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. 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.

💡 Research Summary

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The paper “The Augmented Complex Kernel LMS” introduces a novel framework for adaptive filtering of complex‑valued signals in reproducing kernel Hilbert spaces (RKHS). Building on the authors’ earlier work that defined a complex‑valued RKHS and extended Wirtinger calculus to general Hilbert spaces, this manuscript applies the extended calculus to derive widely‑linear (also called augmented) kernel adaptive filters.

Two main methodological paths are discussed. The first maps complex data directly into a complex RKHS using a pure complex kernel, such as the complex Gaussian kernel
(\kappa_{\sigma,\mathbb{C}^d}(z,w)=\exp!\big