On probability density functions for complex variables

• Published in: IEEE transactions on information theory Vol. 52; no. 3; pp. 1212 - 1217
• Main Author: Olhede, S.C.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-03-2006; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Applied sciences; Bivariate signals; circularity; Complex systems; complex valued signals; complex valued variables; Complex variables; Conjugates; Density; Density functional theory; Earthquakes; Exact sciences and technology; Fluid flow measurement; Formulations; Fourier transforms; Information theory; Information, signal and communications theory; Mathematical functions; Probability density function; Probability density functions; proper random variables; Random processes; Random variables; Scalars; Signal analysis; Signal processing; Telecommunications and information theory; Time series analysis; Variables; circularity; Second order; proper random variables; Complex signal; complex valued signals; complex valued variables; Complex variable method; Statistical method; Roundness; Probability density function; Bivariate signals; Random variable; Information theory
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2005.864451

Complex random variables arise naturally in many settings and their properties are of general interest. Past work on complex variables has mainly focused on their second-order structure, as well as that of their conjugates, whereas the main purpose of this correspondence is to clarify the concept of a density function for a complex random variable, and to discuss its properties. Two different functions play the role that the density of a real univariate random variable holds. Only one of these two functions can be correctly interpreted as a density, but both functions clarify the nature of a complex variable. The role played by the complex conjugate of the variable in this formulation is clarified, and the complex scalar nature of Z is discussed. As the properties of complex random variables are most naturally specified in terms of the complex quantities directly, and given in terms of the distribution of the complex variables rather than formulated in terms of the real and imaginary parts, ensuring that an interpretable complex formulation exists is important