Full Mean Square Performance Bounds on Quaternion Estimators for Improper Data

Abstract

Novel physical insights are provided into the mean square performance bounds of quaternion-valued widely linear (WL), semi-widely linear (SWL), and strictly linear estimators for the generality of quaternion-valued Gaussian data. This is achieved by first defining three kinds of complementary mean square errors (CMSEs) of these estimators and by further exploiting the corresponding degrees of $\mathbb {H}$ -improperness (second-order noncircularity). Next, the investigation of the bounds of the attainable CMSEs by these classes of estimators shows that only a joint consideration of the proposed CMSE analysis and the standard MSE analysis provides enough degrees of freedom for a detailed account of the MSE performance. The so-established framework for the analysis of estimators for $\mathbb {H}$ -improper data is shown to be capable of measuring error power distribution for each data channel, an important finding, which is not possible to obtain through the standard MSE analysis only. Simulations in the system identification setting support the analysis.