A novel reduced-complexity widely linear QLMS algorithm

The widely linear quaternion least mean square (WL-QLMS) algorithm is used to capture complete second order statistics in quaternion valued mean-square estimation (MSE). However, quaternion operations during the signal processing have high computational cost, since each operation involves the calculation of one real part and three different imaginary elements. We propose a reduced-complexity WL-QLMS (RC-WL-QLMS) algorithm, and we show that using a real data vector, the RC-WL-QLMS achieves the same MSE of the WL-QLMS, with lower computational cost. The new algorithm also presents faster convergence when compared to the WL-QLMS, which is shown analytically and through simulations.