Improperness measures for quaternion random vectors

It has been recently proved that the two main kinds of quaternion improperness require two different kinds of widely linear processing. In this work, we show that these definitions satisfy some important properties, which include the invariance to quaternion linear transformations and right Clifford translations, as well as some clear connections with the case of proper complex vectors. Moreover, we introduce a new kind of quaternion properness, which clearly relates the two previous definitions, and propose three measures for the degree of improperness of a quaternion vector. The proposed measures are based on the Kullback-Leibler divergence between two zero-mean quaternion Gaussian distributions, one of them with the actual augmented covariance matrix, and the other with its closest proper version. These measures allow us to quantify the entropy loss due to the improperness of the quaternion vector, and they admit an intuitive geometrical interpretation based on Kullback-Leibler projections onto sets of proper augmented covariance matrices.