A PDE characterization of the intrinsic mode functions

For the first time, a proof of the sifting process (SP) and so the empirical mode decomposition (EMD), is given. For doing this, lower and upper envelopes are modeled in a more convenient way that helps us prove the convergence of the SP towards a solution of a partial differential equation (PDE). We also prove that such a PDE has a unique solution, which ensures the uniqueness of the EMD decomposition. The new formulation of envelopes has another benefit. In fact, it removes interpolation problems and related issues. Not only helps the modelization of envelopes to give a mathematical framework on the EMD, but also, as confirmed by the numerical simulations, the PDE-based EMD improves a lot the classical EMD.