Widely linear estimators and adaptive filters for real-valued quantities in complex-valued environments

In this work, mixed real- and complex-valued models are considered. First, the task of estimating a real-valued parameter vector based on complex-valued measurements in a classical set-up is investigated. The application of standard estimators in general results in complex-valued estimates of the real-valued parameter vector. To avoid this systematic error, several widely linear classical estimators that produce real-valued estimates are proposed. The proposed estimators in general outperform standard estimators and they only require half as many complex-valued measurements. Second, adaptive filtering algorithms are proposed for the case that the optimum filter coefficients are real-valued while the input and desired signal are complex-valued. The suggested least mean squares (LMS) and recursive least squares (RLS) type of filters are deployed in a practical application, where they significantly outperform their standard counterparts.