*2.1. Hilbert–Huang Transform*

In 1998, N.E. Huang proposed the Hilbert–Huang transform (HHT) [9]. HHT can process nonstationary and nonlinear signals adaptively and is made up of EMD and Hilbert transform. First, EMD decomposes the original signal into several intrinsic mode functions (IMFs) adaptively. Secondly, the instantaneous frequency is obtained by the Hilbert transform to each intrinsic mode function (IMF). The formula of Hilbert transform for each IMF is as follows:

$$H[c\_i(t)] = \frac{1}{\pi} \int\_{-\infty}^{\infty} \frac{c\_i(\tau)}{t - \tau} d\tau \tag{1}$$

where *ci*(*t*) is the intrinsic mode component, *i* = 1, 2, ... *n*, and *H*[ ] is the symbol of HHT, *τ*. The analytic signal of each IMF is computed by Equation (2).

$$z\_i(t) = c\_i(t) + jH[c\_i(t)] = a\_i(t)e^{j\varphi\_i(t)}\tag{2}$$

The amplitude function and phase function are defined as

$$a\_i(t) = \sqrt{c\_i^2(t) + H[c\_i(t)]^2} \tag{3}$$

$$\varphi\_i(t) = \arctan \frac{H[c\_i(t)]}{c\_i(t)} \tag{4}$$

where *ai*(*t*) is the amplitude function, and *ϕi*(*t*) means the phase function.

The instantaneous frequency can be obtained by the differential processing of phase function as follows:

$$
\omega\_i(t) = \frac{d\rho\_i(t)}{dt} \tag{5}
$$

Then, the Hilbert spectrum of signal *x*(*t*) can be indicated as

$$H(\omega, t) = \begin{cases} \text{Re}\sum\_{i=1}^{n} a\_i(t)e^{j\int \omega\_i(t)dt}\omega\_i(t) = \omega\\ 0 & \text{otherwise} \end{cases} \tag{6}$$

Even as the core of Hilbert–Huang transform, EMD is limited in the application because of its drawbacks, such as mode mixing, the endpoint effect, uncertain center frequency, and bandwidth of intrinsic mode. Furthermore, since the iterative calculation of EMD is very complicated, the computational efficiency of HHT is extremely low, failing to satisfy real-time project applications [12].
