Augmented second-order statistics of quaternion random signals

Second order statistics of quaternion random variables and signals are revisited in order to exploit the complete second order statistical information available. The conditions for Q‐properQ‐proper (second order circular) random processes are presented, and to cater for the non-vanishing pseudocovariance of such processes, the use of ı‐j‐κ‐covariancesı‐j‐κ‐covariances is investigated. Next, the augmented statistics and the corresponding widely linear model are introduced, and a generic multivariate Gaussian distribution is subsequently derived for both Q‐properQ‐proper and Q‐improperQ‐improper processes. The maximum entropy bound and an extension of mutual information to multivariate processes are derived in order to provide a complete description of joint information theoretic properties of general quaternion valued processes. A comparative analysis with the corresponding second order statistics of quadrivariate real valued processes supports the approach.