# Design of Tracking, Telemetry, Command (TT&C) and Data Transmission Integrated Signal in TDD Mode
## 4 Performance Analysis of the TT&C Signal and Data Transmission Signal
### Tracking Performance for the TT&C Signal  
In the tracking process, the pseudo-range and Doppler frequency are the main measurements [42]. The ranging error refers to the code loop tracking error, and its main sources are thermal noise error and dynamic stress error. The latter is eliminated by carrier-assisted PN code technology, and the thermal noise error can be calculated as (12):  
\[\sigma_{Ranging}=T_{chip}\sqrt{\frac{2d^{2}B_{LD}}{CNR}\bigg{[}2(1-d)+\frac{4d }{T_{col}\cdot CNR}\bigg{]}} \tag{12}\]  
Here, \(T_{chip}\) represents the width of a chip, which is 58.6 m for BOCcos(30,5). \(B_{LD}\) is the bandwidth of the code tracking loop. CNR is the carrier-to-noise ratio of the TT&C receiving signal. \(d\) represents the chip interval of the delay locked loop (DLL). \(T_{col}\) represents the integration time [45].  
For the Doppler frequency, the thermal noise error of the three-order phase locked loop (PLL) is calculated as (13).  
\[\sigma_{PLL}=\sqrt{\frac{B_{LP}}{CNR}\bigg{(}1+\frac{1}{2T_{col}CNR}\bigg{)}} \tag{13}\]  
The Doppler frequency can be calculated as (14).  
\[\sigma_{Doppler}=\frac{\sqrt{2}\sigma_{PLL}}{2\pi\cdot\Delta t} \tag{14}\]  
Here, \(\Delta t\) is the refresh interval time of the Doppler measurement, which is equal to 0.1 s [46; 47].  
Considering the convergence speed and tracking error of the DLL and PLL, we set \(B_{L}=1\) Hz and \(B_{LP}=10\) Hz. In this situation, the ranging error is 0.0755 m when CNR = 55 dBHz, and the Doppler frequency measurement error is 0.0127 Hz. If it is necessary to improve the measurement accuracy, it is very effective to average the measurement results, increase the \(T_{col}\) or improve the CNR.  
In Section 3.2, the extrapolation method is used in the transmission stage of the TDD mode, which will cause errors. The acceleration measurement error and velocity measurement error are assumed to be \(\Delta a\) and \(\Delta v\), respectively. Therefore, the velocity measurement error caused by the acceleration error is \(\Delta a\times T_{sp}\), and the ranging error is as shown in (15).  
\[\sigma_{ce}=\Big{(}\Delta v+\Delta a\times T_{sp}/2\Big{)}\times T_{sp}. \tag{15}\]  
The correction error \(\sigma_{ce}\) shows that the shorter the switching period is, the smaller the extrapolation error in the TDD process. For example, if \(\Delta v\leq 1\) m/s, \(\Delta a\leq 1\times 10^{-3}\) m/s\({}^{2}\) and \(T_{sp}=100\) ms, the pseudo-range correction error by extrapolation is less than 0.10005 m. If \(T_{sp}=1\) s, the correction error is less than 1.005 m. Considering the bit transitions, the relationship between the correction errors and whether the tracking process requires recapturing is shown in Table 3. The threshold is related to the parameter selection of the frequency-locked loop and the DLL.  
Figure 6: _Cont._  
Excessive extrapolation errors of the Doppler frequency and ranging during the TDD switching period will cause loop unlocking and reacquisition. The simulation shows that if the Doppler  
Figure 6: The measurement results when CNR = 60 dBHz.  
extrapolation error exceeds \(\pm\frac{1}{4T_{cah}}\) and the distance extrapolation error exceeds \(\pm 0.25T_{chip}\cdot c\) at the receiving and transmitting switch time, the tracking loop will unlock.