  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.
