About QLMS Derivations

In this letter, a review of the quaternionic least mean squares (QLMS) algorithm is proposed. Three versions coming from three derivation ways exist: the original QLMS [1] based on componentwise gradients, HR-QLMS [2] based on a quaternion gradient operator and iQLMS [3] based on an involutions-gradient. Noting and investigating the differences between the three QLMS formulations, we show that the original QLMS suffers from a mistake in the derivation calculus. Thus, we propose to derive rigorously the criterion following the first way, giving the correct version of QLMS. A comparison with the other QLMS versions validates these results on simulated data.