Widely and semi-widely linear processing of quaternion vectors

In this paper the two main definitions of quaternion properness (or second order circularity) are reviewed, showing their connection with the structure of the optimal quaternion linear processing. Specifically, we present a rigorous generalization of the most common multivariate statistical analysis techniques to the case of quaternion vectors, and show that the different kinds of quaternion improperness require different kinds of widely linear processing. In general, the optimal linear processing is \emph{full-widely linear}, which requires the joint processing of the quaternion vector and its involutions over three pure unit quaternions. However, in the case of jointly $\mathbbQ$-proper and $\mathbbC^{\eta}$-proper vectors, the optimal processing reduces, respectively, to the \emphconventional and \emph{semi-widely linear processing}, with the latter only requiring to operate on the quaternion vector and its involution over the pure unit quaternion $\eta$. Finally, a simulation example poses some interesting questions for future research.