## 3 Materials and Methods
### Proposed Method  
The processing of very long baseline interferometry data typically involves three steps: initial delay model compensation, correlation, and frequency synchronization. First, the initial time delay model is calculated by using the target's prior ephemeris and station coordinates, which is to compensate the raw data recorded at the stations. Then, the original data after compensation is correlated to obtain the cross-spectrum and interferometry fringes. Finally, frequency synchronization is performed to estimate the high-precision interferometry time delay. The initial cross-phases of the DOR beacons are typically located in the range of [\(-\pi\), \(\pi\)). Since the time delay model is usually not accurate enough, there is integer ambiguity at the cross-spectrum phases of \(\pm\)DOR\({}_{2}\), and even of \(\pm\)DOR\({}_{1}\), which implies that the initial cross-phases of \(\pm\)DOR\({}_{2}\) and \(\pm\)DOR\({}_{1}\) should be corrected by integer times of 2\(\pi\). As a result, the ambiguity should be resolved by using coarse time delay estimation that is unambiguous before the final delay estimation. Taking the X-band as an example, the four beacons including the downlink carrier, \(\pm\)DOR2, \(-\)DOR\({}_{1}\), or +DOR\({}_{1}\) are usually sampled and recorded for VLBI data processing. First, the time delay is coarsely estimated by using the downlink carrier and one of the  
Figure 3: The X-band downlink signal spectrum construction of a typical deep spacecraft. \(\pm\)DOR\({}_{1}\) beacons to resolve the integer ambiguity of the cross-phase of the \(\pm\)DOR\({}_{2}\) signal. Second, the cross-phase of the \(\pm\)DOR\({}_{2}\) signal undergoes integer times of 2\(\pi\) correction. Finally, the frequency synchronization is conducted by jointly using the four beacons to obtain the high-accuracy time delay estimation. As shown, the characteristics of the pseudo-code clock component signal are quite consistent with that of the DOR beacons. Its frequency space away from the carrier is close to the S-band DOR beacon or the X-band DOR\({}_{1}\) beacon. Based on these considerations, we propose employing the clock component of the pseudo-code signal and carrier to roughly estimate the time delay in order to resolve potential integer ambiguity in the cross-phase of the \(\pm\)DOR2 beacons, thereby achieving precise time delay estimation. The ambiguity resolving process of the X-band is shown in Figure 4, and the detailed process can be found in references [7; 31]. Suppose that \(f_{1},f_{2}\) correspond to the two frequencies of the signals in Figure 3, for instance, the carrier and \(-\)DOR\({}_{1}\) signal, and \(\varphi_{1}\), \(\varphi_{2}\) are the corresponding cross-spectrum phases, then, the time delay can be estimated by Formula (2):  
\[\hat{\tau}=\frac{\varphi_{2}-\varphi_{1}}{2\pi(f_{2}-f_{1})} \tag{2}\]  
During the data processing, the cross-spectrum phases are normalized to the main value zone, \([-\pi,\pi)\). If there is ambiguity in the cross-spectrum phase, the estimated time delay will be biased. Therefore, one of the preconditions before frequency synchronization is that the cross-spectrum phases of the \(\pm\)DOR1 signal have no ambiguity. The process of resolving the time delay ambiguity in VLBI imposes stringent requirements on the accuracy of the prior time delay model [21], which almost equals half of the reciprocal of the ambiguity resolving signal bandwidth. For the X-band, the ambiguity resolving signal is usually the DOR\({}_{1}\) beacon, and the corresponding signal bandwidth is about 3.8 MHz. Accordingly, the prior time delay model should be more accurate than about 130 ns. Suppose the chip rate of the regenerated pseudo-code ranging signal is 2 Mchips/s, the frequency interval between the clock component of the pseudo-code ranging signal and the carrier will be about \(\pm\)1 MHz, and consequently, the prior time delay model should be more accurate than about 500 ns.