# Integration of Wavelet Denoising and HHT Applied to the Analysis of Bridge Dynamic Characteristics
## 3 Time-Frequency Analysis Method Based on Wavelet Threshold Denoising and HHT
### Time-Frequency Analysis Method Combining Wavelet Threshold Denoising and HHT  
Although the wavelet threshold method can eliminate white noise in the original signal, the noise reduction preprocessing may not sufficiently eliminate the interference of noise in practical applications due to the noise interference of many different properties. Therefore, the IMF correlation must be adopted for appropriate post-processing, and it may improve the recognition accuracy of the mainvibration frequency. The denoising method based on wavelet threshold denoising and EMD is called the wavelet-EMD method, and its specific steps are as follows:  
(1) Wavelet threshold denoising: To eliminate the influence of noise, the original noise-containing non-stationary signal \(x(t)\) is decomposed by the wavelet threshold, and the signal is reconstructed according to the energy of the decomposed frequency band. The denoising signal that eliminates interference noise is then obtained. This denoising process via the wavelet threshold can decrease the influence of high-frequency noise and the decomposition layers of EMD. Hence, it may provide a relatively \"clean\" input signal for EMD. (2) Multi-scale EMD is carried out on the denoised signal \(\widehat{x}(t)\) processed by the wavelet threshold denoising method to obtain the IMF components of the denoised signal, \(c_{1},c_{2},\cdots,c_{n}\). (3) The correlation value of each IMF component is calculated to identify the vibration component of the bridge structure. For the decomposition results of large-scale building dynamic observation data, the high-frequency parts of the IMF component are generally dominated by noise, and the low-frequency part is dominated by the vibration signal. The investigation of the decomposition results of EMD indicates that several components have little energy after decomposition and cannot represent the original signal, which is not conducive to analyzing the spectral characteristics of the signal. To identify the effective vibration component easily, the component correlation \(\rho_{i}\) according to the correlation between each component and the original signal is defined to distinguish the effective IMF component from the unwanted IMF component. \[\rho_{i}=\frac{c_{i}(t)s(t)}{\left[c_{i}(t)^{2}s(t)^{2}\right]^{-0.5}}. \tag{15}\]  
where \(c_{i}(t)\) represents the \(i\)th IMF component of the signal and \(s_{i}(t)\) represents the corresponding residual signal. The residual signal corresponding to the \(i\)th component is obtained as:  
\[s_{i}(t)=x(t)-c_{i}(t). \tag{16}\]  
where \(x(t)\) represents the original signal. The definition implies that the correlation of the IMF components reflects the correlation of each component with the original signal and can be used to identify components with different physical meanings. (4) Reconstruction of the vibration component. The decomposed components are selected based on the correlation value and actuquaal demand. The useless IMF component is eliminated, and the effective IMF component is used for reconstruction to obtain the denoised vibration signal. The flow of time-frequency analysis via the wavelet-EMD method is shown in Figure 2.  
Figure 2: Flowchart of the wavelet-EMD denoising method. The quality of denoising performance is the key to evaluating the advantages and disadvantages of a denoising algorithm in practical applications. Therefore, an objective denoising performance evaluation index must be selected. Similar to the evaluation index of image denoising [33], the root mean squared error (RMSE) and SNR are usually chosen for such an evaluation. RMSE and SNR are expressed as follows:  
\[RMSE=\sqrt{\sum_{i=1}^{N}{\left[{\left(x(i)-y(i)\right)^{2}}/{N}\right.}}. \tag{17}\]  
\[SNR=10\log[\sum_{i=1}^{N}{x^{2}(i)}/{\sum_{i=1}^{N}{\left[{\left(x(i)-y(i) \right)^{2}}\right]}}. \tag{18}\]  
where \(y(i)\) is the denoised signal, \(x(i)\) is the original signal that serves as the standard signal, and \(N\) is the length of the signal.