Time-frequency distribution has received a growing utilization for analysis and interpretation of nonlinear and nonstationary processes in a variety of fields. Among them, two methods, such as, the empirical mode decomposition (EMD) with Hilbert transform (HT) which is termed as the Hilbert-Huang Transform (HHT) and the Hilbert spectrum based on maximal overlap discrete wavelet package transform (MODWPT), are fairly noteworthy. Comparisons of HHT and MODWPT in analyzing several typical nonlinear systems and examinations of the effectiveness using these two methods are illustrated. This study demonstrates that HHT can provide comparatively more accurate identifications of nonlinear systems than MODWPT.

Time-frequency (TF) analysis has experienced a number of qualitative and quantitative changes during the last three decades, and has gradually received growing attentions and further applications in a variety of fields such as radar, water waves [

Huang and Shen put forward that all the analyses in terms of a priori established basis have drained all the physics out of the analyzed results, because any a priori basis could not possibly fit all the variety of data from different driving mechanisms [

Recently, the EMD algorithm, acting as a manner with highly data-driven characteristic of data decomposition, plays a role of either nonlinear or nonstationary processes. With these nice properties, the EMD has been used to calculate the Hurst index of long-range dependence processes (LRD) and network traffic [

Wavelet transforms are one of the fast-evolving mathematical and signal processing tools [

Duffing equation, Lorenz system, and Rössler system are three of the typical nonlinear examples. The network traffic data is also one kind of practical nonlinear processes [

The development of HHT provides an alternative view of the time-frequency-energy paradigm of nonlinear and nonstationary data. To examine data from real-world nonlinear and nonstationary processes, the detailed dynamics in the processes from the data need to be determined because the intrawave frequency modulation, which indicates the instantaneous frequency changes within one oscillation cycle, is one of the typical characteristics of a nonlinear system. As Huang et al. [

The purpose is to separate function

Physically speaking, the necessary conditions to define a meaningful instantaneous frequency are that the signal must be symmetric concerning the local zero mean, and have the same numbers of zero crossings and extrema. This means that, in an IMF function, the number of extrema and the number of zero crossings must be either equal or different at most by one in the whole data set, and the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero at every point. All these conditions are so strict that the determined IMF may not satisfy them precisely. Consequently, the resultant IMF is nearly a monocomponent function.

The EMD is developed based on the assumption that any signal consists of a set of different IMFs. The procedures to decompose signal

Find all the local maxima from

Find all the local minima from

Let the mean

Subtract the difference

Ideally, the difference

Repeat the sifting process

The criterion suggested by Huang for stopping the sifting process is

Remove

Summing up all the IMFs and the final residue, we should be able to reconstruct the original signal

Then, the HT of

Let

Assume that we sample a continuous-time signal at intervals

The maximal overlap discrete wavelet transform (MODWT) can be considered as a revised version of the DWT [

The MODWT creates new filters at each stage by inserting

The coefficients at level

For any signal, the analytic form can be represented as

Generally speaking, the instantaneous frequency changes within one oscillation cycle for nonlinear systems, and can be used to describe intrawave frequency modulation. To discuss the characteristics of data from a nonlinear system by EMD + HT and MODWPT, we will take several typical examples to aid the discussions.

The Duffing oscillator under harmonic excitation described by a second-order differential equation is one of the well-known nonlinear examples

Embodied by a Duffing oscillator with

(a) Numerical solution of the Duffing equation (200s). (b) Power spectral density estimate.

After subjected to the EMD, the numerical result yields the IMF components and the corresponding Fourier spectrum of each IMFs as shown in Figure

The IMF components of the Duffing equation from EMD in (a) and the corresponding power spectrum in (b).

Diagrams at left represent the HHT result and the MODWPT result (at level 3) of the Duffing equation separately; the corresponding Hilbert Marginal spectrum results of EMD and MODWPT are on the right side.

The Lorenz system has also been widely studied and is described by

The numerical solution of the Lorenz equation (a) and the Fourier spectrum of the given

The IMF components of the Lorenz equation from EMD in (a) and the corresponding power spectrum in (b).

The HHT spectrum (a) and the MODWPT spectrum at level 3 (b) for the Lorenz equation solution.

In the diagram of HHT, the transient nature of both components is perfectly located, with the main component being intrawave modulated and a fairly clear indication of the nonlinear effect of the oscillation. Comparatively speaking, the result of MODWPT is not so satisfactory. Only some blurry frequency components can be recognized near 1.4 Hz. The oscillation of the frequency, supposed to be demonstrating the nonlinearity, is not represented here in the spectrum of the MODWPT.

Another example investigated here is the Rössler System denoted as

The waveform of the

The result of the Fourier spectrum of

Comparison of the TF distribution of Rossler equation using HHT and MODWPT (level 3) on left and the corresponding marginal spectrum on right.

This paper focuses on the performance of nonlinear processes using the HHT and the MODWPT with the aim of bringing a better choice of time-frequency decomposition in nonlinear data analysis. Compared with the HHT, the MODWPT shows some weakness in nonlinear data analysis and the study of identifying the computational burden required by these two methods is still at the exploratory stage.

HHT, which decomposes data through EMD, offers a potentially viable method for nonlinear and nonstationary data analysis. Verified by three typical nonlinear systems, the HHT can not only perform and locate main frequency components but also force function frequency details. Furthermore, the intrawave modulation, which is the important characteristic of nonlinear system, can as well be obtained in the distribution of HHT. Compared with the MODWPT that decomposes data into a number of components alike, the EMD gives results much sharper and more supportable to nonlinear system identification.

This work was supported in part by the National Natural Science Foundation of China (NSFC) under the project Grants numbers 60573125 and 60873264.