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


Menguc E. C.

IET SIGNAL PROCESSING, vol.12, no.4, pp.487-495, 2018 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 12 Issue: 4
  • Publication Date: 2018
  • Doi Number: 10.1049/iet-spr.2017.0340
  • Title of Journal : IET SIGNAL PROCESSING
  • Page Numbers: pp.487-495
  • Keywords: calculus, adaptive filters, least squares approximations, statistical analysis, convergence, signal processing, quaternion-valued least-mean kurtosis adaptive filtering algorithm, GHR calculus, QLMK adaptive filtering algorithm, four-dimensional processes, three-dimensional processes, generalised Hamilton-real calculus, error signal, cost function, steady-state error, convergence rate, noise signals, misadjustment conditions, quaternion statistics, quaternion-valued signal processing, SIZE LMS ALGORITHM, STEP-SIZE, MULTILAYER PERCEPTRONS, STOCHASTIC-ANALYSIS, GRADIENT OPERATOR, GAUSSIAN INPUTS, NEURAL-NETWORK, COMPLEX, CLASSIFICATION, CONVERGENCE

Abstract

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.