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We propose a novel filter-type equalizer to improve the solution of the linear minimum-mean squared-error (LMMSE) turbo equalizer, with computational complexity constrained to be quadratic in the filter length. When high-order modulations and/or large memory channels are used the optimal BCJR equalizer is unavailable, due to its computational complexity. In this scenario, the filter-type LMMSE turbo equalization exhibits a good performance compared to other approximations. In this paper, we show that this solution can be significantly improved by using expectation propagation (EP) in the estimation of the a posteriori probabilities. First, it yields a more accurate estimation of the extrinsic distribution to be sent to the channel decoder. Second, compared to other solutions based on EP the computational complexity of the proposed solution is constrained to be quadratic in the length of the finite impulse response (FIR). In addition, we review previous EP-based turbo equalization implementations. Instead of considering default uniform priors we exploit the outputs of the decoder. Some simulation results are included to show that this new EP-based filter remarkably outperforms the turbo approach of previous versions of the EP algorithm and also improves the LMMSE solution, with and without turbo equalization.

Usually, complex-valued RKHS are presented as an straightforward application of the real-valued case. In this paper we prove that this procedure yields a limited solution for regression. We show that another kernel, here denoted as pseudokernel, is needed to learn any function in complex-valued fields. Accordingly, we derive a novel RKHS to include it, the widely RKHS (WRKHS). When the pseudo-kernel cancels, WRKHS reduces to complex-valued RKHS of previous approaches. We address the kernel and pseudo-kernel design, paying attention to the kernel and the pseudo-kernel being complex-valued. In the experiments included we report remarkable improvements in simple scenarios where real a imaginary parts have different similitude relations for given inputs or cases where real and imaginary parts are correlated. In the context of these novel results we revisit the problem of non-linear channel equalization, to show that the WRKHS helps to design more efficient solutions.

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We investigate a turbo soft detector based on the expectation propagation (EP) algorithm for large-scale multipleinput multiple-output (MIMO) systems. Optimal detection in MIMO systems becomes computationally unfeasible for highorder modulations and/or large number of antennas. In this situation, the linear minimum mean square error (LMMSE) exhibits a low-complexity with a good performance, however far from optimal. To improve the performance, the EP algorithm can be used. In this paper, we review previous EP-based detectors and enhance their estimation in terms of complexity and performance. Specifically, we improve the convergence of the self-iterated EP stage by replacing the uniform prior by a non-uniform one, which better characterizes the information returned by the decoder once the turbo procedure starts. We also review the EP parameters to avoid instabilities when using high-order modulations and to reduce the computational complexity. Simulation results illustrate the robustness and enhanced performance of this novel detector in comparison with previous approaches found in the literature. Results also show that the proposed detector is robust in the presence of imperfect channel state information (CSI).