One or two frequencies? The Iterative Filtering answers

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.