The geometry of proper quaternion random variables

TL;DR

This paper introduces a new, more general definition of properness for quaternion random variables based on invariance under 4D rotations, extending previous concepts and providing geometric insights.

Contribution

It proposes the $(rac12,rac12)$-properness definition for quaternion variables, generalizing prior notions and analyzing its covariance symmetry properties.

Findings

The new properness concept is invariant under $SO(4)$ rotations.

Covariance matrices of $(rac12,rac12)$-proper quaternion variables exhibit specific symmetry properties.

Simulations visually demonstrate the geometric interpretation of the new properness definition.

Abstract

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.