LAUSR

Quaternion adaptive filters have been widely used in processing three-dimensional ($\mathbf {3}$-D) and $\mathbf {4}$-D signals. However, the performance of existing quaternion adaptive filtering algorithms will be deteriorated when both the input and output signals are disturbed by noises. To solve this problem, this paper first proposes a quaternion error-in-variables (QEIV) model in which the input and output signals are disturbed by noises, and develops a maximum total quaternion correntropy (MTQC) criterion to resist non-Gaussian noises. Then, combined with the total least squares (TLS) method, an MTQC algorithm is proposed based on the stochastic gradient method and quaternion generalized Hamilton-real (GHR) calculus for the proposed QEIV model. Furthermore, a variable kernel width strategy is given to avoid the problem of kernel width selection and a variable kernel width MTQC (VKWMTQC) algorithm is therefore proposed, simultaneously. More importantly, the local stability, the convergence condition, and steady-state performance of the proposed MTQC algorithm are theoretically analyzed by the quaternion calculus and quaternion matrix theories. Finally, simulation results verify the correctness of the theoretical analysis and demonstrate the superiority of the proposed MTQC algorithm for the QEIV model.