Extension of Wirtinger's Calculus to Reproducing Kernel Hilbert Spaces and the Complex Kernel LMS

Over the last decade, kernel methods for nonlinear processing have successfully been used in the machine learning community. The primary mathematical tool employed in these methods is the notion of the Reproducing Kernel Hilbert Space. However, so far, the emphasis has been on batch techniques. It is only recently, that online techniques have been considered in the context of adaptive signal processing tasks. Moreover, these efforts have only been focussed on real valued data sequences. To the best of our knowledge, no adaptive kernel-based strategy has been developed, so far, for complex valued signals. Furthermore, although the real reproducing kernels are used in an increasing number of machine learning problems, complex kernels have not, yet, been used, in spite of their potential interest in applications that deal with complex signals, with Communications being a typical example. In this paper, we present a general framework to attack the problem of adaptive filtering of complex signals, using either real reproducing kernels, taking advantage of a technique called \textit{complexification} of real RKHSs, or complex reproducing kernels, highlighting the use of the complex gaussian kernel. In order to derive gradients of operators that need to be defined on the associated complex RKHSs, we employ the powerful tool of Wirtinger's Calculus, which has recently attracted attention in the signal processing community. To this end, in this paper, the notion of Wirtinger's calculus is extended, for the first time, to include complex RKHSs and use it to derive several realizations of the Complex Kernel Least-Mean-Square (CKLMS) algorithm. Experiments verify that the CKLMS offers significant performance improvements over several linear and nonlinear algorithms, when dealing with nonlinearities.Comment: 15 pages (double column), preprint of article accepted in IEEE Trans. Sig. Pro

Preamble-Based Channel Estimation for CP-OFDM and OFDM/OQAM Systems: A Comparative Study

In this paper, preamble-based least squares (LS) channel estimation in OFDM systems of the QAM and offset QAM (OQAM) types is considered, in both the frequency and the time domains. The construction of optimal (in the mean squared error (MSE) sense) preambles is investigated, for both the cases of full (all tones carrying pilot symbols) and sparse (a subset of pilot tones, surrounded by nulls or data) preambles. The two OFDM systems are compared for the same transmit power, which, for cyclic prefix (CP) based OFDM/QAM, also includes the power spent for CP transmission. OFDM/OQAM, with a sparse preamble consisting of equipowered and equispaced pilots embedded in zeros, turns out to perform at least as well as CP-OFDM. Simulations results are presented that verify the analysis

Robust Linear Regression Analysis - A Greedy Approach

The task of robust linear estimation in the presence of outliers is of particular importance in signal processing, statistics and machine learning. Although the problem has been stated a few decades ago and solved using classical (considered nowadays) methods, recently it has attracted more attention in the context of sparse modeling, where several notable contributions have been made. In the present manuscript, a new approach is considered in the framework of greedy algorithms. The noise is split into two components: a) the inlier bounded noise and b) the outliers, which are explicitly modeled by employing sparsity arguments. Based on this scheme, a novel efficient algorithm (Greedy Algorithm for Robust Denoising - GARD), is derived. GARD alternates between a least square optimization criterion and an Orthogonal Matching Pursuit (OMP) selection step that identifies the outliers. The case where only outliers are present has been studied separately, where bounds on the \textit{Restricted Isometry Property} guarantee that the recovery of the signal via GARD is exact. Moreover, theoretical results concerning convergence as well as the derivation of error bounds in the case of additional bounded noise are discussed. Finally, we provide extensive simulations, which demonstrate the comparative advantages of the new technique