The degree of impropriety (noncircularity) of complex random vectors

A complex random vector is called improper (noncircular) if it is correlated with its complex conjugate. We consider measures for the degree of impropriety that are invariant under linear transformation. These measures are functions of the canonical correlations between the vector and its complex conjugate, which have been termed the circularity coefficients. However, we show that these circularity coefficients do not tell the whole story: Two random vectors with identical covariance matrix and identical circularity coefficients can still behave differently in second-order estimation and detection.