  Time-frequency analysis for non-linear and non-stationary signals is
extraordinarily challenging. To capture features in these signals, it is
necessary for the analysis methods to be local, adaptive and stable. In recent
years, decomposition based analysis methods, such as the empirical mode
decomposition (EMD) technique pioneered by Huang et al., were developed by
different research groups. These methods decompose a signal into a finite
number of components on which the time-frequency analysis can be applied more
effectively.
  In this paper we consider the iterative filters (IFs) approach as an
alternative to EMD. We provide sufficient conditions on the filters that ensure
the convergence of IFs applied to any $L^2$ signal. Then we propose a new
technique, the Adaptive Local Iterative Filtering (ALIF) method, which uses the
IFs strategy together with an adaptive and data driven filter length selection
to achieve the decomposition. Furthermore we design smooth filters with compact
support from solutions of Fokker-Planck equations (FP filters) that can be used
within both IFs and ALIF methods. These filters fulfill the derived sufficient
conditions for the convergence of the IFs algorithm. Numerical examples are
given to demonstrate the performance and stability of IFs and ALIF techniques
with FP filters. In addition, in order to have a complete and truly local
analysis toolbox for non-linear and non-stationary signals, we propose a new
definition for the instantaneous frequency which depends exclusively on local
properties of a signal.
