Second-order description of complex random vectors

Abstract

In this chapter, we discuss in detail the second-order description of a complex random vector x . We have seen in Chapter 2 that the second-order averages of x are completely described by the augmented covariance matrix R xx . We shall now be interested in those second-order properties of x that are invariant under two types of transformations: widely unitary and nonsingular strictly linear. The eigenvalues of the augmented covariance matrix R xx constitute a maximal invariant for R xx under widely unitary transformation. Hence, any function of R xx that is invariant under widely unitary transformation must be a function of these eigenvalues only. In Section 3.1, we consider the augmented eigenvalue decomposition (EVD) of R xx for a complex random vector x . Since we are working with an augmented matrix algebra, this EVD looks somewhat different from what one might expect. In fact, because all factors in the EVD must be augmented matrices, widely unitary diagonalization of R xx is generally not possible. As an application for the augmented EVD, we discuss rank reduction and transform coding. In Section 3.2, we introduce the canonical correlations between x and x * , which have been called the circularity coefficients . These constitute a maximal invariant for R xx under nonsingular strictly linear transformation. They are interesting and useful for a number of reasons. They determine the loss in entropy that an improper Gaussian random vector incurs compared with its proper version (see Section 3.2.1). […]