Duality between widely linear and dual channel adaptive filtering

We address the duality between adaptive filtering in C and R² and provide a comparison between the well understood dual channel real valued least mean square (DCRLMS) algorithm in R² and the corresponding algorithms in C. These include the complex LMS (CLMS) and the recently introduced augmented CLMS (ACLMS), a widely linear algorithm designed for the processing of noncircular complex valued signals. The analysis shows that the standard CLMS and DCRLMS in general provide different adaptive filtering solutions, whereas the ACLMS and DCRLMS are isomoprhic and can be made equivalent. The analysis is supported by simulations on noncircular real world signals.