Conjectures on spectral properties of ALIF algorithm

Barbarino, Giovanni; Cicone, Antonio

doi:10.48550/arxiv.2009.00582

Cited by 1 publication

(2 citation statements)

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“…Nonetheless, we still cannot assert the absence of negative or complex eigenvalues which are not fulfilling the relation (5). Recent studies [10,6] show that for large n and continuous functions k(z), ℓ(x), almost all eigenvalues of the matrix K are real and nonnegative, but it is still not enough to establish the convergence of the method. Moreover, it has been ascertained experimentally that such cases may arise, especially whit a fast changing function ℓ(x).…”

Section: Discrete Alif and Stabilizationmentioning

confidence: 85%

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Stabilization and Variations to the Adaptive Local Iterative Filtering Algorithm: the Fast Resampled Iterative Filtering Method

Barbarino¹,

Cicone²

2021

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Non-stationary signals are ubiquitous in real life. Many techniques have been proposed in the last decades which allow decomposing multi-component signals into simple oscillatory mono-components, like the groundbreaking Empirical Mode Decomposition technique and the Iterative Filtering method. When a signal contains mono-components that have rapid varying instantaneous frequencies, we can think, for instance, to chirps or whistles, it becomes particularly hard for most techniques to properly factor out these components. The Adaptive Local Iterative Filtering technique has recently gained interest in many applied fields of research for being able to deal with non-stationary signals presenting amplitude and frequency modulation. In this work, we address the open question of how to guarantee a priori convergence of this technique, and propose two new algorithms. The first method, called Stable Adaptive Local Iterative Filtering, is a stabilized version of the Adaptive Local Iterative Filtering that we prove to be always convergent. The stability, however, comes at the cost of a higher complexity in the calculations. The second technique, called Resampled Iterative Filtering, is a new generalization of the Iterative Filtering method. We prove that Resampled Iterative Filtering is guaranteed to converge a priori for any kind of signal. Furthermore, in the discrete setting, by leveraging on the mathematical properties of the matrices involved, we show that its calculations can be accelerated drastically. Finally, we present some artificial and real-life examples to show the powerfulness and performance of the proposed methods.

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Section: Discrete Alif and Stabilizationmentioning

confidence: 85%

“…The method, albeit being very powerful and having already been utilized in a variety of applications, still lacks a theoretical analysis proving the convergence of (4), except in a few notorious cases [6,10,12]. In the next sections, we report some of the available convergence results for the discrete version of the algorithm.…”

Section: Linear Alifmentioning

confidence: 99%