Widely nonlinear quaternion-valued second-order Volterra recursive least squares filter

Recently, a family of the quaternion-valued second-order Volterra LMS (QSOV-LMS) algorithms were proposed to deal with 3d and 4d signals. However, it should be noted that the QSOV-LMS algorithms suffer from low convergence speed and high steady-state properties. To this end, a widely nonlinear quaternion recursive least square algorithm is proposed to train the second-order Volterra filter (WNL-QSOVRLS) for enhancing the performance of QSOV-LMS algorithms. Furthermore, to avoid the immense computation complexity, we also present a novel widely nonlinear quaternion Volterra recursive least square dichotomous coordinate descent filtering model (WNL-QSOVRLS-DCD) which introduce the quaternion dichotomous coordinate descent (QDCD) algorithm for the first time. Moreover, based on the augmented model of quaternion second-order Volterra filter, the two proposed algorithms can process quaternion-valued circular and non-circular signals well. Finally, experiments verify the performance of the proposed algorithms.