# The Generalized Complex Kernel Least-Mean-Square Algorithm

## TL;DR

This paper introduces the generalized complex kernel LMS (gCKLMS), an advanced adaptive regression method for complex signals that improves convergence and representation by incorporating a pseudo-kernel, outperforming previous algorithms in nonlinear channel equalization.

## Contribution

The paper presents the gCKLMS algorithm, which includes a pseudo-kernel for better complex signal modeling, unifies previous complex KLMS variants, and offers improved convergence and flexibility.

## Key findings

- gCKLMS outperforms previous algorithms in convergence speed.
- The pseudo-kernel enhances modeling of complex signals with different real and imaginary properties.
- Experimental results confirm significant performance improvements in nonlinear channel equalization.

## Abstract

We propose a novel adaptive kernel based regression method for complex-valued signals: the generalized complex-valued kernel least-mean-square (gCKLMS). We borrow from the new results on widely linear reproducing kernel Hilbert space (WL-RKHS) for nonlinear regression and complex-valued signals, recently proposed by the authors. This paper shows that in the adaptive version of the kernel regression for complex-valued signals we need to include another kernel term, the so-called pseudo-kernel. This new solution is endowed with better representation capabilities in complex-valued fields, since it can efficiently decouple the learning of the real and the imaginary part. Also, we review previous realizations of the complex KLMS algorithm and its augmented version to prove that they can be rewritten as particular cases of the gCKLMS. Furthermore, important conclusions on the kernels design are drawn that help to greatly improve the convergence of the algorithms. In the experiments, we revisit the nonlinear channel equalization problem to highlight the better convergence of the gCKLMS compared to previous solutions. Also, the flexibility of the proposed generalized approach is tested in a second experiment with non-independent real and imaginary parts. The results illustrate the significant performance improvements of the gCKLMS approach when the complex-valued signals have different properties for the real and imaginary parts.