The Generalized Complex Kernel Least-Mean-Square Algorithm

Abstract

We propose a novel adaptive kernel based regression method for complex-valued signals: the generalized complexvalued 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 complexvalued signals we need to include another kernel term, the socalled 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.

Spline-Based Adaptive Cancellation of Even-Order Intermodulation Distortions in LTE-A/5G RF Transceivers

“…This form is obtained by factoring out all common terms and using the identity a + a * = 2 {a}. The gradient of the norm constraint (43), as well as the limiter (56), remain unaffected by the complex control points and can be directly reused. Combining all results, the update equations of the CIO-WSAF are…”

“…The second method we use for comparison is the kernel recursive least squares (KRLS) algorithm, a very general adaptive learning concept [42]. We choose a real-valued Gaussian kernel with a complex-valued input [43], which allows the KRLS to model a wide range of complex-valued nonlinearities without the single-tap scaler. In order to limit its complexity, the KRLS needs an additional sparsification method, in our case the approximate linear dependency (ALD) criterion.…”

The Generalized Complex Kernel Affine Projection Algorithms

“…The inverse operation of G + δI 2P can no longer be solved by the iteration method in [23]. However, the directly inverse operation of the augmented Gram matrix requires the complexity of O(8P 3 ). We propose a decomposition method to reduce the complexity.…”

“…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.…”

“…In [2], the widely linear reproducing kernel Hilbert space (WL-RKHS) theory was developed, in which a new pseudo-kernel was introduced to complement the standard kernel for a complete representation of the complex Hilbert space. In [3], the authors borrowed from the results on the WL-RKHS for nonlinear regression and proposed the generalized CKLMS (GCKLMS). That algorithm can provide a better representation of complex-valued signal than the pure complex and complexified methods and perform the independent learning of the real and the imag-inary parts.…”