## 2 Principle
When the bandwidth of the pseudo-random code is finite, its correlation function is [23]  
\[E_{\text{L}}(t)=\sqrt{1-r_{1}}E_{0}e^{i(2\pi f_{0}t+\phi_{0}-\frac{\pi}{2})}, \tag{2}\]0  
where \(H_{\mathrm{BB}}(jf)\) is the transfer function of the LPF. Figure 2 shows the correlation functions of the theoretical pseudo-random code and the pseudo-random code with a bandwidth of 250 MHz. As shown in Figure 2, the correlation function of the theoretical pseudo-random code is a triangle. The LPF changes the correlation function of the pseudo-random code, and the correlation function of the band-limited pseudorandom code is no longer a triangle. Therefore, the early-late discriminant models of pseudo-random codes with different bandwidths are also different. The actual ranging system is band-limited, and the devices used in the system are not ideal. When the theoretical phase discrimination model is applied to the actual ranging system, it can result in ranging deviations that decrease the accuracy of the ranging. Therefore, the characteristic parameters of the early-late discriminator model need to be calibrated to improve the ranging accuracy. To solve this problem, this study calibrates the characteristic parameters of the phase detection model utilized in the real-world system. After this calibration, the fine time delay is calculated according to the calibrated parameters, thereby improving the accuracy of the distance measurements. The flow of the ranging process is illustrated in Figure 1c. The data position \(m_{\mathrm{R}}\) corresponding to the correlation peak is obtained via cross-correlation. The coarse distance is \(L_{\mathrm{coar}}=(m_{\mathrm{R}}-1)T_{\mathrm{S}}c/2\). A segment of data \(X_{2}(i+m_{\mathrm{R}})\), with a length of \(N_{2}\), is extracted along the time axis according to \(m_{\mathrm{R}}\). Then, \(X_{2}(i+m_{\mathrm{R}})\) is correlated with \(Y_{\mathrm{E2}}(m)\) and \(Y_{\mathrm{L2}}(m)\) to obtain \(Z\). Finally, the fine distance \(L_{\mathrm{fine}}=\Delta\tau T_{\mathrm{S}}c/2\) (\(0\leq\Delta\tau<1\)) is obtained via an inverse calculation using the phase discrimination model \(Z(T)=a_{2}T+b_{2}\) that matches the actual system. The distance of the target is \(L=L_{\mathrm{coar}}+L_{\mathrm{fine}}\). Because the characteristic parameters \(a_{2}\) and \(b_{2}\) in the phase discrimination model \(Z(T)=a_{2}T+b_{2}\) are calibrated based on the actual system, the model matches the actual system. The proposed method can significantly improve the ranging accuracy. The proposed methodology initially employs this ranging system to establish a characteristic parameter calibration system with variable time delays. It calculates the value \(Z\) for different time delays, where the actual time delay interval \(T_{1}=L_{1}/c\) is less than the ADC sampling period \(T_{\mathrm{S}}\). This principle is depicted at the top of \[E_{\text{L}}(t)=\sqrt{1-r_{1}}E_{0}e^{i(2\pi f_{0}t+\phi_{0}-\frac{\pi}{2})}, \tag{2}\]1 The position of the sampling point of the DSO in the chip is changed by adjusting the retroreflector position, which effectively changes the phase of the sampled data. Within the distance \(L_{2}=cT_{\mathrm{S}}/2\) corresponding to the sampling period, the retroreflector steps forward \(L_{1}\) each time for a total  
\[E_{\text{L}}(t)=\sqrt{1-r_{1}}E_{0}e^{i(2\pi f_{0}t+\phi_{0}-\frac{\pi}{2})}, \tag{2}\]2 Correlation functions of pseudo-random codes with different bandwidths. of \(N_{3}\) steps. The pseudo-random code synchronization pulse and the target echo are sampled simultaneously at each position, and \(N_{4}\) cycles of pseudo-random code are sampled. A pseudo-random sequence \(X_{3}^{(N_{3}-1)L_{1}}(j_{1})=\{D(j_{1}),D(j_{1}+1),\,\ldots,D(j_{1}+N_{2}-1)\}\) of length \(N_{2}\) is extracted at each position from the target echo data \(D(m)\), where \(X_{3}^{(N_{3}-1)L_{1}}(j_{1})\) is aligned with \(Y_{\rm P}(m)\). The mean value of \(Z\{(N_{3}-1)L_{1}\}\) at each position is computed.