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On the Log-Likelihood Ratio Evaluation of CWCU Linear and Widely Linear MMSE Data Estimators

In soft decoding of data bits, the log-likelihood ratios are evaluated from the estimated data symbols. For proper constellation diagrams such as QPSK or 16-QAM, these data symbols are often estimated using the linear minimum mean square error (LMMSE) estimator. The LMMSE estimator only fulfills the weak Bayesian unbiasedness constraint. Recently, estimators fulfilling the more stringent component-wise conditionally unbiased (CWCU) constraints have been investigated, such as the CWCU LMMSE estimator. In this paper, we prove that the CWCU LMMSE estimates result in the very same log-likelihood ratios as the LMMSE estimates. For improper constellation diagrams such as 8-QAM, widely linear estimators are used. For this case, we show that the widely linear versions of the LMMSE estimator and the CWCU LMMSE estimator also yield identical log-likelihood ratios. Finally, we give a simulation example which illustrates a number of interesting properties of the discussed widely linear estimators.