On intrinsic mode function

Empirical Mode Decomposition (EMD) has been widely used to analyze non-stationary and nonlinear signal by decomposing data into a series of intrinsic mode functions (IMFs) and a trend function through sifting processes. For lack of a firm mathematical foundation, the implementation of EMD is still empirical and ad hoc. In this paper, we prove mathematically that EMD, as practiced now, only gives an approximation to the true envelope. As a result, there is a potential conflict between the strict definition of IMF and its empirical implementation through natural cubic spline. It is found that the amplitude of IMF is closely connected with the interpolation function defining the upper and lower envelopes: adopting the cubic spline function, the upper (lower) envelope of the resulting IMF is proved to be a unitary cubic spline line as long as the extrema are sparsely distributed compared with the sampling data. Furthermore, when natural spline boundary condition is adopted, the unitary cubic spline line degenerates into a straight line. Unless the amplitude of the IMF is a strictly monotonic function, the slope of the straight line will be zero. It explains why the amplitude of IMF tends to be a constant with the number of sifting increasing ad infinitum. Therefore, to get physically meaningful IMFs the sifting times for each IMF should be kept low as in the practice of EMD. Strictly speaking, the resolution of these difficulties should be either to change the EMD implementation method and eschew the spline, or to define the stoppage criterion more objectively and leniently. Short of the full resolution of the conflict, we should realize that the EMD as implemented now yields an approximation with respect to cubic spline basis. We further concluded that a fixed low number of iterations would be the best option at this time, for it delivers the best approximation.