An alternative kernel adaptive filtering algorithm for quaternion-valued data

Nonlinear adaptive filters are getting more common and are useful especially where performance of linear adaptive filters may be unacceptable. Such areas include communications, image processing and biological systems. Quaternion valued data has also been drawing recent interest in various areas of statistical signal processing, including adaptive filtering, image pattern recognition, and modeling and tracking of motion. The benefit for quaternion valued processing includes performing data transformations in a 3 or 4-dimensional space in a more convenient fashion than using vector algebra. In this paper we describe an alternative kernel adaptive filter for quaternion valued data we refer to as the involution Quaternion Kernel Least Mean Square (iQuat-KLMS) algorithm. The approach is based on the Quaternion KLMS (Quat-KLMS) algorithm obtained previously, as well as the recently-developed involution gradient (i-gradient). A modified HR Calculus for Hilbert spaces is used for finding cost function gradients defined on a quaternion RKHS. Simulation tests with a synthetic quaternion channel are used to verify the benefit of iQuat-KLMS in convergence compared to Quat-KLMS.