Widely Linear Quaternion-Valued Least-Mean Kurtosis Algorithm

Menguc E. C. , Acir N., Mandic D. P.

IEEE TRANSACTIONS ON SIGNAL PROCESSING, vol.68, pp.5914-5922, 2020 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 68
  • Publication Date: 2020
  • Doi Number: 10.1109/tsp.2020.3029959
  • Page Numbers: pp.5914-5922
  • Keywords: Quaternions, Signal processing algorithms, Calculus, Covariance matrices, Cost function, Adaptation models, Algebra, Adaptive filters, circular and noncircular quaternion signals, least mean kurtosis, strictly linear, widely linear, PERFORMANCE ANALYSIS, LMS ALGORITHM, TREMOR, CLMS


A widely linear quaternion-valued least-mean kurtosis (WL-QLMK) algorithm is introduced for adaptive filtering of quaternion-valued circular and noncircular signals. In the design, kurtosis-based cost function is first defined in the quaternion domain by integrating the widely linear model, and augmented statistics, and then minimized using the recently developed generalized Hamilton-real (GHR) calculus. In this way, the novel WL-QLMK algorithm is obtained for training quaternion-valued adaptive filter structures. Furthermore, its steady-state performance is theoretically analyzed to determine the bounds of the step size, which provides a theoretical justification for simulations. The simulation results over both benchmark system identification scenarios, and one-step-ahead predictions of real-world 4D pathological resting tremors show that the proposed WL-QLMK algorithm, by virtue of its newly defined cost function, significantly enhances the performance compared to the recently developed quaternion-valued algorithms, especially for noncircular signals.