The Generalized HR $q$-Derivative and Its Application to Quaternion Least Mean Square Algorithm

Quaternion adaptive filters have been widely used in processing three-dimensional and four-dimensional signals. To improve the performance of quaternion adaptive filtering algorithms, this letter first proposes a novel derivation rule named generalized HR (GHR)
$q$
-derivative based on the concept of
$q$
-derivative and quaternion GHR derivative. Then, the product rules of GHR
$q$
-derivative are deduced, and the results of GHR
$q$
-derivative regarding some common univariate and multivariable functions are given. In addition, based on the proposed GHR
$q$
-derivative, the cost function of the quaternion least mean square (QLMS) algorithm is minimized to generate the
$q$
-QLMS algorithm. Finally, an example of Lorentz chaotic time-series prediction verifies the effectiveness of the proposed
$q$
-QLMS algorithm.