Low-Complexity Quaternion Adaptive Filters
  A general representation of the quaternion gradients presented in the literature is proposed, and an universal update equation for QLMS-like algorithms is obtained. The general update law is used to study the convergence of widely linear (WL) algorithms. It is proved that techniques obtained with a gradient similar to the i-gradient are the fastest-converging in two situations: 1) When the correlation matrix contains elements only in 2 axis (1 and $i$, for instance), and 2) When the algorithms use a real data vector, obtained staking up the real and imaginary parts of the original quaternion input vector. The general update law is also used to study the convergence of WL-QLMS-based algorithms, and an accurate second-order model is developed for quaternion algorithms using real-data input. Based on the proposed analysis, we obtain the fastest-converging WL-QLMS algorithm with real-regressor vector, which is also less costly than the reduced-complexity WL-QLMS (RC-WL-QLMS) algorithm proposed in our previous work. It is shown that the new method corresponds to the four-channel LMS algorithm written in the quaternion domain, and that they have the same computational complexity. Simulations illustrate the performance of the new technique and the accuracy of the analysis. 
Fernando G. Almeida Neto and V\'itor H. Nascimento
math.OC