  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 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.
