On the Behavior of MEMD in Presence of Multivariate Fractional Gaussian Noise

Multivariate empirical mode decomposition (MEMD) has been introduced to make standard EMD suitable for direct multichannel signals processing. Unlike EMD, MEMD is able to align sifted intrinsic mode functions (IMFs) from multiple data channels. The aim of this work is to analyze the behavior of MEMD under multivariate fGn (MfGn) with different Hurst exponents, <inline-formula><tex-math notation="LaTeX">$H$</tex-math></inline-formula>, and strengths of link between pairs of channels, <inline-formula><tex-math notation="LaTeX">$\rho$</tex-math></inline-formula>, of each sifted IMF. We report results supporting the claim that, regardless of <inline-formula><tex-math notation="LaTeX">$\rho$</tex-math></inline-formula> values and for both MfGns long-range and short-range dependent, MEMD acts as filter bank on each channel of the input multivariate signal. Whatever the <inline-formula><tex-math notation="LaTeX">$\rho$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$H$</tex-math></inline-formula> values, this equivalent filter bank structure is dyadic with constant-Q band-pass filters. The observed self-similar filter bank structure leads to a deeper statistical studies of the variance distribution and zero-crossings alignment in order to express this self-similarity in terms of spectral densities of multidimensional IMFs. These statistical properties generalize what was previously conducted for EMD to MEMD and estimation strategy of <inline-formula><tex-math notation="LaTeX">$H$</tex-math></inline-formula> exponent is proposed. The filter bank behavior of MEMD is illustrated on real turbulent flow data and the estimated <inline-formula><tex-math notation="LaTeX">$H$</tex-math></inline-formula> exponent brings out the long-range-dependent nature of the turbulent flow data. An application to EEG data is also proposed.

[1]  Jean-Christophe Cexus,et al.  Analyse des échos de cibles Sonar par Transformation de Huang-Teager (THT)* , 2008 .

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

[3]  Gabriel Rilling,et al.  Empirical mode decomposition as a filter bank , 2004, IEEE Signal Processing Letters.

[4]  Jean-Luc Starck,et al.  Astronomical image and data analysis , 2002 .

[5]  Anne Philippe,et al.  Basic properties of the Multivariate Fractional Brownian Motion , 2010, 1007.0828.

[6]  Patrice Abry,et al.  A Wavelet-Based Joint Estimator of the Parameters of Long-Range Dependence , 1999, IEEE Trans. Inf. Theory.

[7]  Gabriel Rilling,et al.  Empirical mode decomposition, fractional Gaussian noise and Hurst exponent estimation , 2005, Proceedings. (ICASSP '05). IEEE International Conference on Acoustics, Speech, and Signal Processing, 2005..

[8]  Abdel-Ouahab Boudraa,et al.  Audio Watermarking Via EMD , 2013, IEEE Transactions on Audio, Speech, and Language Processing.

[9]  Abdel-Ouahab Boudraa,et al.  EMD-Based Filtering Using Similarity Measure Between Probability Density Functions of IMFs , 2014, IEEE Transactions on Instrumentation and Measurement.

[10]  Abdel-Ouahab Boudraa,et al.  Speech enhancement using empirical mode decomposition and the Teager-Kaiser energy operator. , 2014, The Journal of the Acoustical Society of America.

[11]  Ali Mansour,et al.  Blind separation of ECG signals from noisy signals affected by electrosurgical artifacts , 2020 .

[12]  Jianjun Cui,et al.  Equidistribution on the Sphere , 1997, SIAM J. Sci. Comput..

[13]  Gabriel Rilling,et al.  Bivariate Empirical Mode Decomposition , 2007, IEEE Signal Processing Letters.

[14]  Abdel-Ouahab Boudraa,et al.  On signals compression by EMD , 2012 .

[15]  Danilo P. Mandic,et al.  Empirical Mode Decomposition for Trivariate Signals , 2010, IEEE Transactions on Signal Processing.

[16]  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.

[17]  Bruno Torrésani,et al.  HHT-based audio coding , 2015, Signal Image Video Process..

[18]  E. Bullmore,et al.  A Resilient, Low-Frequency, Small-World Human Brain Functional Network with Highly Connected Association Cortical Hubs , 2006, The Journal of Neuroscience.

[19]  Toshihisa Tanaka,et al.  Complex Empirical Mode Decomposition , 2007, IEEE Signal Processing Letters.

[20]  Danilo P. Mandic,et al.  Empirical Mode Decomposition-Based Time-Frequency Analysis of Multivariate Signals: The Power of Adaptive Data Analysis , 2013, IEEE Signal Processing Magazine.

[21]  Muhammad Altaf,et al.  Rotation Invariant Complex Empirical Mode Decomposition , 2007, 2007 IEEE International Conference on Acoustics, Speech and Signal Processing - ICASSP '07.

[22]  Abdel-Ouahab Boudraa,et al.  Nonstationary signals analysis by Teager-Huang Transform (THT) , 2006, 2006 14th European Signal Processing Conference.

[23]  B. Mandelbrot,et al.  Fractional Brownian Motions, Fractional Noises and Applications , 1968 .

[24]  Thierry Chonavel,et al.  Audio coding via EMD , 2020, Digit. Signal Process..

[25]  Pierre-Olivier Amblard,et al.  On multivariate fractional brownian motion and multivariate fractional Gaussian noise , 2010, 2010 18th European Signal Processing Conference.

[26]  Jean-Yves Billard,et al.  An Experimental Study of Unsteady Partial Cavitation , 2004 .

[27]  Danilo P. Mandic,et al.  Filter Bank Property of Multivariate Empirical Mode Decomposition , 2011, IEEE Transactions on Signal Processing.

[28]  D. P. Mandic,et al.  Multivariate empirical mode decomposition , 2010, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[29]  Hui Tian,et al.  A Study of the Characteristics of MEMD for Fractional Gaussian Noise , 2016, IEICE Trans. Fundam. Electron. Commun. Comput. Sci..

[30]  Danilo P. Mandic,et al.  Emd via mEMD: multivariate noise-Aided Computation of Standard EMD , 2013, Adv. Data Sci. Adapt. Anal..

[31]  S. Hambric,et al.  Low-wavenumber turbulent boundary layer wall-pressure measurements from vibration data on a cylinder in pipe flow , 2010 .

[32]  Pierre-Olivier Amblard,et al.  Identification of the Multivariate Fractional Brownian Motion , 2011, IEEE Transactions on Signal Processing.

[33]  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.

[34]  Abdel-Ouahab Boudraa,et al.  On the Behavior of EMD and MEMD in Presence of Symmetric $\alpha $ -Stable Noise , 2015, IEEE Signal Processing Letters.

[35]  Abdel-Ouahab Boudraa,et al.  EMD-Based Signal Filtering , 2007, IEEE Transactions on Instrumentation and Measurement.