Signal analysis via instantaneous frequency estimation of signal components

Original Paper


The empirical mode decomposition (EMD) algorithm, introduced by Huang et al. (Proc Roy Soc Lond Ser A Math Phys Eng Sci 454(1971):903–995, 1998), is arguably the most popular mathematical scheme for non-stationary signal decomposition and analysis. The objective of EMD is to separate a given signal into a number of components, called intrinsic mode functions (IMF’s) after which the instantaneous frequency (IF) and amplitude of each IMF are computed through Hilbert spectral analysis (HSA). On the other hand, the synchrosqueezed wavelet transform (SST), introduced by Daubechies and Maes (Wavelets in Medicine and Biology, pp. 527–546, 1996) and further developed by Daubechies et al. (Appl Comput Harmon Anal 30:243–261, 2011), is applied to estimate the IF’s of all signal components of the given signal, based on one single reference “IF function”, under the assumption that the signal components satisfy certain strict properties of a so-called adaptive harmonic model, before the signal components of the model are recovered. The objective of our paper is to develop a hybrid EMD-SST computational scheme by applying a “modified SST” to each IMF of the EMD, as an alternative approach to the original EMD-HSA method. While our modified SST assures non-negative instantaneous frequencies of the IMF’s, application of the EMD scheme eliminates the dependence on a single reference IF value as well as the guessing work of the number of signal components in the original SST approach. Our modification of the SST consists of applying vanishing moment wavelets (introduced in a recent paper by C.K. Chui, Y.-T. Lin and H.-T. Wu) with stacked knots to process signals on bounded or half-infinite time intervals, and spline curve fitting with optimal smoothing parameter selection through generalized cross-validation. In addition, we formulate a local cubic spline interpolation scheme for real-time realization of the EMD sifting process that improves over the standard global cubic spline interpolation, both in quality and computational cost, particularly when applied to bounded and half-infinite time intervals.


Time–frequency analysis Instantaneous frequency Empirical mode decomposition Intrinsic mode functions Synchrosqueezing wavelet transform Hilbert transform Hilbert spectrum Adaptive harmonic model Local spline interpolation 

Mathematics Subject Classification

94A12 41A05 41A15 



The research of Charles K. Chui was supported by ARO Grant # W911 NF-11-1-0426.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of StatisticsStanford UniversityStanfordUSA
  2. 2.Department of Mathematics and Computer ScienceUniversity of Missouri-St. LouisSt. LouisUSA

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