A theoretical bound for the performance advantage of quaternion widely linear estimation

The quaternion widely linear model was recently introduced for optimal second order estimation of noncircular 3D and 4D data. Its superiority over the standard strictly linear model was shown experimentally, however, a rigorous proof giving performance bounds has been lacking. To this end, we here present a mathematical proof for the degree of performance benefits obtained when using the widely linear model in the context of minimum mean square error estimation.