On the complex Kernel-based adaptive filter

Ogunfunmi, Tokunbo; Paul, Thomas

doi:10.1109/iscas.2011.5937800

Cited by 15 publications

(16 citation statements)

References 7 publications

(14 reference statements)

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“…It is straightforward to show that previous approaches in [4], [5], [6] belong to the SRKHS type with kernels as described in the following section and, therefore, are limited compared to the WRKHS. An interesting singular case of this SRKHS formulation corresponds to the scenario where the real and imaginary parts are not related.…”

Section: B Connection To Previous Approachesmentioning

confidence: 99%

“…The non-linear processing of complex-valued signals has been addressed from the point of view of neural networks, [2] and, recently, using reproducing kernel Hilbert spaces (RKHS). Some complex kernel-based algorithms have been lately proposed for classification [3], regression [4], [5], [6] and mainly for kernel principal component analysis [7]. Regarding regression, in [5] the authors propose a complex-valued kernel based in the results in [3] and face the derivative of cost functions by using Wirtinger's derivatives.…”

Section: Introductionmentioning

confidence: 99%

“…Regarding regression, in [5] the authors propose a complex-valued kernel based in the results in [3] and face the derivative of cost functions by using Wirtinger's derivatives. Same kernel is adopted in [4]. And in [8] the augmented version of the algorithm is proposed.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Widely Linear Complex-Valued Kernel Methods for Regression

Boloix-Tortosa

Murillo-Fuentes

Santos

et al. 2017

IEEE Trans. Signal Process.

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Usually, complex-valued RKHS are presented as an straightforward application of the real-valued case. In this paper we prove that this procedure yields a limited solution for regression. We show that another kernel, here denoted as pseudokernel, is needed to learn any function in complex-valued fields. Accordingly, we derive a novel RKHS to include it, the widely RKHS (WRKHS). When the pseudo-kernel cancels, WRKHS reduces to complex-valued RKHS of previous approaches. We address the kernel and pseudo-kernel design, paying attention to the kernel and the pseudo-kernel being complex-valued. In the experiments included we report remarkable improvements in simple scenarios where real a imaginary parts have different similitude relations for given inputs or cases where real and imaginary parts are correlated. In the context of these novel results we revisit the problem of non-linear channel equalization, to show that the WRKHS helps to design more efficient solutions.

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Section: B Connection To Previous Approachesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Widely Linear Complex-Valued Kernel Methods for Regression

Boloix-Tortosa

Murillo-Fuentes

Santos

et al. 2017

IEEE Trans. Signal Process.

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“…In nonlinear electromagnetic calculation problems, the amplitude and phase of an electromagnetic wave can also be equivalent to a complex-value form. Therefore, several complex KAFs have recently been proposed [3][4][5][6][7][8]22,24]. The complex KLMS (CKLMS) was first proposed in [5,6], which uses Wirtinger's calculus to generalize a complex RKHS.…”

Section: Introductionmentioning

confidence: 99%

The Generalized Complex Kernel Affine Projection Algorithms

Wang

Zhang

et al. 2021

Circuits Syst Signal Process

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The complex kernel adaptive filter (CKAF) has been widely applied to the complex-valued nonlinear problem in signal processing and machine learning. However, most of the CKAF applications involve the complex kernel least mean square (CKLMS) algorithms, which work in a pure complex or complexified reproducing kernel Hilbert space (RKHS). In this paper, we propose the generalized complex kernel affine projection (GCKAP) algorithms in the widely linear complex-valued RKHS (WL-RKHS). The proposed algorithms have two main notable features. One is that they provide a complete solution for both circular and non-circular complex nonlinear problems and show many performance improvements over the CKAP algorithms. The other is that the GCKAP algorithms inherit the simplicity of the CKLMS algorithm while reducing its gradient noise and boosting its convergence. The second-order statistical characteristics of WL-RKHS have also been developed. An augmented Gram matrix consists of a standard Gram matrix and a pseudo-Gram matrix. This decomposition provides more underlying information when the real and imaginary parts of the signal are correlated and learning is independent. In addition, some online sparsification criteria are compared comprehensively in the GCKAP algorithms, including the novelty criterion, the coherence criterion, and the angle criterion. Finally, two nonlinear channel equalization experiments with non-circular complex inputs are presented to illustrate the performance improvements of the proposed algorithms.

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“…Regarding regression, in [11] the authors propose a complexvalued kernel based on the results in [6] and face the derivative of cost functions by using Wirtinger's derivatives. Same kernel is adopted in [7]. In [9] the authors review the kernel design to improve the previous solutions.…”

Section: Introductionmentioning

confidence: 99%

Complex Gaussian Processes for Regression

Boloix-Tortosa

Murillo-Fuentes

Payán-Somet

et al. 2018

IEEE Trans. Neural Netw. Learning Syst.

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In this paper, we propose a novel Bayesian solution for nonlinear regression in complex fields. Previous solutions for kernels methods usually assume a complexification approach, where the real-valued kernel is replaced by a complex-valued one. This approach is limited. Based on the results in complex-valued linear theory and Gaussian random processes, we show that a pseudo-kernel must be included. This is the starting point to develop the new complex-valued formulation for Gaussian process for regression (CGPR). We face the design of the covariance and pseudo-covariance based on a convolution approach and for several scenarios. Just in the particular case where the outputs are proper, the pseudo-kernel cancels. Also, the hyperparameters of the covariance can be learned maximizing the marginal likelihood using Wirtinger's calculus and patterned complex-valued matrix derivatives. In the experiments included, we show how CGPR successfully solves systems where the real and imaginary parts are correlated. Besides, we successfully solve the nonlinear channel equalization problem by developing a recursive solution with basis removal. We report remarkable improvements compared to previous solutions: a 2-4-dB reduction of the mean squared error with just a quarter of the training samples used by previous approaches.

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On the complex Kernel-based adaptive filter

Cited by 15 publications

References 7 publications

Widely Linear Complex-Valued Kernel Methods for Regression

Widely Linear Complex-Valued Kernel Methods for Regression

The Generalized Complex Kernel Affine Projection Algorithms

Complex Gaussian Processes for Regression

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