  Although the real reproducing kernels are used in an increasing number of
machine learning problems, complex kernels have not, yet, been used, in spite
of their potential interest in applications such as communications. In this
work, we focus our attention on the complex gaussian kernel and its possible
application in the complex Kernel LMS algorithm. In order to derive the
gradients needed to develop the complex kernel LMS (CKLMS), we employ the
powerful tool of Wirtinger'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, the notion of Writinger's calculus is extended to include complex RKHSs.
Experiments verify that the CKLMS offers significant performance improvements
over the traditional complex LMS or Widely Linear complex LMS (WL-LMS)
algorithms, when dealing with nonlinearities.
