Patrick Flandrin

Empirical mode decomposition as a filter bank

Empirical mode decomposition (EMD) has recently been pioneered by Huang et al. for adaptively representing nonstationary signals as sums of zero-mean amplitude modulation frequency modulation components. In order to better understand the way EMD behaves in stochastic situations involving broadband noise, we report here on numerical experiments based on fractional Gaussian noise. In such a case, it turns out that EMD acts essentially as a dyadic filter bank resembling those involved in wavelet decompositions. It is also pointed out that the hierarchy of the extracted modes may be similarly exploited for getting access to the Hurst exponent.

Improving the readability of time-frequency and time-scale representations by the reassignment method

In this paper, the use of the reassignment method, first applied by Kodera, Gendrin, and de Villedary (1976) to the spectrogram, is generalized to any bilinear time-frequency or time-scale distribution. This method creates a modified version of a representation by moving its values away from where they are computed, so as to produce a better localization of the signal components. We first propose a new formulation of this method, followed by a thorough theoretical study of its characteristics. Its practical use for a large variety of known time-frequency and time-scale distributions is then addressed. Finally, some experimental results are reported to demonstrate the performance of this method. >

Wigner-Ville spectral analysis of nonstationary processes

The Wigner-Ville spectrum has been recently introduced as the unique generalized spectrum for time-varying spectral analysis. Its properties are revised with emphasis on its central role in the analysis of second-order properties of nonstationary random signals. We propose here a general class of spectral estimators of the Wigner-Ville spectrum: this class is based on arbitrarily weighted covariance estimators and its formal description corresponds to the general class of conjoint time-frequency representations of deterministic signals with finite energy. Classical estimators like short-time periodograms and the recently introduced pseudo-Wigner estimators are shown to be special cases of the general class. The generalized framework allows the calculation of the moments of general spectral estimators and comparing the results emphasizes the versatility of the new pseudo-Wigner estimators. The effective numerical implementation, by an N-point FFT, of pseudo-Wigner estimators of 2N points is indicated and various examples are given.

On the spectrum of fractional Brownian motions

Fractional Brownian motions (FBMs) provide useful models for a number of physical phenomena whose empirical spectra obey power laws of fractional order. However, due to the nonstationary nature of these processes, the precise meaning of such spectra remains generally unclear. Two complementary approaches are proposed which are intended to clarify this point. The first one, based on a time-frequency analysis, takes into account the nonstationary nature of FBM and puts emphasis on time-averaged measurements; the second one, based on a time-scale analysis, is matched to self-similarity properties of FBM and reveals an underlying stationary structure relative to each time-scaling. >

One or Two Frequencies? The Empirical Mode Decomposition Answers

This paper investigates how the empirical mode decomposition (EMD), a fully data-driven technique recently introduced for decomposing any oscillatory waveform into zero-mean components, behaves in the case of a composite two-tones signal. Essentially two regimes are shown to exist, depending on whether the amplitude ratio of the tones is greater or smaller than unity, and the corresponding resolution properties of the EMD turn out to be in good agreement with intuition and physical interpretation. A refined analysis is provided for quantifying the observed behaviors and theoretical claims are supported by numerical experiments. The analysis is then extended to a nonlinear model where the same two regimes are shown to exist and the resolution properties of the EMD are assessed.

A complete ensemble empirical mode decomposition with adaptive noise

In this paper an algorithm based on the ensemble empirical mode decomposition (EEMD) is presented. The key idea on the EEMD relies on averaging the modes obtained by EMD applied to several realizations of Gaussian white noise added to the original signal. The resulting decomposition solves the EMD mode mixing problem, however it introduces new ones. In the method here proposed, a particular noise is added at each stage of the decomposition and a unique residue is computed to obtain each mode. The resulting decomposition is complete, with a numerically negligible error. Two examples are presented: a discrete Dirac delta function and an electrocardiogram signal. The results show that, compared with EEMD, the new method here presented also provides a better spectral separation of the modes and a lesser number of sifting iterations is needed, reducing the computational cost.

Bivariate Empirical Mode Decomposition

The empirical mode decomposition (EMD) has been introduced quite recently to adaptively decompose nonstationary and/or nonlinear time series. The method being initially limited to real-valued time series, we propose here an extension to bivariate (or complex-valued) time series that generalizes the rationale underlying the EMD to the bivariate framework. Where the EMD extracts zero-mean oscillating components, the proposed bivariate extension is designed to extract zero-mean rotating components. The method is illustrated on a real-world signal, and properties of the output components are discussed. Free Matlab/C codes are available at http://perso.ens-lyon.fr/patrick.flandrin.

Time-scale energy distributions: a general class extending wavelet transforms

The theory of a new general class of signal energy representations depending on time and scale is developed. Time-scale analysis has been introduced recently as a powerful tool through linear representations called (continuous) wavelet transforms (WTs), a concept for which an exhaustive bilinear generalization is given. Although time scale is presented as an alternative method to time frequency, strong links relating the two are emphasized, thus combining both descriptions into a unified perspective. The authors provide a full characterization of the new class: the result is expressed as an affine smoothing of the Wigner-Ville distribution, on which interesting properties may be further imposed through proper choices of the smoothing function parameters. Not only do specific choices allow recovery of known definitions, but they also provide, via separable smoothing, a continuous transition from Wigner-Ville to either spectrograms or scalograms (squared modulus of the WT). This property makes time-scale representations a very flexible tool for nonstationary signal analysis. >

Multiscale nature of network traffic

The complexity and richness of telecommunications traffic is such that one may despair to find any regularity or explanatory principles. Nonetheless, the discovery of scaling behavior in teletraffic has provided hope that parsimonious models can be found. The statistics of scaling behavior present many challenges, especially in nonstationary environments. In this article, we overview the state of the art in this area, focusing on the capabilities of the wavelet transform as a key tool for unraveling the mysteries of traffic statistics and dynamics.

Some features of time-frequency representations of multicomponent signals

The Wigner-Ville Distribution (WVD) is now known to be a convenient tool for the time-frequency analysis of non-stationary signals, and especially monocomponent ones. However, in the case of multicomponent signals, its bilinear structure is also known to create cross-terms without any physical significance. Starting with the general formulation of time-frequency representations, which only depend on an arbitrary kernel function, we first characterize properties of such cross-terms and then propose appropriate smoothings of the WVD in order to reduce their influence. Such suitable and versatile approximations are compared on synthetic and natural signals and an extension to time-frequency filtering is proposed.