The Gaussian process (GP) is a well-established Bayesian nonparametric tool for inference in nonlinear estimation problems. When GPs are used for regression, the goal is to estimate a target signal${y}$from an input vector$\mathbf {x}$without assuming that they are linearly related, but with a probabilistic model$p({y}|\mathbf {x})$that is Gaussian distributed. Therefore, GPs can be understood as a natural nonlinear extension to MMSE estimation. For real-valued GPs, this has been analyzed in the existing literature, and it is concluded that they are the natural nonlinear Bayesian extension to the linear minimum mean-squared error (LMMSE) estimation. In this letter, we show that, consequently, complex-valued GP regression (GPR) models are the natural nonlinear Bayesian extension of the widely linear minimum mean squared-error (WLMMSE) estimation. As in the real-valued case, complex-valued GPs are able to better model many regression problems by making use of the information that the complementary kernel or pseudo-kernel provides.

In this paper, we propose a novel Bayesian solution for nonlinear regression in complex fields. Previous solutions for kernels methods usually assume a complexification approach, where the real-valued kernel is replaced by a complex-valued one. This approach is limited. Based on the results in complex-valued linear theory and Gaussian random processes, we show that a pseudo-kernel must be included. This is the starting point to develop the new complex-valued formulation for Gaussian process for regression (CGPR). We face the design of the covariance and pseudo-covariance based on a convolution approach and for several scenarios. Just in the particular case where the outputs are proper, the pseudo-kernel cancels. Also, the hyperparameters of the covariance can be learned maximizing the marginal likelihood using Wirtinger's calculus and patterned complex-valued matrix derivatives. In the experiments included, we show how CGPR successfully solves systems where the real and imaginary parts are correlated. Besides, we successfully solve the nonlinear channel equalization problem by developing a recursive solution with basis removal. We report remarkable improvements compared to previous solutions: a 2-4-dB reduction of the mean squared error with just a quarter of the training samples used by previous approaches.

In probabilistic equalization of channels with intersymbol interference, the BCJR algorithm and its approximations become intractable for high-order modulations, even for moderate channel dispersions. In this paper, we introduce a novel soft equalizer to approximate the symbol a posteriori probabilities (APP), where the expectation propagation (EP) algorithm is used to provide an accurate estimation. This new soft equalizer is presented as a block solution, denoted as block-EP (BEP), where the structure of the matrices involved is exploited to reduce the complexity order to O(LN2), i.e., linear in the length of the channel, L, and quadratic in the frame length, N. The solution is presented in complex-valued formulation within a turbo equalization scheme. This algorithm can be cast as a linear minimum-mean-squared-error (LMMSE) turbo equalization with double feedback architecture, where constellations being discrete is a restriction exploited by the EP that provides a first refinement of the APP. In the experiments included, the BEP exhibits a robust performance, regardless of the channel response, with gains in the range 1.5-5 dB compared with the LMMSE equalization.

This letter presents a novel time-of-arrival (TOA) blind estimation technique for ultra wideband energy detection receivers with reduced complexity. The proposed method is blind in the sense that it does not exploit any information about channel or noise power. This new approach is based on a set of approximations of the exact likelihood function (ELF) of the observed energy. Even though these approximations achieve an important reduction of complexity, the shape of the new approximated function is accurate enough compared to the ELF. Application of a threshold to the differential of the approximated log-likelihood function completes the procedure. Simulations show that the performance of the proposed method in terms of the cumulative distribution function of the estimation error approaches that of a method based on the ELF and a genie-aided algorithm with perfect knowledge of the optimal threshold.

In this letter we focus on the application of independent component analysis (ICA) to a class of overdetermined blind source separation (BSS) problems. The mixing matrix in the BSS model is the product of an unknown square diagonal matrix and a projection matrix. The last matrix performs a known projection to the same or larger dimensional space. We demonstrate the conditions for the model to be strictly separable and identifiable under the statistical independence condition, paying attention to permutations and relative scalings. These results find application, e.g., in the channel estimation of ZP-OFDM and precoded-OFDM systems.

In the above letter (ibid., vol. 17, no. 3, pp. 285-288, Mar. 10), equation (12) on. p. 287 was shown incorrectly. The correct version is presented here.

In this paper, we explore the significance of second- and higher-order statistics learning in communication systems. The final goal in spread-spectrum communication systems is to receive a signal of interest completely free from interference caused by other concurrent signals. To achieve this end, we exploit the structure of the interference by designing second-order statistics detectors, such as the minimum square error, in conjunction with higher-order statistics (HOS) techniques, such as the blind source separation (BSS). This hybrid higher-order statistics (HyHOS) approach enables us to alleviate BSS algorithms of one of their main problems, that is, their sensitiveness to high levels of noise. In addition, we benefit from remarkable properties of BSS in learning such as fast learning (superefficiency) and independence of the initial settings of the problem (equivariance). We successfully applied the results of this approach to the design of multiuser detectors in code-division multiple access channels.