Notes

2.2 Empirical Wavelet Transform
Section
EWT [17] is a newly proposed technique to adaptively detect the different modes of the signal and accordingly build the empirical wavelets to represent the signal by different modes detected. Empirical wavelets means building a set of wavelets adapted to the processed signal, i.e. in Fourier domain means building a set of band-pass filters. Adaptation here lies in detecting filter's supports according to the information located in the processed signal. Modes can be thought of as the principal components (referred to as amplitude modulated--frequency modulated (AM-FM) components [18]) of the signal which represent the signal completely. Following are the steps involved in EWT – Fourier transform and segmentation, filter construction, and empirical transform.
The adaptability in this transform is provided by the segmentation of Fourier axis. Segmentation of Fourier axis is done in a way so as to separate different portions of spectrum which correspond to modes that are centered around a specific frequency and of compact support. To find such boundaries we find (N − 1) local maximas in the Fourier spectrum. For this set of maximas along with 0 and π, we define boundaries ωn of each segment as the centre between two consecutive maximas, where and . Each segment is denoted as therefore . Centred around each ωn, we define a transition phase Tn of width 2γ ωn where γ is chosen according to the properties of intrinsic mode function (IMF) [19] to get a tight frame, from [17] γ is given in Eq. (2).

(2)
By utilizing the idea used in the construction of Littlewood–Paley and Meyer's wavelets [20], a set of band-pass filters or empirical wavelets is constructed. Empirical scaling function and empirical wavelet function used for the EWT are as given in literature [17].
The detailed coefficients obtained by EWT, as given in Eq. (3), are defined by the inner products with the empirical wavelets.

(3)
And the approximation coefficients , given in Eq. (4), are defined by inner product with the scaling function.

(4)
And the reconstructed signal as given in [17] is obtained by the following equation:

(5)
Figure 2 shows standard ECG signal 117m with 4000 samples, sampled at 360 Hz, along with three modes decomposed by this transform, after preprocessing of the signal.