  Over the last decade, kernel methods for nonlinear processing have
successfully been used in the machine learning community. However, so far, the
emphasis has been on batch techniques. It is only recently, that online
adaptive techniques have been considered in the context of signal processing
tasks. To the best of our knowledge, no kernel-based strategy has been
developed, so far, that is able to deal with complex valued signals. In this
paper, we take advantage of a technique called complexification of real RKHSs
to attack this problem. In order to derive gradients and subgradients of
operators that need to be defined on the associated complex RKHSs, we employ
the powerful tool ofWirtinger's Calculus, which has recently attracted much
attention in the signal processing community. Writinger's calculus simplifies
computations and offers an elegant tool for treating complex signals. To this
end, in this paper, the notion of Writinger's calculus is extended, for the
first time, to include complex RKHSs and use it to derive the Complex Kernel
Least-Mean-Square (CKLMS) algorithm. Experiments verify that the CKLMS can be
used to derive nonlinear stable algorithms, which offer significant performance
improvements over the traditional complex LMS orWidely Linear complex LMS
(WL-LMS) algorithms, when dealing with nonlinearities.
