  Complex-valued signals are used in the modeling of many systems in
engineering and science, hence being of fundamental interest. Often, random
complex-valued signals are considered to be proper. A proper complex random
variable or process is uncorrelated with its complex conjugate. This assumption
is a good model of the underlying physics in many problems, and simplifies the
computations. While linear processing and neural networks have been widely
studied for these signals, the development of complex-valued nonlinear kernel
approaches remains an open problem. In this paper we propose Gaussian processes
for regression as a framework to develop 1) a solution for proper
complex-valued kernel regression and 2) the design of the reproducing kernel
for complex-valued inputs, using the convolutional approach for
cross-covariances. In this design we pay attention to preserve, in the complex
domain, the measure of similarity between near inputs. The hyperparameters of
the kernel are learned maximizing the marginal likelihood using Wirtinger
derivatives. Besides, the approach is connected to the multiple output learning
scenario. In the experiments included, we first solve a proper complex Gaussian
process where the cross-covariance does not cancel, a challenging scenario when
dealing with proper complex signals. Then we successfully use these novel
results to solve some problems previously proposed in the literature as
benchmarks, reporting a remarkable improvement in the estimation error.
