## 3 Materials and Methods
### Proposed Method
In other words, the accuracy requirement of the delay model for integer ambiguity resolution using the clock component of the pseudo-code ranging signal can be relatively reduced. Figure 4: The cross-phase ambiguity resolving of X-band downlink signal in PN code ranging system. Based on the partial derivation of Formula (2), the relationship between the accuracy of the time-delay estimation and the performance of the cross-phase estimation can be obtained, expressed as Formula (3):  
\[\sigma_{\tau}=\frac{\sqrt{2}}{2\pi(f_{2}-f_{1})}\sigma_{\varphi}=\frac{\sqrt{2}}{2 \pi\Delta f}\sigma_{\varphi} \tag{3}\]  
where \(\Delta f\) means the effective bandwidth, and \(\sigma_{\varphi}\) and \(\sigma_{\tau}\) are the accuracies of the cross-phase and interferometry delay, respectively. Formula (3) indicates that the wider the effective bandwidth is, and the more accurately the cross-phase is estimated, the more accurate the time delay estimation will be. According to the power allocation of the downlink signal, the power of the pseudo-code clock component is slightly stronger than that of the DOR\({}_{1}\) signal, but with a narrower bandwidth. Therefore, the precision of the estimated time delay by using the clock component of the pseudo-code ranging signal may be worse than that of using the \(-\)DOR\({}_{1}\) or \(+\)DOR\({}_{1}\) signal. On the basis of Formula (2), the integer ambiguity of the cross-phase of the \(\pm\)DOR\({}_{2}\) signal can be obtained. \[N_{\text{ambi}}=[f_{\text{DOR2}}\tau] \tag{4}\]  
where \([x]\) denotes the integer value closest to \(x\). The coarsely estimated time delay can be expressed as Formula (5), where \(\tau\) denotes the real time delay and \(\Delta\tau\) is the estimation bias. Usually, \(\Delta\tau\) can be modeled as a stochastic variety, the mean square error of which represents the time delay accuracy in Formula (3). \[\tau=\tau+\Delta\tau \tag{5}\]  
According to the 3\(\sigma\) rule, the delay estimation bias, \(\Delta\tau\), will fall into the value zone of \([\tau-3\sigma_{\tau},\tau+3\sigma_{\tau}]\) with a probability of 99.7%, which can be described using Formula (6):  
\[P(|\Delta\tau|\leq 3\sigma_{\tau})=99.7\% \tag{6}\]  
Therefore, in order to ensure the correctness of the integer ambiguity resolution, the accuracy requirement of the time delay estimation can be obtained from Formulas (4)-(6):  
\[\sigma_{\tau}<\frac{1}{6f_{\text{DOR2}}} \tag{7}\]  
The accuracy requirement of the cross-phase estimation can be obtained by combining Formulas (3) and (7):  
\[\sigma_{\varphi}<\frac{\pi\Delta f}{3\sqrt{2}f_{\text{DOR2}}} \tag{8}\]  
It can be seen that the wider the effective bandwidth, the lower the requirement of the phase estimation accuracy. Therefore, the cross-phase estimation accuracy of the clock component should be higher than that of the \(-\)DOR\({}_{1}\) or \(+\)DOR\({}_{1}\) signal when the integer ambiguity is resolved. For the two groups of the X-band DOR beacon signals, \(\Delta f=3.8\) MHz, \(f_{\text{DOR2}}=19.2\) MHz, the phase estimation accuracy is not less than 0.1466 rad. Suppose the pseudo-code rate is 2 Mchips/s, its clock component is 1 MHz away from the carrier, that is, \(\Delta f=1\) MHz, and \(f_{\text{DOR2}}\) remains unchanged, the corresponding phase estimation accuracy is no worse than 0.0386 rad. In summary, using the X-band data processing as a representative example, when the pseudo-code ranging signal (\(\pm\)PNc) and DOR beacon signal are processed jointly, the flow of the proposed interferometry method is as follows:  
1. sample and record the four signals including \(-\)DOR\({}_{2}\), \(-\)PNc or \(+\)PNc, carrier, and \(+\)DOR\({}_{2}\). 2. compensate the raw data recorded on the stations by using the initial delay model, and then conduct correlation to obtain the cross-phases of the four signals in Step 1. 3. obtain the coarse time delay estimation by jointly processing the \(-\)PNc or \(+\)PNc signal and the carrier signal.