Complex-Valued Nonlinear Adaptive Filters With Applications in $\alpha$-Stable Environments

A nonlinear adaptive filtering framework for processing complex-valued signals is derived. The introduced adaptive filter extends the fractional-order framework of the authors for dealing with real-valued signals to the complex domain via the augmented statistical approach to complex-valued signal processing. This results in a versatile class of adaptive filtering techniques, which allows the classical Gaussian assumption to be extended to the generalized context of α-stables. For rigor, the performance of the introduced adaptive filtering framework is analyzed, its convergence criteria is established, and its application in tracking signals of chaotic systems is demonstrated using simulations.