## 2 Displacement Reconstruction Using Measured Acceleration Data
### Digital Filtering and Frequency Domain Integration  
The moving average method can effectively eliminate random fluctuations induced by high-frequency interferences through multiple corrections. In this work, a high-pass filter with low-frequency attenuation was applied to remove trend terms induced by low-frequency noises. The actual sampling signal can be expressed by the algorithm of Fourier transformation in the frequency domain:  
\[X(k)=\frac{1}{N}\sum_{r=0}^{N-1}x(r)e^{-j2\pi k\frac{r}{N}} \tag{7}\]  
where \(N\) is the number of sampled data, \(x(r)\) is the sampling sequence signal of \(N\) points, and \(x(k)\) is the discrete spectrum.  
The frequency spectrum of the fast Fourier transform (FFT) of the acceleration sequence can be expressed as [21]:  
\[a(k)=\frac{2}{N}\sum_{r=0}^{N-1}a(r)e^{-j2\pi k\frac{r}{N}}=a_{k}+b_{k}j \tag{8}\]  
where \(a(r)\) is the acceleration sequence with \(N\) number of sampled data and \(a(k)\) is the complex sequence of \(a(r)\) in the frequency domain after Fourier transformation. Further, \(a(k)\) can be represented as:  
\[a(k)=A_{k}\cos(2\pi f_{k}t+\varphi_{k}) \tag{9}\]  
where \(A_{k}=\sqrt{a_{k}^{2}+b_{k}^{2}}\), \(\varphi_{k}=\arctan\frac{b_{k}}{a_{k}}\), and \(f_{k}\) is the frequency corresponding to \(a(k)\).  
Figure 1: Comparison of noise-added signal and de-noising signals: (**a**) noise-added signal and (**b**) de-trending term and de-noising signal.  
The structural displacement in the time domain can be cumulatively expressed as:  
\[s(t)=\sum_{k=1}^{N}s(k)= \sum_{k=1}^{N}\frac{A_{k}}{\omega_{k}^{2}}\cos(2\pi f_{k}t+\varphi_{ k}-\pi)=\sum_{k=1}^{N}\chi\cos(2\pi f_{k}t+\varphi_{k}-\pi) \tag{10}\]  
In order to eliminate low-frequency trend interferences, the low-frequency attenuation method, which could retain components near the target frequency to the maximum extent, was applied. Therefore, the displacement in the frequency domain can be calculated as [21]:  
\[s(t)=-F^{-1}\bigg{[}\frac{\omega^{2}}{\omega^{4}+\delta}F(a(t))\bigg{]} \tag{11}\]  
where \(\delta\) is the regularization coefficient to control the trend term error of the measured acceleration:  
\[\delta=\frac{1-\alpha}{\alpha}(2\pi f_{t})^{4} \tag{12}\]  
where \(\alpha\) is the target accuracy coefficient and \(f_{t}\) is the target frequency.  
The low-frequency attenuation algorithm was applied by setting a high-pass filter with a target frequency. The amplitude-frequency response characteristics of a filter with a target frequency of 2 Hz are presented in Figure 2. It is noticeable that the amplitude of the transfer function approaches 1 in the high-frequency band and is equal to 0 in the low-frequency band. Therefore, this high-pass filter could effectively retain information near the target frequency and attenuate low-frequency information simultaneously. In this work, the analysis target precision coefficient was set to 0.95 to ensure better integration precision in the small target frequency. Meanwhile, the choice of target frequency is difficult for the low-frequency attenuation algorithm. The target frequency is affected by the level of low-frequency noises, and an appropriate value should be selected to eliminate the trend term error and drift error. In this paper, the value of the target frequency is selected to be 1 Hz, where good integration accuracies can be obtained in engineering applications.