On the Circularity of a Complex Random Variable

Abstract

An important characteristic of a complex random variable z is the so-called circularity property or lack of it. We study the properties of the degree of circularity based on second-order moments, called circularity quotient, that is shown to possess an intuitive geometrical interpretation: the modulus and phase of its principal square-root are equal to the eccentricity and angle of orientation of the ellipse defined by the covariance matrix of the real and imaginary part of z. Hence, when the eccentricity approaches the minimum zero (ellipse is a circle), the circularity quotient vanishes; when the eccentricity approaches the maximum one, the circularity quotient lies on the unit complex circle. Connection with the correlation coefficient rho is established and bounds on rho given the circularity quotient (and vice versa) are derived. A generalized likelihood ratio test (GLRT) of circularity assuming complex normal sample is shown to be a function of the modulus of the circularity quotient with asymptotic chi <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> distribution.