Wavelet analysis of non-stationary signals with applications

The empirical mode decomposition (EMD) algorithm, introduced by N.E. Huang et al in 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 I. Daubechies and S. Maes in 1996 and further developed by I. Daubechies, J. Lu and H.-T. Wu in 2011, is rst applied to estimate the IF's of all signal components of the given signal, based on one single frequency reassignment rule, under the assumption that the signal components satisfy certain strict properties of the so-called adaptive harmonic model, before the signal components of the model are recovered, based on the estimated IF's. The objective of this dissertation is to develop a hybrid EMDSST computational scheme by applying a modi ed SST to each IMF produced by a modi ed EMD, as an alternative approach to the original EMD-HSA method. While our modi ed SST assures nonnegative instantaneous frequencies of the IMF's, application of the EMD scheme eliminates the dependence on a single frequency reassignment rule as well as the guessing work of the number of signal components in the original SST approach. Our modi cation of the SST consists of applying analytic 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-in nite time intervals, and spline curve tting with optimal smoothing parameter selection through generalized cross-validation. In addition, we modify EMD by formulating a local spline interpolation scheme for bounded intervals, for real-time realization of the EMD sifting process. This scheme improves over the standard global cubic spline interpolation, both in quality and computational cost, particularly when applied to bounded and half-in nite time intervals. Van der Walt, Maria, 2015, UMSL, p.ii

[1]  Vassilis G. Papanicolaou,et al.  Some Results on Ordinary Differential Operators with Periodic Coefficients , 2014, 1412.5423.

[2]  Charles K. Chui,et al.  Real-time dynamics acquisition from irregular samples -- with application to anesthesia evaluation , 2014, 1406.1276.

[3]  C. D. Boor,et al.  Spline approximation by quasiinterpolants , 1973 .

[4]  C. D. Boor,et al.  On Calculating B-splines , 1972 .

[5]  Ingrid Daubechies,et al.  A Nonlinear Squeezing of the Continuous Wavelet Transform Based on Auditory Nerve Models , 2017 .

[6]  N. Huang,et al.  The empirical mode decomposition and the Hilbert spectrum for nonlinear and non-stationary time series analysis , 1998, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[7]  Charles K. Chui,et al.  Signal decomposition and analysis via extraction of frequencies , 2016 .

[8]  Boualem Boashash,et al.  Time Frequency Analysis , 2003 .

[9]  S. S. Shen,et al.  A confidence limit for the empirical mode decomposition and Hilbert spectral analysis , 2003, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[10]  J. D. Villiers Mathematics of Approximation , 2012 .

[11]  J. Mayer,et al.  On the Quantum Correction for Thermodynamic Equilibrium , 1947 .

[12]  Patrick Flandrin,et al.  Improving the readability of time-frequency and time-scale representations by the reassignment method , 1995, IEEE Trans. Signal Process..

[13]  Charles K. Chui,et al.  A General framework for local interpolation , 1990 .

[14]  François Auger,et al.  Time-Frequency Reassignment: From Principles to Algorithms , 2018, Applications in Time-Frequency Signal Processing.

[15]  Norden E. Huang,et al.  On Instantaneous Frequency , 2009, Adv. Data Sci. Adapt. Anal..

[16]  R. Sharpley,et al.  Analysis of the Intrinsic Mode Functions , 2006 .

[17]  E. Bedrosian A Product Theorem for Hilbert Transforms , 1963 .

[18]  Karlheinz Gröchenig,et al.  Foundations of Time-Frequency Analysis , 2000, Applied and numerical harmonic analysis.

[19]  Ruqiang Yan,et al.  Wavelets: Theory and Applications for Manufacturing , 2010 .

[20]  L. Schumaker Spline Functions: Basic Theory , 1981 .

[21]  Hau-Tieng Wu,et al.  Non‐parametric and adaptive modelling of dynamic periodicity and trend with heteroscedastic and dependent errors , 2014 .

[22]  Hau-tieng Wu,et al.  Instantaneous frequency and wave shape functions (I) , 2011, 1104.2365.

[23]  Patrick Flandrin,et al.  Time-Frequency/Time-Scale Analysis , 1998 .

[24]  F. Hlawatsch,et al.  Linear and quadratic time-frequency signal representations , 1992, IEEE Signal Processing Magazine.

[25]  Sylvain Meignen,et al.  Time-Frequency Reassignment and Synchrosqueezing: An Overview , 2013, IEEE Signal Processing Magazine.

[26]  Yuesheng Xu,et al.  A B-spline approach for empirical mode decompositions , 2006, Adv. Comput. Math..

[27]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[28]  Norden E. Huang,et al.  A review on Hilbert‐Huang transform: Method and its applications to geophysical studies , 2008 .

[29]  Peter Craven,et al.  Smoothing noisy data with spline functions , 1978 .

[30]  Xianhong Xie,et al.  Optimal spline smoothing of fMRI time series by generalized cross-validation , 2003, NeuroImage.

[31]  Gabriel Rilling,et al.  EMD Equivalent Filter Banks, from Interpretation to Applications , 2005 .

[32]  Dennis Gabor,et al.  Theory of communication , 1946 .

[33]  N. Huang,et al.  A new view of nonlinear water waves: the Hilbert spectrum , 1999 .

[34]  Hau-Tieng Wu,et al.  Synchrosqueezing-Based Recovery of Instantaneous Frequency from Nonuniform Samples , 2010, SIAM J. Math. Anal..

[35]  I. J. Schoenberg On Pólya Frequency Functions , 1988 .

[36]  Carl de Boor,et al.  A Practical Guide to Splines , 1978, Applied Mathematical Sciences.

[37]  I. Daubechies,et al.  Synchrosqueezed wavelet transforms: An empirical mode decomposition-like tool , 2011 .

[38]  G. Hardy ON HILBERT TRANSFORMS , 1932 .

[39]  N. Huang,et al.  A study of the characteristics of white noise using the empirical mode decomposition method , 2004, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[40]  C. Chui Wavelets: A Mathematical Tool for Signal Analysis , 1997 .

[41]  Norden E. Huang,et al.  Ensemble Empirical Mode Decomposition: a Noise-Assisted Data Analysis Method , 2009, Adv. Data Sci. Adapt. Anal..

[42]  Gene H. Golub,et al.  Generalized cross-validation as a method for choosing a good ridge parameter , 1979, Milestones in Matrix Computation.

[43]  Qingtang Jiang,et al.  Applied Mathematics: Data Compression, Spectral Methods, Fourier Analysis, Wavelets, and Applications , 2013 .