Extracting oscillating components from nonstationary time series: A wavelet-induced method

This paper consists in the description and application of a method called wavelet-induced mode extraction (WIME) in the context of time-frequency analysis. WIME aims to extract the oscillating components that build amplitude modulated-frequency modulated signals. The essence of this technique relies on the successive extractions of the dominant ridges of wavelet-based time-frequency representations of the signal under consideration. Our tests on simulated examples indicate strong decomposition and reconstruction skills, trouble-free handling of crossing trajectories in the time-frequency plane, sharp performances in frequency detection in the case of mode-mixing problems, and a natural tolerance to noise. These results are compared with those obtained with empirical mode decomposition. We also show that WIME still gives meaningful results with real-life data, namely, the Oceanic Niño Index.

Figure 1

(a) Original signal $f$ (blue) and reconstructed signal (red) of Sec. 3a; (b) $(t,a)\mapsto |W_f(t,a)|$ (stands for the TF representation of $f$) and ridge detected (black line); (c) wavelet spectrum of $f$ and its segmentation (red lines) to isolate the highest local maximum (used to set a starting point for the ridge); (d) first component $c_1$ extracted following the ridge; (e) original first component $f_1$; figures (f) to (i) are the same figures with $f-c_1$ instead of $f$. Clearly, $c_1$ and $c_2$ successively extracted match $f_1$ and $f_2$.

Figure 2

As for Fig. 1, (a) displays the original signal $f$ (blue) and the reconstructed one (red) of Sec. 3b, and the other rows are, for the successive signals under consideration: (b), (f), (j) $|W_f(t,a)|$ and ridges (black line); (c), (g), (k) wavelet spectra; (d), (h), (l) components extracted following the ridges; (e), (i), (m) original components. The influence of the multiple crossings in the TF plane remains limited because WIME respects the energy-based hierarchical order of the components.

Figure 3

(a), (c), (e) IMFs extracted with EMD from the signal of Sec. 3b; (b), (d), (f) their TF representation obtained with the CWT. It can be seen that EMD works following a decreasing-frequency based approach, regardless the amplitudes of the components or IMFs to be extracted. Consequently, the IMFs only match the original components piecewise.

Figure 4

As previously, (a) displays the original signal $f$ (blue) and the reconstructed one (red) (Sec. 3c); the other rows are, for the successive signals under consideration: (b), (f), (j), (n) $|W_f(t,a)|$ and ridges (black line); (c), (g), (k), (o) wavelet spectra; (d), (h), (l), (p) components extracted following the ridges; (e), (i), (m), (q) original components. The period detection skill of WIME is high.

Figure 5

IMFs obtained with EMD from the signal of Sec. 3c. They are considerably different from the original components; the first IMF itself (a) represents almost the whole signal. The estimated frequencies $[\approx$ 1/41 (a), 1/75 (b), 1/165 (c), 1/284 (d) Hz] are far from the targets ($\approx$ 1/23, 1/31, 1/47, 1/65).

Figure 6

(a) Original chirp (red) and extracted component (blue) associated with the ridge of the TF representation of the signal (b). Remaining rows: the same with $f+X$ (c), (d), $f+2X$ (e), (f), $f+3X$ (g), (h), where $X$ is a Gaussian white noise with zero mean and variance 1. One can see that, despite the high level of noise, WIME is still able to retrieve the initial chirp. Moreover, it does not require any pre- or post-processing treatment or extra computational power.

Figure 7

(a) Original ONI signal (red) and reconstructed signal (blue); (b) the difference between them. Then the first six successive TF representations (c), (e), (g), (i), (k), (m) and components (d), (f), (h), (j), (l), (n) obtained with WIME.