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

Abstract

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.

New theoretical insights in the decomposition and time-frequency representation of nonstationary signals: the IMFogram algorithm

“…The only alternative method proposed so far in the literature which is based on iterations, and hence does not require any a priori assumption on the signal under analysis, is Iterative Filtering (IF) algorithm [32], its fast implementation based on FFT, named Fast Iterative Filtering (FIF) [11], and their generalizations, the Adaptive Local Iterative Filtering (ALIF) and Resampled Iterative Filtering (RIF) algorithms [4,5] for the handling of signals containing strongly nonstationarities, like chirps, whistles and multipaths. These alternative iterative methods, although published only recently, have already been used effectively in a wide variety of applied fields like, for instance, in [57,45,30,37,47,35,38,41,46,40].…”