Two different real representations of complex matrices for describing widely linear systems

There is a well-known equivalent real representation of complex matrices which has proven useful in many signal processing applications. In this paper, we discuss why the class of such matrices with block-skew-circulant structure is not sufficient for the study of widely linear processing and improper signals, and why a less known alternative real representation with block-Hankel-skew-circulant structure is needed in addition. We compare the two different real representations of complex matrices and show situations in which they each are appropriate. Several new results for both types of matrices are derived and an application example is presented.