Performance advantage of quaternion widely linear estimation: An approximate uncorrelating transform approach

Widely linear processing has been shown to be superior to the traditional strictly linear processing in quaternion minimum mean square error (MMSE) estimation. However, a quantifiable performance difference between strictly and widely linear processing and the relationship between the performance and quaternion impropriety are still lacking. To this end, we present a proof for the performance advantage of widely linear estimation and relate the performance bounds to signal properties by exploiting the approximate joint diagonalisation of quaternion covariance matrices. In that sense, this work can be seen as a generalisation of complex-valued MMSE estimation, and can thus also be applied to the complex-valued case. Simulations on synthetic signals support the analysis.