Elsevier

Signal Processing

Volume 91, Issue 12, December 2011, Pages 2783-2792
Signal Processing

Multivariate empirical mode decomposition and application to multichannel filtering

https://doi.org/10.1016/j.sigpro.2011.01.018Get rights and content

Abstract

Empirical Mode Decomposition (EMD) is an emerging topic in signal processing research, applied in various practical fields due in particular to its data-driven filter bank properties. In this paper, a novel EMD approach called X-EMD (eXtended-EMD) is proposed, which allows for a straightforward decomposition of mono- and multivariate signals without any change in the core of the algorithm. Qualitative results illustrate the good behavior of the proposed algorithm whatever the signal dimension is. Moreover, a comparative study of X-EMD with classical mono- and multivariate methods is presented and shows its competitiveness. Besides, we show that X-EMD extends the filter bank properties enjoyed by monovariate EMD to the case of multivariate EMD. Finally, a practical application on multichannel sleep recording is presented.

Introduction

The Empirical Mode Decomposition (EMD) is raising great interest since its first appearance in the nineties [1]. Indeed, it received wide attention in various fields such as biomedical engineering [2], [3], space research [4], hydrology [5], synthetic aperture radar [6], speech enhancement [7], watermarking [8], etc. EMD aims at decomposing sequentially a given signal into the sum of Amplitude and Frequency Modulated (AM/FM) zero mean oscillatory signals, called Intrinsic Mode Functions (IMFs), plus a non-zero mean low-degree polynomial remainder. Each IMF is computed by using an iterative procedure, named sifting process, applied to the residual multicomponent signal. One sifting iteration consists: (i) in computing the local mean of the residual signal and (ii) in subtracting it from the residue. This process is repeated until convergence to a designated IMF. Generally, the local mean is calculated as the half sum of the upper and the lower envelopes, obtained by interpolating between local minima and maxima, respectively, using cubic splines [1].

Since the original work on EMD [1], several studies have been presented. Some ones aim at giving a theoretical framework to monovariate EMD [9], [10], [11] in order to make easier its analysis. Other ones underline the behavior of original EMD [1] on simulated data such as its data-driven filter bank structure in the case of broadband noise [12], [13]. Eventually, EMD procedure devoted to bivariate [14], [15], [16], trivariate [17] and, recently, multivariate signals [18], [19], [20] has been proposed. The first bivariate approach in [14] uses the real parts of analytic and anti-analytic components of the bivariate signal which leads to two sets of complex-valued IMFs corresponding to the positive and negative frequency components of the signal, respectively. The second bivariate technique presented in [15] defines the extrema of a complex signal as points where the angle of the first derivative of the signal vanishes. In [16], two bivariate algorithms based on more geometrical considerations are developed. These two last approaches compute a local mean by averaging the monodimensional EMDs of projections of the original bivariate signal onto different angular planes. Rehman et al. proposed a trivariate approach [17] which is based on the computation of the extrema by projecting the input signal in multiple directions in 3D spaces via a quaternion rotation framework. The multivariate extension of EMD, developed in [18], can be seen as a generalization of the concept employed in Rilling's bivariate [16] and Rehman's trivariate [17] EMD. The mean envelope is obtained by averaging multiple envelopes, generated by taking projections of the multivariate signal along multiple directions on hyperspheres. Nevertheless, none of these techniques is a straightforward extension of the original monovariate EMD concept [1] to multivariate signals, say signals defined from R to RD with D>1. It is probably due to the difficulty of interpreting the notion of extremum when multivariate signals are considered.

