## 4 Numerical Simulation
### Synthesized Ideal Discrete Acceleration  
Consider a composite-simulated sinusoidal signal that includes the frequencies of 10, 15, and 20 Hz with the corresponding acceleration amplitudes of 10, 20, and 30 g, respectively. The acceleration signal can be expressed as:  
\[a(t)=10\sin(20\pi t)+20\sin(30\pi t)+30\sin(40\pi t) \tag{25}\]  
Now, taking the sampling frequency of the above signal sequence as 1000 Hz, the theoretical displacement signal can be obtained as:  
\[x(t)=-\frac{10}{\left(20\pi\right)^{2}}\sin(20\pi t)-\frac{20}{\left(30\pi \right)^{2}}\sin(30\pi t)-\frac{30}{\left(40\pi\right)^{2}}\sin(40\pi t) \tag{26}\]  
Figure 4 presents the effects of the improved algorithm on the integral displacement. Figure 4a depicts the difference between integral and standard displacements. The error was large at 0.2 s due to the uncertainty of the initial value and the truncation error, and it fitted well after 0.2 s. Displacements at 1.175-1.195 s are locally amplified in Figure 4b, and it is noticeable that these displacement signals had the same frequency and waveform, with slightly different amplitudes. The frequency spectrum shown in Figure 4c was obtained by performing FFT transformation on theoretical and reconstructed displacement signals. It is evident that theoretical and reconstructed displacement signals had the same frequency spectrum components, which included three main frequency components of 10, 15, and 20 Hz. However, the effect of interference on signal amplitude existed under 10 Hz. Next, the integral error evaluation index was applied to calculate the error between integral and standard displacements (Figure 4a) [29]:  
\[Err=\frac{\left|d(t_{i})-y(t_{i})\right|}{range(d)} \tag{27}\]  
Figure 3: Schematic diagram of window size and step length. where \(d(t_{i})\) and \(y(t_{i})\) are the standard and integral displacements at time \(t_{i}\) respectively, and _range(d)_ is the range of the standard displacement in the whole period and represents the difference between the maximum and minimum standard displacements. The error diagram of reconstructed and locally amplified displacement signals in the interval of 1-1.5 s is exhibited in Figure 4d. The error between reconstructed and locally amplified displacement signals was large at 0.2 s and then dropped fast to 0.02, proving the validity of the improved algorithm. The value of the error is related to the selection of target frequency and the precision coefficient, and the maximum value of error is shown in Table 1 by changing these two parameters. It was found that the better precision was obtained until 1 Hz. Generally, singular values of a matrix possess some important hidden information. Hankel matrices consisting of acceleration and displacement data were analyzed to verify the high recognition ability of the improved algorithm. First, 3000 data points were \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Maximum Error**} & \multicolumn{6}{c}{**Frequency**} \\ \cline{2-7}  & **0.2 Hz** & **0.5 Hz** & **1 Hz** & **4 Hz** & **8 Hz** \\ \hline Precision & 0.92 & 0.059 & 0.025 & 0.013 & 0.016 & 0.015 \\ Coefficient & 0.95 & 0.054 & 0.023 & 0.012 & 0.015 & 0.016 \\  & 0.98 & 0.047 & 0.021 & 0.011 & 0.017 & 0.020 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The maximum value of error after 0.5 s by changing the target frequency and target precision coefficient. Figure 4: (**a**) Comparison between theoretical and reconstructed displacements, (**b**) locally amplified displacement signals at 1.175–1.195 s, (**c**) spectrum comparison between theoretical and reconstructed displacements, and (**d**) error diagram of reconstructed and locally amplified displacement signals in the interval of 1–1.5 s.  
selected, and 1500 \(\times\) 1500 Hankel matrices were constructed. The singular values and ranks of different Hankel matrices were obtained and normalized [19].