# Phase-Modulated Continuous-Wave Coherent Ranging Method and Anti-Interference Evaluation
## 4 Anti-Interference Capability
Briefly, if the interference has no influence on the PhMCW system, the difference of the measured precision with and without interference is thus a white noise, i.e., \(\Delta_{\sigma}=\sigma_{\text{W}}/-\sigma_{\text{W}/\text{O}}\) is Gaussian distributed random measurement error, where \(\sigma_{\text{W}}/\) and \(\sigma_{\text{W}/\text{O}}\) are the measured precisions with and without interference, respectively. Subsequently, the Ljung-Box statistic is calculated using \(\text{Q}(\text{m})=\text{T}(\text{T}+2)\sum\limits_{\text{n}=1}^{m}\frac{ \rho_{\text{n}}}{\text{I}-\text{m}}\), where \(\text{T}=6\) is the sample size, \(\text{m}=1\) is the lag rank or degree of freedom, and \(\rho_{\text{n}}=\frac{\text{T}^{\text{T}}\text{n}}{\text{I}}\frac{\text{T}^{ \text{T}}_{\text{n}+1}(\text{x}_{\text{k}}-\text{T})(\text{x}_{\text{k}}- \text{T})}{\text{I}\sum\limits_{\text{k}=1}^{m}(\text{x}_{\text{k}}-\text{T}) ^{2}}\) is the n-order autocorrelation coefficient. Finally, the \(p\)-value is calculated by \(\text{p}=\text{P}(\chi^{2}\geq\text{Q}(\text{m}))\), where \(\chi^{2}\) is the chi-square random variable. If the \(p\)-value is greater than 0.05, we cannot reject the null hypothesis that \(\Delta_{\sigma}\) is white noise and thus interference does not influence the LiDAR system. The measured precision values are summarized in Table 1 and the \(p\)-value is calculated to be 0.0589, demonstrating the anti-interference capability against ToF LiDARs. For a commercial LiDAR, the echo waveform may not be measured during the anti-interference evaluation. To resolve this problem, we can first place the victim Lidar on an optical table and then observe the stationary scanning pattern by using an infrared camera. Subsequently, we can aim the interference source (AFG1 + PLD + COL1 in Figure 7) to a particular voxel and record the ranging results of the corresponding voxel as a function of time. The interference source can be amplitude-, frequency- or phase-modulated to mimic any possible interference in realistic applications. Finally, the Ljung-Box test is performed to quantitively determine the anti-interference capability, as mentioned above. In addition to ToF, we also validated the immunity against coherent LiDARs. The diagram of the setup is illustrated in Figure 10. As a demonstration, the aggressor light source was another PhMCW laser operating at 1550 nm. The phase-modulation frequency of the aggressor laser was chosen to be 10 MHz, which is two orders of magnitude higher than that of the victim PhMCW system, guaranteeing that the phase-modulation interference \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**d (m)** & **2** & **3** & **4** & **5** & **6** & **7** \\ \hline \(\sigma_{\text{W}/}\) (cm) & 2.393 & 2.407 & 2.129 & 2.302 & 2.261 & 2.223 \\ \hline \(\sigma_{\text{W}/\text{O}}\) (cm) & 2.463 & 2.336 & 2.199 & 2.160 & 2.348 & 2.215 \\ \hline \(\Delta_{\sigma}\) (cm) & \(-0.070\) & 0.071 & \(-0.070\) & 0.142 & \(-0.087\) & 0.008 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Precision as a function of distance with and without ToF LiDAR interference. Figure 9: (a) Superimposed image of the interfered RX signal. For better visibility, only 10 RX signals are shown. (b) Precision as a function of distance with and without interference. distance of 50 m, and a ranging error as low as 2.2 cm was achieved, which is comparable to that of commercial LiDARs. In addition, we modeled the ranging precision of the PhMCW system. Furthermore, a one-dimensional scanning LiDAR system that is capable of detecting targets at 28 m was built, demonstrating the validation of the PhMCW method. Subsequently, we proposed a quantitative method for evaluating the anti-interference capability of lidar systems. Specifically, the ranging precision was measured with and without interference, and the Ljung-Box test was performed to quantitatively illustrate the minimal influence of interference.