In recent works [19], [20], we proposed an EMD algorithm, called 2T-EMD (Turning Tangent EMD), able to process mono- and multivariate signals whatever the signal dimension D is, without any change in the algorithm. In addition, 2T-EMD algorithm seems to require a smaller amount of calculations than classical EMD algorithms. To do so, the signal mean trend is redefined as the signal obtained by interpolating the barycenters of particular oscillations called elementary oscillations (see Section II of [19], [20]). More precisely, 2T-EMD algorithm is based on two main steps: (i) identification of all elementary oscillations and (ii) computation of the barycenters of each associated elementary oscillations and interpolation between all these barycenters to obtain the signal mean trend. Recall that, in [19], [20], an elementary oscillation of a given function s with values in RD (D1) is considered as a piece of s defined between two consecutive oscillation extrema of s.

In this paper, we propose to compute the local mean trend without calculation of the oscillation barycenters (as proposed in 2T-EMD), giving rise to a new multivariate EMD algorithm, named X-EMD (eXtended EMD). Indeed, in X-EMD algorithm (as described in Section 2), the local mean is directly computed by averaging two envelopes generated directly from oscillation extrema. In other words, the exploitation of oscillation extrema is clearly different between X-EMD and 2T-EMD algorithms and makes X-EMD algorithm easier both to implement and to use. The numerical complexity of the proposed approach is then analyzed in Section 3 and compared with the complexity of four existing EMD methods. A performance study is proposed in Section 4 on synthetic mono-, bi-, and trivariate signals showing: (i) the competitive behavior of X-EMD versus existing methods, and (ii) its ability to process any output signal dimension. Finally, data-driven spectrum analyzer properties satisfied by X-EMD are highlighted and a biomedical application is considered in Section 5. Indeed, empirical filter bank structure enjoyed by monovariate EMD [13] is shown to be preserved by X-EMD in multivariate context. Such results are used to analyze a quadrivariate ElectroEncephaloGraphic (EEG) sleep recording.

Section snippets

An eXtended EMD approach: X-EMD

The proposed X-EMD method aims at providing a unique tool able to process mono- and multivariate signals without any modification. The notion of oscillation extrema, originally presented in [19], [20], is exploited to redefine the local mean operator. However, before going further, let us briefly recall the definition of oscillation extrema.

Oscillation extremum: In [1], the local mean is given by taking average of the upper and the lower envelopes, which are computed by interpolating between

Numerical complexity analysis

This section aims at relating the X-EMD algorithm and some existing EMD methods from a computational complexity point of view. Numerical complexity is calculated in terms of number of floating point operations (flops). A flop corresponds to a multiplication followed by an addition. But in practice, only multiplications are counted since, in general, there are about as many (and slightly more) multiplications as additions. The first considered EMD method is Huang's solution [1] and will be named

Performance evaluation of X-EMD

The objective of this section is to compare the performance of X-EMD algorithm with classical approaches and to illustrate its ability to process multivariate signals whatever their output dimension. Before applying X-EMD on test signals, it is prudent to give some details about the set of signals that X-EMD can decompose successfully. As previously mentioned, the considered signal has to be in class C1 or at least, in the case of irregular signals, an appropriate numerical estimation of the

Filter bank structure and application to multichannel sleep recording

One important property of the classical monovariate EMD [1] especially highlighted in [12] is its empirical filter bank structure observed when EMD is applied on broadband noises.

Filter bank structure: The goal of this section is to show how this property is preserved by X-EMD for higher output signal dimensions. Nevertheless, the notion of frequential content for multivariate signals may be ambiguous. As suggested in [13], one alternative way to deal with IMFs frequency in a monovariate

Conclusion and perspectives

A new algorithm called X-EMD based on a redefinition of the local mean operator is proposed. The obtained results show that, under certain assumptions on the processed signal, this alternative definition enables to decompose both mono- and multivariate signals without any modification in the core of the algorithm. This last point is the main difference regarding the existing approaches of the literature. The comparative study also suggests that X-EMD seems to offer competitive performance

Acknowledgments

We acknowledge all the reviewers and the associate editor for their valuable comments. This work is supported by the French Government under an ANR Contract, namely mv-EMD (BLAN07-0314-02).

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