The geometry of proper quaternion random variables

Second order circularity, also called properness, for complex random variables is a well known and studied concept. In the case of quaternion random variables, some extensions have been proposed, leading to applications in quaternion signal processing (detection, filtering, estimation). Just like in the complex case, circularity for a quaternion-valued random variable is related to the symmetries of its probability density function. As a consequence, properness of quaternion random variables should be defined with respect to the most general isometries in $4D$, i.e. rotations from $SO(4)$. Based on this idea, we propose a new definition of properness, namely the $(\mu_1,\mu_2)$-properness, for quaternion random variables using invariance property under the action of the rotation group $SO(4)$. This new definition generalizes previously introduced properness concepts for quaternion random variables. A second order study is conducted and symmetry properties of the covariance matrix of $(\mu_1,\mu_2)$-proper quaternion random variables are presented. Comparisons with previous definitions are given and simulations illustrate in a geometric manner the newly introduced concept.Comment: 14 pages, 3 figure

Generalized Likelihood Ratios for Testing the Properness of Quaternion Gaussian Vectors

In a recent paper, the second-order statistical analysis of quaternion random vectors has shown that there exist two different kinds of quaternion widely linear processing, which are associated with the two main types of quaternion properness. In this paper, we consider the problem of determining, from a finite number of independent vector observations, whether a quaternion Gaussian vector is proper or not. Specifically, we derive three generalized likelihood ratio tests (GLRTs) for testing the two main kinds of quaternion properness and show that the GLRTs reduce to the estimation of three previously proposed quaternion improperness measures. Interestingly, the three GLRT statistics (improperness measures) can be interpreted as an estimate of the entropy loss due to the quaternion improperness. Additionally, we analyze the case in which the orthogonal basis for the representation of the quaternion vector is unknown, which results in the problem of estimating the principal C-properness direction, i.e., the pure unit quaternion minimizing the C-improperness measure. Although this estimation problem is not convex, we propose a technique based on successive convex approximations, which can be solved in closed form. Finally, some simulation examples illustrate the performance and practical application of the proposed tests