On gradient calculation in quaternion adaptive filtering

A novel way to calculate the gradient of real functions of quaternion variables, typical cost functions in quaternion signal processing, is proposed. This is achieved by revisiting quaternion involutions and by simplifying the existing HR derivatives. This has allowed us to express the class of quaternion least mean square (QLMS) algorithms in a more compact form while keeping the same generic form of LMS. Simulations in the prediction setting support the approach.