COMPLEX KERNELS FOR PROPER COMPLEX-VALUED SIGNALS: A REVIEW Rafael Boloix-Tortosa,F. Javier Pay´ an-Somet, Eva Arias-de-Reyna and Juan Jos´ e Murillo-Fuentes Department of Signal Theory and Communications, University of Seville, Spain. e-mail: rboloix@us.es ABSTRACT In this paper we investigate the conditions that complex ker- nels 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 fil- tering of complex-valued signals assume that those signals are proper, i.e, they are not correlated with their complex conju- gate. 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. Index Terms— Gaussian processes, regression, proper complex processes, kernel methods. 1. INTRODUCTION Complex-valued signals model a vast range of nowadays sys- tems in science and engineering. The nonlinear processing of complex-valued signals has been recently addressed using re- producing kernel Hilbert spaces (RKHS) [5]. Some complex kernels have been lately proposed for classification [6], kernel principal component analysis and regression [1–4]. Regard- ing regression, in [2] the authors propose a complex-valued kernel based on the results in [6]. The same kernel is adopted in [1], and its convergence behavior is studied in [7]. As discussed later in this paper, the resulting approach involves properness of the complex-valued signals, i.e., they are un- correlated with their complex conjugate. Besides, the kernel used is neither stationary nor isotropic, and it may suffer from numerical problems. In [4] the authors review the kernel de- sign to improve the previous solution with a kernel they de- note as independent. The resulting kernel yields also proper complex-valued outputs. The kernel is stationary, but again it is not isotropic in the complex-valued input space, as the real and imaginary parts of the input are split and fed to dif- ferent real valued kernels. Hence, these previous works do not report results for isotropic and stationary kernels that may better model the underlying physics of some systems. Also, the structure of the kernel remains more rigid than needed. Thanks to Spanish government (Ministerio de Educaci´ on y Ciencia TEC2012-38800-C03-02) and European Union (FEDER) for funding. These drawbacks make these solutions not powerful enough to learn a wide range of systems. We study in this paper the conditions that complex pos- itive definite kernels must satisfy for proper complex-valued signals, to improve previous solutions. The starting point is the complex nonlinear regression problem y = f (x)+ ǫ, where the output, y ∈ C, the input vector, x ∈ C d , and the unknown nonlinear latent function, f ∈ C, are com- plex valued, and the error, ǫ is modeled as additive zero- mean complex Gaussian noise. We analyze the structure of the covariance matrix of the complex-valued vector y = [y(x(1)), ..., y(x(n))] ⊤ when it is proper. The covariance function or kernel must produce the entries of that covari- ance matrix. We will show that the real part of the kernel is given by the covariance of the real part plus the covariance of the imaginary part of the outputs, while the imaginary part of the kernel describes the cross-covariance between real and imaginary parts of the outputs. We prove that the real and imaginary parts of the kernel can be designed with different features. But we conclude that the imaginary part, in addition to be skew-symmetric, must be constructed to ensure the whole covariance to be semi-definite positive, i.e. a reproducing kernel or covariance matrix [8]. We also pay attention to the modeling of physical systems. As an example, we propose the construction of a complex kernel that explains a positive and negative correlation of the real part of the output with the imaginary part for a positive and negative delay, canceling at the origin. Also, in order to measure similarity between inputs our example makes use of the absolute value of the complex difference between inputs. Therefore, the kernel is isotropic and stationary. We resort to the convolution approach [9, 10] to ensure that the produced kernel is a valid covariance function. The procedure followed could help in other complex-valued kernel designs for proper complex-valued outputs. 2. COMPLEX COVARIANCE FUNCTIONS Consider a zero-mean complex vector y = y r + jy j ∈ C n , with y r its real part and y j its imaginary part. The covariance matrix K = E  yy H  is [11]: K = K rr + K jj + j (K jr - K rj ) , (1) 23rd European Signal Processing Conference (EUSIPCO) 978-0-9928626-3-3/15/$31.00 ©2015 IEEE 2416