## 2 Principle
By performing a least squares fitting on \(N_{3}\) sets of data \(Z\), one can derive the characteristic parameters \(a_{2}\) and \(b_{2}\) of the phase discrimination model \(Z\{(N_{3}-1)L_{1}\}=a_{2}(N_{3}-1)L_{1}+b_{2}\) that matches the actual ranging system, as shown in the upper right area of \[E_{\text{L}}(t)=\sqrt{1-r_{1}}E_{0}e^{i(2\pi f_{0}t+\phi_{0}-\frac{\pi}{2})}, \tag{2}\]3 During the ranging process, the peak position \(m_{\rm R}\) is obtained using the cross-correlation method, and the data \(X_{2}(i+m_{\rm R})\) of length \(N_{2}\) are obtained according to \(m_{\rm R}\). \(Z\) is obtained by performing the cross-correlation operation and normalization operation on \(X_{2}(i+m_{\rm R})\), \(Y_{\rm E2}(m)\), and \(Y_{\rm L2}(m)\) according to Figure 1c. The phase difference \(\Delta\tau\) is inversely calculated by taking \(Z\) into the phase discrimination model \(Z(T)=a_{2}T+b_{2}\). The final distance of the target is \(L=(m_{\rm R}-1)T_{\rm S}c/2+\Delta\tau T_{\rm S}c/2\). Figure 4 depicts the procedure for determining the velocity. Initially, a fast Fourier transform (FFT) operation is performed to obtain the frequency \(f_{\rm FFT}\). The local orthogonal digital waveforms \(\cos(2\pi f_{\rm FFT}mT_{\rm S}+\theta_{2})\) and \(\sin(2\pi f_{\rm FFT}mT_{\rm S}+\theta_{2})\) are then constructed according to \(f_{\rm FFT}\), where \(\theta_{2}\) is the initial phase. Next, heterodyne demodulation and phase unwrapping are conducted on \(U(m)\) to derive the phase data \(2\pi f_{\rm resid}mT_{\rm S}+\alpha\frac{\pi}{2}b_{2}\) (\(mT_{\rm S}-\tau_{0})+\theta_{3}\), along with the residual frequency difference \(f_{\rm resid}\) and the residual phase \(\theta_{3}\). Subsequently, curve fitting is performed on the phase data to determine the residual frequency \(f_{\rm resid}\). The speed of the target is expressed as \(V=f_{\rm D}\lambda/2=(f_{\rm FFT}+f_{\rm resid}-f_{1})\lambda/2\), where \(\lambda\) represents the wavelength of the light wave. In summary, it can be seen from Equations (9) and (10) and Figure 2 that the correlation functions of the theoretical pseudo-random code and bandwidth-limited pseudo  
\[E_{\text{L}}(t)=\sqrt{1-r_{1}}E_{0}e^{i(2\pi f_{0}t+\phi_{0}-\frac{\pi}{2})}, \tag{2}\]4 Velocity solving flowchart. \[E_{\text{L}}(t)=\sqrt{1-r_{1}}E_{0}e^{i(2\pi f_{0}t+\phi_{0}-\frac{\pi}{2})}, \tag{2}\]5 Calibration of the characteristic parameters. random code are different. The correlation function determines the early-late discriminator model [21, 23]. In the traditional method, an early-late discriminator model with theoretical characteristic parameters is used for ranging within the phase range of \((0,T_{\text{C}})\) or \((-T_{\text{C}}/2,T_{\text{C}}/2)\) by changing the local pseudo-random code. For the actual bandwidth-limited ranging system, this method cannot solve the problem of mutual constraints between ranging accuracy and ranging precision. Therefore, the situation of high ranging precision and poor ranging accuracy occurs [19]. In this paper, the characteristic parameter calibration method and data-shifting method are adopted. An early-late discriminator model with calibrated characteristic parameters is used for ranging within the phase range of \((-T_{\text{C}}/20,T_{\text{C}}/20)\). This method solves the problem of mutual restriction between ranging accuracy and ranging precision in the traditional method and achieves good ranging precision and ranging accuracy.