The Augmented Complex Kernel LMS

Abstract

Abstract-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.

The quaternion kernel least squares

“…Kernel regression algorithms [3][4][5][6][7][8][19][20][21] aim to approximate y = f (x) by mapping the input x onto a reproducing kernel Hilbert space (RKHS) H according to x −→ φ(x), in order to yield the estimate of the function f (·) through a linear transformation in the feature space, that iŝ…”

“…Recent results show clear advantage of kernel regression algorithms for complex-valued signals [6][7][8], in which the kernels themselves are also complex-valued. We next show that the quaternion-valued kernel will inherit this property by continuity, since both the real, complex and quaternion domain are normed division algebras.…”

“…The generic nature of kernel algorithms has also allowed for complexvalued extensions of real-valued kernels algorithms [6][7][8]; these inherit the advantages offered by the complex division algebra in general bivariate estimation applications. On the other hand, the recent progress in sensor technology and data acquisition systems has brought to light three-and four-dimensional sensor signals, in areas including wind prediction, econometric signal estimation and spatial motion tracking.…”

Multikernel least squares estimation

“…The promising results of augmented complex KLMS [6], [7] considering pairs of complex conjugate functions as feature elements, suggest that the multikernel approach inherits benefits from the enhanced dimensionality RKHS. We focus on nonlinear estimation by optimally combining multiple kernels to fit the available training set.…”

“…Kernel-based algorithms are also being developed for complex-valued signals where feature spaces of enhanced dimensionality have been considered: the work in [6] addresses wind profile prediction, while [7] focuses on channel identification and equalisation. A key concept in the enhanced dimensionality of complex-valued kernel algorithms is the use of a pair of conjugate kernels; this motivates the extension of real-valued single-kernel algorithms to multiple kernel (multikernel) approaches, in which each kernel is designed to account for different nonlinear features of the data at hand.…”