A B-spline approach for empirical mode decompositions

Abstract

We propose an alternative B-spline approach for empirical mode decompositions for nonlinear and nonstationary signals. Motivated by this new approach, we derive recursive formulas of the Hilbert transform of B-splines and discuss Euler splines as spline intrinsic mode functions in the decomposition. We also develop the Bedrosian identity for signals having vanishing moments. We present numerical implementations of the B-spline algorithm for an earthquake signal and compare the numerical performance of this approach with that given by the standard empirical mode decomposition. Finally, we discuss several open mathematical problems related to the empirical mode decomposition.

Piecewise linear spectral sequences

“…Since g 0 = 0 and 1] , it is easy to see e 2πig0 , e 2πig1 = 0. We assume that e 2πigj , e 2πig l = 0, for j, l ∈ J n and show that it holds for j, l ∈ J n+1 .…”

“…Using assumption (2.3) and the definition of g n , it can be verified that Since the system e n , n ∈ Z, forms an orthonormal basis for both spaces L 2 (I) and L 2 [1,2], it follows from (2.4) that f = 0.…”

“…In particular, when a signal is non-linear and nonstationary, the classical Fourier basis does not give a satisfactory representation of such a signal [4]. Finding a better basis has been a main research subject in harmonic analysis, approximation theory and signal analysis, such as window Fourier bases, Gabor bases, wavelet bases (cf., [2,3,6,10]) and intrinsic mode functions recently studied in [1,4].…”

Algorithmic model development of B-spline non-linear electric signal decomposition

“…The difference of the developed algorithm from a classical one [2] is that identification of mode functions is made not by cubic splines, but by B-splines. The distinctive feature of Bsplines is formulation of smoothed diagrams allowing one to reduce errors, including mode function oscillations, and consequently to improve accuracy and effectiveness of spectral time analysis of non-linear and nonstationary processes [3,4]. The first operation is offered to carry out with the help of the parabolas method in «flexible» interval Ta [5], based on step-by-step interval interchange (sliding) of Ta along h j (t) with simultaneous extremum calculation at each step of interval transfer.…”

Aeroelastic Flight Data Analysis with the Hilbert-Huang Algorithm

“…This fundamental problem of the empirical mode decomposition has to be resolved since only with the intrinsic mode function can nonlinear distorted waveforms be resolved from nonlinear processes. There have been attempts to circumvent the mathematical difficulties in the EMD with some success by casting the IMFs in terms of B-splines, 3 and applying towards mechanical system fault-detection. 15 System identification of the IMFs as a multi-component system is suggestive in the light of multiresolution system identification procedures such as multiresolution singular value decompositions, Kalman filters, and subspace algorithms.…”