Quaternion-Valued Nonlinear Adaptive Filtering

A class of nonlinear quaternion-valued adaptive filtering algorithms is proposed based on locally analytic nonlinear activation functions. To circumvent the stringent standard analyticity conditions which are prohibitive to the development of nonlinear adaptive quaternion-valued estimation models, we use the fact that stochastic gradient learning algorithms require only local analyticity at the operating point in the estimation space. It is shown that the quaternion-valued exponential function is locally analytic, and, since local analyticity extends to polynomials, products, and ratios, we show that a class of transcendental nonlinear functions can serve as activation functions in nonlinear and neural adaptive models. This provides a unifying framework for the derivation of gradient-based learning algorithms in the quaternion domain, and the derived algorithms are shown to have the same generic form as their real- and complex-valued counterparts. To make such models second-order optimal for the generality of quaternion signals (both circular and noncircular), we use recent developments in augmented quaternion statistics to introduce widely linear versions of the proposed nonlinear adaptive quaternion valued filters. This allows full exploitation of second-order information in the data, contained both in the covariance and pseudocovariances to cater rigorously for second-order noncircularity (improperness), and the corresponding power mismatch in the signal components. Simulations over a range of circular and noncircular synthetic processes and a real world 3-D noncircular wind signal support the approach.