Kernel methods for nonlinear identification, equalization and separation of signals

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Authors Steven Van Vaerenbergh
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Paper Abstract In the last decade, kernel methods have become established techniques to perform nonlinear signal processing. Thanks to their foundation in the solid mathematical framework of reproducing kernel Hilbert spaces (RKHS), kernel methods yield convex optimization problems. In addition, they are universal nonlinear approximators and require only moderate computational complexity. These properties make them an attractive alternative to traditional nonlinear techniques such as Volterra series, polynomial filters and neural networks. Kernel methods also exhibit certain drawbacks that must be addressed properly in every application, including complexity issues for large data sets and overfitting problems. In this work we propose a set of kernel-based algorithms to solve a number of related, nonlinear problems in signal processing and communications. In particular, we deal with the identification and equalization of nonlinear systems, and with nonlinear blind source separation (BSS). First, the identification of nonlinear systems is addressed. After discussing supervised kernel-based techniques for identifying black-box nonlinear systems, we focus on the family of online kernel algorithms, which are usually posed as adaptive filtering algorithms in the kernel feature space. Nevertheless, most online kernel methods show difficulties that are not encountered in classical adaptive filtering and have not been satisfactorily solved yet. Specifically, they require a growing memory and are unable to track time-varying nonlinear systems. As a first contribution we present a set of kernel recursive least-squares (KRLS) algorithms that deal with both problems by fixing the memory size and adjusting the stored data adequately. These algorithms are also used as building blocks in later chapters. In order to limit the complexity of the nonlinear mapping we then study the block-based nonlinear Wiener and Hammerstein systems. Despite their limited modeling capability, these systems are sufficient to represent many nonlinearities that appear in practice. By applying a suitable restriction in the chosen identification diagram, we show how a kernel canonical correlation analysis (KCCA) solution emerges. Additionally, by including the previously proposed KRLS techniques in this framework, we obtain a set of adaptive KCCA algorithms suitable for online identification and equalization of Wiener and Hammerstein systems. After a further analysis of block-based systems, we demonstrate how oversampling allows blind identification and blind equalization of Wiener systems by applying a KCCA-based technique. The proposed technique is inspired by a linear blind identification method which we extend into feature space, and it can be applied to any scenario where multiple Wiener systems are excited by the same input signal. In the second part of this thesis we treat blind source separation problems that allow for clustering approaches. This is the case when the source signals either belong to a finite alphabet or show a high degree of sparseness. However, when the mixture process is time-varying or nonlinear, the algorithms based on classical clustering do not hold. We study two such problems and we show that they can be tackled by designing specific versions of spectral clustering, which has an interpretation as kernel principal component analysis. The first scenario is found in the blind decoding problem of fast time-varying multiple-input multiple-output (MIMO) systems. While the scatter plot data form overlapping clusters here, which prohibits the application of conventional clustering algorithms, we observe that they can be untangled by including temporal information. For the resulting problem we present a spectral clustering algorithm whose kernel is designed to favor the expected cluster shape and to exploit the constellation geometry. The second scenario is a nonlinear blind source separation problem in which the sources are sparse. We deal with the very restrictive underdetermined case, in which the number of available mixtures is less than the number of sources, and we develop a clustering algorithm capable of identifying the nonlinear mixture process, which allows to recover the original source signals. In summary, this dissertation presents several techniques for related applications in nonlinear signal processing and communications. The proposed methods contribute to the state of the art in nonlinear system identification and equalization, and in nonlinear blind source separation.
Date of publication 2010
Code Programming Language MATLAB

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