# Runway Pavement Structural Analysis Using Remote Laser Doppler Vibrometers
## 3 Compensation Methods
### Vibration Compensation
#### 3.2.2 Experimental Data Processing and Analysis  
* Fourier Transform  
The Fourier transform plays a pivotal role in signal processing. Researchers frequently transform complex time-domain signals into more tractable signals in the frequency domain. When a continuous time-domain signal satisfies the absolute integrability condition, it can be depicted by a series of sine and cosine functions, as represented in Equation (2) [37]:  
\[X(t)=\frac{a_{0}}{2}+\sum_{n=1}^{\infty}a_{n}\cos(\frac{2\pi nt}{T})+\sum_{n=1 }^{\infty}b_{n}\sin(\frac{2\pi nt}{T}) \tag{2}\]  
This implies that the Fourier transform facilitates the transformation of a continuous pavement vibration function into a continuous frequency spectrum. Owing to the complexity of real-world pavement vibration conditions, the time-domain function of the pavement vibration is converted into the frequency domain via Equation (3). This conversion aids in the analysis of the pavement's vibration characteristics. \[X(\textit{iw})=\int\limits_{-\infty}^{+\infty}X(t)e^{-\textit{jwt}}dt \tag{3}\]  
where \(t\) and \(w\) are the continuous spatial variable and continuous time variable. * Filter Selection  
Filters can be classified into various types based on the filter approximation method, including Butterworth filters [38], Bessel filters [39], and Chebyshev filters [40]. Notably, the Butterworth filter is characterized by its uniformly flat amplitude-frequency traits within the passband and monotonically decreasing amplitude-frequency characteristics in the transition band. As such, it is extensively employed in the realm of signal processing. Consequently, for the processing of vibration signals in this study, the Butterworth filter was chosen. * Vibration Data Processing and Analysis  
Through our research and analysis, the raw data of RLDV measurements consist of fixed end beams and pavement vibration. The vibration of the fixed-end beams can be detected by means of accelerometers. The quadratic integration of the accelerometer data revealed that the peak vibration impact of the fixed-end beams generated by the external  
Figure 5: Compensation algorithm experimental validation. conditions was 13 \(\upmu\)m under 100 kN, as shown in Figure 6. The peak value of pavement vibration for the control group was 101.88 \(\upmu\)m according to Table 2, and the effect of fixed end-beam vibration accounted for 12.76%, which could not be ignored. Figure 7 shows the flowchart for vibration compensation of RLDV test data. Each step can be further described as follows. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Loads(kN)** & **Experimental Group (\(\upmu\)m)** &
\begin{tabular}{c} **Control Group** \\ **(\(\upmu\)m)** \\ \end{tabular} & **Error (\%)** \\ \hline
80 & 88.42 & 91.39 & 3.25 \\
100 & 95.47 & 101.88 & 6.29 \\
120 & 108.89 & 112.66 & 3.35 \\
140 & 111.49 & 119.21 & 6.48 \\
160 & 120.86 & 125.53 & 3.72 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Analysis of Vibration compensation experimental results. Figure 6: Fixed-end Beam Vibration Displacement (100 kN). (**a**) Vibration acceleration of fixed-end beam; (**b**) Vibration displacement of fixed-end beam after quadratic integration. Step 1. Apply Fourier transform to the vibration data and analyze the vibrational characteristics of the concrete slab. Figure 8a,b, respectively, depict the collected pavement vibration data under a 100 kN load from both the experimental and control groups using RLDV. The comparison indicates that both sets of vibration data exhibit the same frequency range of 30 to 50 Hz. However, the experimental group registered a substantial vibration signal in the low-frequency band, ranging from 0 to 30 Hz, possibly attributed to the vibration of the fixed-end beam. Step 2. Conduct a Fourier transform on the accelerometer data to analyze the vibrational characteristics of the fixed-end beam. Figure 9 shows the vibration data collected by the accelerometer under a load of 100 kN and the frequency domain results after Fourier transformation.