Modeling of Widely-Linear Quaternion Valued Systems Using Hypercomplex Algorithms

The data-driven optimal modeling and identification of widely-linear quaternion-valued synthetic systems is achieved by using a quaternion-valued gradient based algorithms. To account rigorously for the second-order statistics of the quaternion system, the quaternion least mean square (QLMS) and widely linear quaternion least mean square (WLQLMS) were selected. The QLMS is shown to successfully model the quaternion-valued systems and the WL-QLMS is able to model both quaternion and widely-linear quaternion valued systems taking into account the full second-order statistics of the system. Analysis has proven that both algorithms are able to adapt to non-stationary nature of the systems. This approach is supported by simulations of various synthetic systems.