*4.4. Hilbert Spectrum*

As indicated in Equation (17), an IMF component can be converted from a time-domain to a frequency-domain component; this process is called Hilbert spectral analysis (Figure 7)

$$X(t) = \sum\_{j=1}^{n} a\_j(t) \exp\left(i \int \omega\_j(t)dt\right) \tag{17}$$

Although the Hilbert transform can process a monotonous trend and consider it a part of a longer amplitude, the remaining energy may be excessively strong, considering the uncertainties of longer-term trends and other low-energy elements and information contained in a high-frequency component. The preceding formula provides a time function for each amplitude and frequency component, and this formula can be expanded using a Fourier expression as

$$X(t) = \sum\_{j=1}^{\infty} a\_j e^{i\omega\_j t} \tag{18}$$

where *aj* and *wj* are constants. Comparing Equations (17) and (18) reveals that the IMF can be represented by a generalized Fourier expansion. Variables within the amplitude and the IF cannot only improve the expansion but also render it applicable to unsteady signals. In terms of the expansion of the IMF, the amplitude and the frequency modulation are clearly separated. The amplitude of the time function and the IF can be combined as the time–frequency–amplitude spectrum; this spectrum is referred to as the Hilbert amplitude spectrum.

**Figure 7.** HHT calculation process.
