Technical Report 2017-018

Spectral and Convergence Analysis of the Discrete ALIF Method

Antonio Cicone, Carlo Garoni, and Stefano Serra-Capizzano

September 2017

The Adaptive Local Iterative Filtering (ALIF) method is a recently proposed iterative procedure to decompose a signal into a finite number of "simple components" called intrinsic mode functions. % It is an alternative to the well-known and widely used empirical mode decomposition, a method proposed in 1998 by Huang and his research team at NASA. % In this paper we consider the discrete version of the ALIF method and we perform a spectral analysis of the related iteration matrices, with particular focus on the eigenvalue clustering and the eigenvalue distribution. % Based on the eigenvalue distribution, we formulate a necessary condition for the convergence of the Discrete ALIF method. We also provide a simple criterion to construct appropriate filters satisfying this condition. % Our analysis, which represents a first step toward a complete mathematical understanding of the ALIF method, relies on recent matrix-theoretic results about sampling matrices and, above all, on the theory of generalized locally Toeplitz sequences, which we extend in this paper. % Several numerical examples in support of the theoretical analysis are presented.

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