Quaternion least mean kurtosis algorithm for adaptive filtering of 3D and 4D signal processes

In this paper, a novel quaternion adaptive filtering algorithm is proposed for a unified processing of 3D and 4D data, called quaternion least mean kurtosis (QLMK) algorithm. Multi-dimensional signals exhibit a complex nonlinear relationship and couple among different components. Considering that quaternion has huge advantage in terms of the representation of 3D and 4D signal, quaternion algebra is employed to derive the quaternion least mean square (QLMS) algorithm for hypercomplex signal processes. However, QLMS originates from the least mean square (LMS) algorithm, which may result in performance degradation when the signal is non-Gaussian. Due to the desirable performance of the least mean kurtosis (LMK) algorithm in non-Gaussian situation, in the present work we extend the original LMK algorithm to quaternion domain to manage the 3D and 4D signal processes. The analysis shows that QLMK provides a solution that is responsive to dynamically changing environments. Simulations on prediction of 4D Saito's chaotic circuit and 3D Lorenz attractor confirm the desirable performance of the proposed method.