# Vibration Compensation of the Frequency-Scanning-Interferometry-Based Absolute Ranging System
## 2 Theory Background
### Method for Vibration Influence Reduction  
Based on Equation (10), a tiny movement of the target has a great influence on the measurement accuracy. For the sake of solving this problem, we propose to use a triangular-wave frequency-modulated tunable laser to generate the alternating forward-scanning and downward-scanning signals, and then to process the beat signals of two adjacent scanning periods to produce a new signal that is immune to vibration. Therefore, a stable frequency can be extracted from the new signal, and the distance value is further calculated. The processing steps are presented below. The beat signals of two adjacent scanning periods can be respectively expressed by:  
\[I_{sweep-up}=\frac{1}{2}\eta{A_{L}}^{2}\cos\{2\pi[(\alpha_{0}\tau_{0}-\frac{2 vf _{0}}{c})t+f_{0}\tau_{0}+\]  
\[\int_{0}^{t}\epsilon(t)dt-\int_{0}^{t-\tau_{0}}\epsilon(t-\tau_{0})dt]\},\tau _{0}\leq t\leq T_{m}/2, \tag{11}\]  
\[ \begin{array}{c}I_{sweep-down}=\frac{1}{2}\eta{A_{L}}^{2}\cos\{2\pi[(-\alpha _{0}\tau_{0}-\frac{2 vf_{0}}{c})t+(f_{0}+\alpha_{0}T_{m})\tau_{1}+\\ \int_{\tau_{m}/2}^{t}\epsilon(t)dt-\int_{\tau_{m}/2}^{t-\tau_{1}}\epsilon(t- \tau_{1})dt]\},\ T_{m}/2+\tau_{1}\leq t\leq T_{m},\end{array}  \tag{12}\]  
where \(I_{sweep-up}\) and \(I_{sweep-down}\) are the beat signals of the adjacent forward-scanning and downward-scanning signals, respectively; \(\tau_{1}\) is the time delay corresponding to the initial distance of the downward-scanning period, and \(\epsilon(t)\) is the nonlinear error of the frequency. Since the target displacement and the vibration frequency caused by an environment are usually tiny, and the scanning period is very short, it can be assumed that the target vibrates at a constant velocity \(v\) during the two adjacent scanning periods. To remove the influence of nonlinear tuning, after two photodetector signals are simultaneously sampled and recorded by an oscilloscope, the beat signal of the measuring optical path is resampled in the equal phases. After being resampled, the beat signals of the two adjacent scanning periods are given by:  
\[I_{sweep-up}^{\prime}=\frac{1}{2}\eta{A_{L}}^{2}\cos\{2\pi[\frac{(\alpha_{0} \tau_{0}-2 vf_{0}/c)}{2\alpha_{0}\tau_{r}}k+f_{0}\tau_{0}]\},k=0,\ 1,\ 2,\ \cdots, \tag{13}\]  
\[I_{sweep-down}^{\prime}=\frac{1}{2}\eta{A_{L}}^{2}\cos\{2\pi[\frac{(-\alpha _{0}\tau_{0}-2 vf_{0}/c)}{2\alpha_{0}\tau_{r}}k+(f_{0}+\alpha_{0}T_{m})\tau_{ 1}]\},k=0,\ 1,\ 2,\ \cdots. \tag{14}\]  
Since the signals given by Equations (13) and (14) are discrete, there is no distinction between variables. To remove the influence of target vibration on ranging, a new signal can be obtained by multiplying Equations (13) and (14), which results in:  
\[ \begin{split} I_{mul}=\frac{1}{8}\eta^{2}{A_{L}}^{4}\{\cos[2\pi( \frac{-2\gamma_{0}f_{0}}{\alpha_{0}\tau_{c}}k+f_{0}\tau_{0}+f_{0}\tau_{1}+\alpha _{0}T_{m}\tau_{1})]+\\ \cos[2\pi(\frac{\tau_{0}k}{\tau_{r}}+f_{0}\tau_{0}-f_{0}\tau_{1}- \alpha_{0}T_{m}\tau_{1})]\}.\end{split}  \tag{15}\]  
It should be noted that Equation (15) consists of two terms. The first represents an interference term with a low frequency, and the second is a single-frequency term related to the target distance to be measured. Fortunately, the Doppler shift in the second term is removed and the phase modulation in different periods is reduced greatly. Therefore, an accurate distance value can be obtained from the second term. Since the frequency of the first term is extremely small, the frequency of the second term can be easily distinguished from the frequency of the first term in different regions of the frequency domain. The specific implementation steps are as follows. First, an oscilloscope is used to sample the beat signals of the measuring optical path and the reference optical path synchronously. Then, an adaptive filtering method is applied to the beat signals of the measuring optical path and the reference optical path to reduce the impact of the erroneous sampling points and interfering signals.