Novel quaternion‐valued least‐mean kurtosis adaptive filtering algorithm based on the GHR calculus

A novel quaternion‐valued least‐mean kurtosis (QLMK) adaptive filtering algorithm is proposed for three‐ and four‐dimensional processes by using the recent generalised Hamilton‐real (GHR) calculus. The proposed QLMK algorithm based GHR calculus minimises the negated kurtosis of the error signal as a cost function in the quaternion domain, thus provides an elegant way to solve a trade‐off problem between the convergence rate and steady‐state error. Moreover, the proposed QLMK algorithm has naturally a robust behaviour for a wide range of noise signals due to its kurtosis‐based cost function. Furthermore, the steady‐state performance of the proposed QLMK algorithm is analysed to obtain convergence and misadjustment conditions. The comprehensive simulation results on benchmark and real‐world problems show that the use of this cost function defined by the quaternion statistics in the proposed QLMK algorithm allows us to process quaternion‐valued signals and thus, significantly enhances the performance of the adaptive filter in terms of both the steady‐state error and the convergence rate, as compared with the quaternion‐valued least‐mean‐square algorithm based on the recent GHR calculus.