Design of quaternion-valued second-order Volterra adaptive filters for nonlinear 3-D and 4-D signals

Abstract

In this paper, the quaternion-valued second-order Volterra adaptive filters (QSOVAFs) are designed for the processing of nonlinear three-dimensional (3-D) and four-dimensional (4-D) signals. In the proposed frameworks, the structure of the strictly nonlinear (SNL), semi-widely nonlinear (SWNL), and widely nonlinear (WNL) QSOVAFs are primarily constructed in the quaternion domain. Then, their loss functions defined by the instantaneous error signals are minimized in the quaternion domain by using the recent generalized Hamilton-real (GHR) calculus. Thus, novel weight update equations are obtained for training the proposed SNL-QSOVAF, SWNL-QSOVAF, and WNL-QSOVAF. Furthermore, the stability bounds for each quaternion-valued kernel functions of them are derived from the convergence in the mean analysis. The comprehensive simulations on the nonlinear system identification and one-step-ahead prediction experiments support that the proposed SWNL-QSOVAF and WNL-QSOVAF can be effectively used in the processing of nonlinear noncircular quaternion signals, whereas that their SNL version can produce optimal results for nonlinear circular quaternion signals.