Complex kernels for proper complex-valued signals: A review

In this paper we investigate the conditions that complex kernels must satisfy for proper complex-valued signals. We study the structure that complex kernels for proper complex-valued signals must have. Also, we demonstrate that complex kernels that have been previously proposed and used in adaptive filtering of complex-valued signals assume that those signals are proper, i.e, they are not correlated with their complex conjugate. We provide an example of how a complex-valued kernel suitable for a particular model is designed, with a procedure that could help in other designs. The experiments included show the good behavior of the proposed kernel in the task of nonlinear channel equalization.