Measuring impropriety in complex and real representations

So-called improper signals, i.e., signals which are correlated with their complex conjugates, can occur in many signal processing applications such as communication systems, medical imaging, audio and speech processing, analysis of oceanographic data, and many more. Being aware of potential impropriety can be crucial whenever we model signals as complex random quantities since an appropriate treatment of improper signals, e.g., by widely linear filtering, can significantly improve the system performance. After a brief introduction into the fundamentals of improper signals, this article focuses on the problem of quantifying the impropriety of complex random vectors and gives a survey of various impropriety measures in both the composite real representation and the augmented complex formulation. Unlike in previous publications, these two frameworks are presented side by side to reveal the differences and common points between them. Moreover, their applicability is compared in several practical examples. As additional aspects, we consider the problem of testing for impropriety based on measurement data, and the differential entropy of Gaussian vectors as an impropriety measure in information theoretic studies. The article includes a tutorial-style introduction, a collection of important formulae, a comparison of various mathematical approaches, as well as some new reformulations.