On the filtering properties of the empirical mode decomposition

The empirical mode decomposition (EMD) based time-frequency analysis has been used in many scientific and engineering fields. The mathematical expression of EMD in the time-frequency-energy domain appears to be a generalization of the Fourier transform (FT), which leads to the speculation that the latter may be a special case of the former. On the other hand, the EMD is also known to behave like a dyadic filter bank when used to decompose white noise. These two observations seem to contradict each other. In this paper, we study the filtering properties of EMD, as its sifting number changes. Based on numerical results of the decompositions using EMD of a delta function and white noise, we conjecture that, as the (pre-assigned and fixed) sifting number is changed from a small number to infinity, the EMD corresponds to filter banks with a filtering ratio that changes accordingly from 2 (dyadic) to 1; the filter window does not narrow accordingly, as the sifting number increases. It is also demonstrated that the components of a delta function resulted from EMD with any prescribed sifting number can be rescaled to a single shape, a result similar to that from wavelet decomposition, although the shape changes, as the sifting number changes. These results will lead to further understandings of the relations of EMD to wavelet decomposition and FT.