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

Abstract

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) <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-derivative based on the concept of <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-derivative and quaternion GHR derivative. Then, the product rules of GHR <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-derivative are deduced, and the results of GHR <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-derivative regarding some common univariate and multivariable functions are given. In addition, based on the proposed GHR <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-derivative, the cost function of the quaternion least mean square (QLMS) algorithm is minimized to generate the <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-QLMS algorithm. Finally, an example of Lorentz chaotic time-series prediction verifies the effectiveness of the proposed <inline-formula><tex-math notation="LaTeX">$q$</tex-math></inline-formula>-QLMS algorithm.