vectors by introducing canonical correlations, which paves the way for a discussion of the Wiener filter and its rank-reduced version. We link the concepts of propriety and joint propriety to eigenanalysis and canonical correlation analysis, respectively. Our treatment is extended to WSS signals. In particular, we give a result on the asymptotic distribution of eigenvalues and examine the connection between WSS, proper, and analytic signals.