# Integration of Wavelet Denoising and HHT Applied to the Analysis of Bridge Dynamic Characteristics
## 2 Basic Principle of HHT  
Hilbert spectrum analysis is performed using HHT from the non-stationary signal. The HHT-based time-frequency analysis method consists of two main steps: (1) multi-scale decomposition, which is implemented on non-stationary signals using the EMD method and whose result is a series of IMFs, and (2) HT, which is performed on each IMF component represented in the joint time-frequency domain. Then, the Hilbert spectrum of the signal is acquired. The steps of EMD method [26; 27; 28] are as follows:  
(1) The extreme points of non-stationary signal \(x\)(t) are fitted using the cubic spline function mentioned in [26] to obtain the upper and lower envelopes of the signal. The mean is calculated as:  
\[m(t)=(e_{\max}(t)+e_{\min}(t))/2. \tag{1}\]  
(2) The remainder is calculated as:  
\[h_{1}(t)=x(t)-m(t). \tag{2}\]  
Afterwards, whether \(h_{1}(t)\) satisfies the definition of IMF is determined [26]. If the definition is satisfied, then \(h_{1}(t)\) is the first IMF component and denoted as:  
\[c_{1}(t)=h_{1k}(t). \tag{3}\]  
where \(h_{1k}(t)\) is the value of \(h_{1}(t)\) after the \(k\)th iteration. Otherwise, iterative calculation continues until the condition mentioned in [26] is met.  
(3) \(c_{1}(t)\) is separated from \(x(t)\). Then, a differential signal with the high-frequency components removed is obtained as follows:  
\[x_{1}(t)=x(t)-c_{1}(t). \tag{4}\]  
(4) The preceding steps are repeated many times by taking \(x_{1}(t)\) (the latter counterpart is recorded as \(x_{n}(t)\)) as a new signal [26], and the individual IMF components are decomposed sequentially. The decomposition process terminates when residual signal \(x\) is a monotonic function. Then, \(x(t)\) is decomposed into \(n\) IMF components and one remainder through the above-mentioned method.  
\[x(t)=\sum_{i=1}^{n}c_{i}(t)+r_{n}(t). \tag{5}\]  
The decomposition process shown above indicates that the EMD model separates the non-stationary signal into several intrinsic mode functions, which are all stationary or stationary-trended signals (i.e., it is an adaptive signal decomposition method). The obtained \(c_{i}(t)\) indicates that the original signal is decomposed into components with different frequency scales. \(r_{n}(t)\) is the tendency part, and the decomposition result is related only to the original signal.  
HT is carried out on each IMF \(h_{i}(t)\) decomposed from Equation (2) as follows:  
\[\widehat{h}_{i}(t)=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{h_{i}(\tau)}{t- \tau}d\tau. \tag{6}\]The analytic signal is constructed as:  
\[z_{i}(t)=h_{i}(t)+j\widehat{h}_{i}(t)=a_{i}(t)e^{i\psi_{i}(t)}. \tag{7}\]  
The amplitude function is obtained as:  
\[a_{i}(t)=\sqrt{h_{i}^{2}(t)+\widehat{h}_{i}^{2}(t)}. \tag{8}\]  
Furthermore, the instantaneous frequency is described as:  
\[f_{i}(t)=\frac{1}{2\pi}\omega_{i}(t)=\frac{1}{2\pi}\times\frac{d\varphi_{i}(t) }{dt}. \tag{9}\]  
Therefore,  
\[x(t)=\text{Re}\!\!\sum_{i=1}^{n}a_{i}(t)e^{i\psi_{i}(t)}=\text{Re}\!\!\sum_{i=1 }^{n}a_{i}(t)e^{i\int\omega_{i}(t)dt}. \tag{10}\]  
where \(Re\) is the real part of the original signal, and residual function \(r_{n}(t)\) can be ignored.  
Equation (10) is the Hilbert spectrum, which is recorded as:  
\[H(\omega,t)=\text{Re}\!\!\sum_{i=1}^{n}a_{i}(t)e^{i\int\omega_{i}(t)dt}. \tag{11}\]  
The Hilbert marginal spectrum is defined as:  
\[h(\omega)=\int_{0}^{T}H(\omega,t)dt. \tag{12}\]  
where \(T\) represents the total length of the signal, \(H(\omega,t)\) indicates that the signal amplitude varies with the change in time and frequency within the entire frequency range, and \(h(\omega)\) shows that the signal amplitude varies with the change in frequency. When the energy of a certain frequency appears in \(H(\omega,t)\) or \(h(\omega)\), the vibration wave of this frequency appears. Thus, the Hilbert marginal spectrum can accurately reflect the actual frequency component of the signal. The flow of non-stationary data processing using the HHT method is shown in Figure 1.  
Figure 1: Flowchart of the HHT algorithm.