## 4 Frequency Estimation of Midspan Deflection Signal Acquisition
### Frequency Characteristics of Bridge Deflection
At the same time, the quasi-static time-history curve of the structure is obtained by driving at a uniform speed of 1 km/h with the same vehicle load, as shown in Figure 8.  
In this case, for the quasi-static curve, the statistical characteristic parameters are as follows: the mean is \(u_{0}=-1.65779\), the standard deviation is \(\varpi_{0}=0.87003\), the variance is \(\sigma_{0}^{2}=0.75695\), and the absolute maximum is \(S_{0}=2.73974\). Based on the obtained midspan deflection time-history curve, the re-collected data with different cutoff frequencies \(f_{1}\) are obtained by low-pass filtering, and then the frequency-domain signal is converted into the time-domain signal by inverse Fourier transform, and the results are obtained, as shown in Figure 9.  
Figure 8: Time-history curve of midspan deflection. According to the Nyquist sampling theorem \(f_{s}\geq\ 2\text{f}_{\text{max}}\)[25], the results show that when \(f_{1}\)\(=\) 0.7 Hz, that is, the acquisition frequency \(f_{s}\)\(=\) 1.4 Hz, the reconstructed signal is far from the original signal, resulting in mixing phenomenon. When \(f_{1}\)\(=\) 1.5 Hz, that is, the acquisition frequency \(f_{s}\)\(=\) 3 Hz, the reconstructed signal has some deviation from the original signal, but the contour curve is almost the same. When \(f_{1}\)\(=\) 7 Hz, that is, when the acquisition frequency \(f_{s}\)\(=\) 14 Hz, the reconstructed signal is almost the same as the original signal, which is basically consistent. The above analysis shows that the low-frequency acquisition frequency (1.4 Hz) cannot obtain the complete response of the structure under live load, and the difference is significant, while the high-frequency acquisition frequency (14 Hz) can obtain the relatively complete response of the structure under live load, but the middle sampling frequency (3.0 Hz) is four times lower than the high-frequency acquisition frequency, and the response under live load can still be obtained relatively completely. Therefore, the acquisition frequency of the structural response should not be too low to obtain the structural response; it should not be too high. Although a complete response can be obtained, there is no obvious difference between the results obtained with a relatively low sampling frequency, which increases the difficulty and cost of data processing. To delve deeper into the concept of reasonable acquisition frequency, it is imperative to employ the fast Fourier transform (FFT) to convert the original time domain signal into a frequency domain signal. This process involves decomposing the signal into a weighted combination of complex exponential functions, thereby establishing a transformational \"bridge\" between the time domain and the frequency domain. The basic essence of fast Fourier transform is to divide the signal into a certain length, and then carry out a discrete Fourier transform on the signals with two lengths of N/2 and then splice them. Compared with the traditional discrete Fourier transform, the above method saves half the workload [43]. At present, there are two main frequency domain analysis methods: amplitude spectrum analysis and power spectrum analysis. The following will take the midspan deflection signal as an example to explore the advantages and disadvantages of the two analysis methods, and finally determine the estimation method of reasonable acquisition frequency in this paper.