The Augmented Complex Kernel LMS

Recently, a unified framework for adaptive kernel based signal processing of complex data was presented by the authors, which, besides offering techniques to map the input data to complex reproducing kernel Hilbert spaces, developed a suitable Wirtinger-like calculus for general Hilbert spaces. In this short paper, the extended Wirtinger´s calculus is adopted to derive complex kernel-based widely linear estimation filters suitable for applications involving noncircular data. Furthermore, we illuminate several important characteristics of the widely linear filters. We show that, although in many cases the gains from adopting widely linear estimation filters, as alternatives to ordinary linear ones, are rudimentary, for the case of kernel based widely linear filters significant performance improvements can be obtained.