Serra Capizzano, Stefano

The Iterative Filtering method is a technique aimed at the decomposition of non-stationary and non-linear signals into simple oscillatory components. This method, proposed a decade ago as an alternative technique to the Empirical Mode Decomposition, has been used extensively in many applied fields of research and studied, from a mathematical point of view, in several papers published in the last few years. However, even if its convergence and stability are now established both in the continuous and discrete setting, it is still an open problem to understand up to what extent this approach can separate two close-by frequencies contained in a signal. In this paper, first we recall previously discovered theoretical results about Iterative Filtering. Afterward, we prove a few new theorems regarding the ability of this method in separating two nearby frequencies both in the case of continuously and discrete sampled signals. Among them, we prove a theorem which allows to construct filters which captures, up to machine precision, a specific frequency. We run numerical tests to confirm our findings and to compare the performance of Iterative Filtering with the one of Empirical Mode Decomposition and Synchrosqueezing methods. All the results presented confirm the ability of the technique under investigation in addressing the fundamental “one or two frequencies” question.

When approximating elliptic problems by using specialized approximation techniques, we obtain large structured matrices whose analysis provides information on the stability of the method. Here we provide spectral and norm estimates for matrix-sequences arising from the approximation of the Laplacian via ad hoc finite differences. The analysis involves several tools from matrix theory and in particular from the setting of Toeplitz operators and Generalized Locally Toeplitz matrix-sequences. Several numerical experiments are conducted, which confirm the correctness of the theoretical findings.

Electroencephalography (EEG) signals are considered one of the oldest techniques for detecting disorders in medical signal processing. However, brain complexity and the non-stationary nature of EEG signals represent a challenge when applying this technique. The current paper proposes new geometrical features for classification of seizure (S) and seizure-free (SF) EEG signals with respect to the Poincaré pattern of discrete wavelet transform (DWT) coefficients. DWT decomposes EEG signal to four levels, and thus Poincaré plot is shown for coefficients. Due to patterns of the Poincaré plot, novel geometrical features are computed from EEG signals. The computed features are involved in standard descriptors of 2‑D projection (STD), summation of triangle area using consecutive points (STA), as well as summation of shortest distance from each point relative to the 45-degree line (SSHD), and summation of distance from each point relative to the coordinate center (SDTC). The proposed procedure leads to discriminate features between S and SF EEG signals. Thereafter, a binary particle swarm optimization (BPSO) is developed as an appropriate technique for feature selection. Finally, k-nearest neighbor (KNN) and support vector machine (SVM) classifiers are used for classifying features in S and SF groups. By developing the proposed method, we have archived classification accuracy of 99.3 % with respect to the proposed geometrical features. Accordingly, S and SF EEG signals have been classified. Also, Poincaré plot of SF EEG signals has more regular geometrical shapes as compared to S group. As a final remark, we notice that the Poincaré plot of coefficients in S EEG signals has occupied more space as compared to SF EEG signals (Tab. 3, Fig. 11, Ref. 57). Text in PDF www.elis.sk Keywords: EEG signal, DWT, Poincaré plot, geometrical feature, BPSO, SVM, KNN.