Component-Wise Conditionally Unbiased Widely Linear MMSE Estimation

Biased estimators can outperform unbiased ones in terms of the mean square error (MSE). The best linear unbiased estimator (BLUE) fulfills the so called global conditional unbiased constraint when treated in the Bayesian framework. Recently, the component-wise conditionally unbiased linear minimum mean square error (CWCU LMMSE) estimator has been introduced. This estimator preserves a quite strong (namely the CWCU) unbiased condition which in effect sufficiently represents the intuitive view of unbiasedness. Generally, it is global conditionally biased and outperforms the BLUE in a Bayesian MSE sense. In this work we briefly recapitulate CWCU LMMSE estimation under linear model assumptions, and additionally derive the CWCU LMMSE estimator under the (only) assumption of jointly Gaussian parameters and measurements. The main intent of this work, however, is the extension of the theory of CWCU estimation to CWCU widely linear estimators. We derive the CWCU WLMMSE estimator for different model assumptions and address the analytical relationships between CWCU WLMMSE and WLMMSE estimators. The properties of the CWCU WLMMSE estimator are deduced analytically, and compared by simulation to global conditionally unbiased as well as WLMMSE counterparts with the help of a parameter estimation example and a data estimation/channel equalization application.