Essential Statistics and Tools for Complex Random Variables

Complex random signals play an increasingly important role in array, communications, and biomedical signal processing and related fields. However, the fundamental properties of complex-valued signals and mathematical tools needed to process them are scattered in literature. We provide a concise, unified, and rigorous treatment of essential properties and tools of complex random variables, and apply these fundamentals to derive complex extensions of Leibniz rule, Faa di Bruno's formula, and Taylor's series. The extensions allow establishing relationships among complex moments and cumulants, and characterizing the circularity property. We propose measures for testing and quantifying circularity, and observe that non-circularity may be more common in practical applications than previously thought. All results are rigorously proved and supplemented with clarifying examples.