On the log-likelihood ratio evaluation of CWCU linear and widely linear MMSE data estimators

In soft decoding, log-likelihood ratios (LLRs) are calculated from estimated data symbols. Data symbols from proper constellation diagrams such as QPSK are often estimated using the linear minimum mean square error (LMMSE) estimator. We prove that the recently introduced component-wise conditionally unbiased (CWCU) LMMSE estimator results in the very same LLRs as the LMMSE estimator for typical model assumptions. For improper constellation diagrams such as 8-QAM, we show that the widely linear versions of the LMMSE and the CWCU LMMSE estimator also yield identical LLRs. In that case, the CWCU estimator allows to reduce the complexity of the LLR determination.