On the convergence of iterative filtering empirical mode decomposition

Abstract

Empirical mode decomposition (EMD), an adaptive technique for data and signal decomposition, is a valuable tool for many applications in data and signal processing. One approach to EMD is the iterative filtering EMD, which iterates certain banded Toeplitz operators in l ^∞(ℤ). The convergence of iterative filtering is a challenging mathematical problem. In this chapter we study this problem, namely for a banded Toeplitz operator T and x∈l ^∞(ℤ) we study the convergence of T ⁿ(x). We also study some related spectral properties of these operators. Even though these operators don’t have any eigenvalue in Hilbert space l ²(ℤ), all eigenvalues and their associated eigenvectors are identified in l ^∞(ℤ) by using the Fourier transform on tempered distributions. The convergence of T ⁿ(x) relies on a careful localization of the generating function for T around their maximal points and detailed estimates on the contribution from the tails of x.