# Analysis and Demonstration of First Cross-Support Interferometry Tracking in China Mars Mission
## 2 Theorem and Methods
### Algorithm Denolping and Analysis  
To take DeltaDOR measurements, spacecraft generally sends dedicated DOR tones signal produced by the transponder or high harmonics of the telemetry subcarrier, the spanned frequency of all the tones is usually several MHz. In contrast, signal from quasar looks like white noise distributed in the whole channel. For this reason, in general, different algorithms are used when extracting the delay in the signal arrival time at the two stations for different signal's characteristics. However, an inter-DeltaDOR interface, including observable definition, is adopted for routine DeltaDOR tracking within LCTC. While this interface does not fit to the cross-support interferometry tracking, as the definition of observables from spacecraft is the same as quasar observation, that is the time difference in the same wavefront arriving at the two stations. So, a new algorithm is developed and adopted to support the cross-support interferometry, especially for the comparison of data processing results between CLTC and ESOC. For spacecraft DOR tones signal processing, algorithm of local correlation is developed and discussed in this chapter [38; 39]. Spacecraft could work under One-way mode or Three-way mode during DeltaDOR measurements. The following description focuses on One-way mode, which is used in China Mars mission cross-support interferometry. The formulation of signal \(\mathrm{S}_{A}(\mathrm{t})\), \(\mathrm{S}_{B}(\mathrm{t})\) received at the tracking stations are expressed as:  
\[\left\{ \begin{array}{l}\mathrm{S}_{A}(\mathrm{t})=\exp\left\{\mathrm{j}(2 \pi f_{LA-DOR}t+\varphi_{DOR})\right\}\exp\left\{-\mathrm{j}2\pi f_{DOR}\tau_{A} (t)\right\}\\ \mathrm{S}_{B}(\mathrm{t})=\exp\left\{\mathrm{j}(2\pi f_{LB-DOR}t+\varphi_{DOR })\right\}\exp\left\{-\mathrm{j}2\pi f_{DOR}\tau_{B}(t)\right\}\end{array} \right. \tag{9}\]  
where \(f_{DOR}\) is the sky frequency of the tone signal sent by spacecraft, \(f_{LA-DOR}\) is the intermediate frequency after signal down-converting, \(\varphi_{DOR}\) is original phase, and \(\tau_{A}(t)\) is the time delay between the spacecraft and the tracking station A, which is the same as \(f_{LB-DOR}\), \(\tau_{B}(t)\). Firstly, a delay model expressed by a polynomial is reconstructed for each station receiving signal. The delay model is described with a polynomial expression, with the order of N. The delay \(\tau_{A}(t)\) and delay rate \(\dot{\tau}_{A}(t)\) are listed as follows:  
\[\left\{ \begin{array}{l}\tau_{A}(t)=\tau_{AN}t^{N}+\tau_{AN-1}t^{N-1}+\ldots+ \tau_{A1}t+\tau_{A0}\\ \dot{\tau}_{A}(t)=N\tau_{AN}t^{N-1}+(N-1)\tau_{AN-1}t^{N-2}+\ldots+\tau_{A1} \end{array} \right. \tag{10}\]  
In order to reconstruct the polynomial coefficient, the fundamental relation between Doppler value and the delay rate is induced; in principle, the Doppler of signal \(f_{d\_est}(t)\) is proportional to delay rate \(\dot{\tau}_{A}(t)\). \[\dot{\tau}_{A}(t)=\frac{f_{d\_est}(t)}{f_{DOR}} \tag{11}\]  
With delay rate \(\dot{\tau}_{A}(t)\) obtained, then delay \(\tau_{A}(t)\) could be computed with the below formulation, where \(C_{A}\) is a constant value. \[\tau_{A}(t)=\int\dot{\tau}_{A}(t)dt+C_{A} \tag{12}\]  
Iteration strategy is designed during model reconstruction. Algorithm is explained as follows:  
1. Coarse Doppler estimation using over lapped FFT, then delay-rate vector with respect to time could be obtained based on Equation (11). 2. Delay rate and delay polynomial model could be obtained, with a constant offset coefficient. 3. A local model signal is computed with Equation (9). 4. Cross correlation with received signal \(\mathrm{S}_{A}(\mathrm{t})\) and local model signal \(\mathrm{S}(\mathrm{t})\). \[S_{corr}(t)=\mathrm{S}(\mathrm{t}).*\mathrm{conj}(\mathrm{S}_{A}(\mathrm{t})) \tag{13}\]
5. Compute the residual phase of \(S_{corr}(t)\) which is used to update the delay polynomial model. 6. Refresh delay polynomial model by iteration, until there is no phase blur in the residual phase.