# A NavCom Signal Authentication Scheme Based on Twice Two-Way Satellite Time Transfer
## 2 A Twice TWSTT NavCom Signal Authentication Scheme
### Signal and Detection Model  
The earlier analysis indicates that the accuracy of the clock difference from the twice TWSTT is significant for security authentication capabilities. As such, it is essential to analyze the error associated with clock difference measurement accuracy. Equation (3) reveals that the accuracy of clock difference measurement is influenced by four main factors: hardware device time delay difference, Sagnac effect error, space propagation time delay difference, and signal arrival time measurement error [20]. (1) Hardware device time delay difference:  
Figure 4: Schematic of twice TWSTT in a repeater spoofing interference scenario. The hardware device time delay error mainly includes the hardware processing time delay error, the hardware storage time delay error, and the hardware variation error with temperature. From the literature [20], these errors add up to about 80 ps. (2) Sagnac effect error:  
The Sagnac effect error is a result of the signal transmission process and arises from the continuous rotation of the Earth, which is caused by the changing relative distance between the ground terminal and the satellite. This error can be calculated using calibration formulas and is typically on the order of 1 ps [21]. (3) Space propagation time delay difference:  
The space propagation delay difference includes ionospheric delay, tropospheric delay, and geometric path delay. Among them, the ionospheric time delay difference is 10 ps, the tropospheric time delay can be negligible after correction by the tropospheric delay model, and as for the geometric path delay, TWSTT can be accomplished within 10 ms due to the low orbital altitude of LEO satellites so that the uplink and downlink can be regarded as completely symmetrical between the satellite and the ground terminal. Therefore, the space propagation time delay difference is about 10 ps [22]. (4) Signal arrival time measurement error:  
Without loss of generality, a detailed analysis using conventional BPSK navigation signals as an example is presented in this section. The measurement of signal arrival time can be performed either through carrier phase measurement or code phase measurement for receivers. The receiver typically utilizes code phase measurement, due to the long convergence time required for carrier phase measurement, and the TWSTT usually uses a short burst signal. In this case, assuming the receiver uses a code-ring discriminator with a noncoherent pre-subtracted post-power method, we can calculate the mean square error of the code phase measurement in terms of pseudo-code code slices using Equation (6) [23]:  
\[\sigma_{IDLL}=\sqrt{\frac{B_{L}}{2\cdot C/N_{0}}D\bigg{(}1+\frac{2}{(2-D)T_{ \mathrm{coh}}\ \cdot C/N_{0}}\bigg{)}} \tag{6}\]  
where \(B_{L}\) is the loop noise bandwidth, \(T_{\mathrm{coh}}\) is the coherence accumulation time, \(D\) is the early-late correlator spacing, and \(C/N_{0}\) is the CNR. If the chip duration is \(T_{\mathrm{C}}\), then the mean square deviation of time measurement error \(\sigma_{time}\) can be determined by  
\[\sigma_{time}=\sigma_{IDLL}\times T_{C} \tag{7}\]  
Let the chip rate of the BPSK navigation signal be 1.023 MHz, then \(T_{c}=1/1023\) ms, \(B_{L}=1\) Hz, \(T_{\mathrm{coh}}=1\) ms, \(D=1\) chip, and \(C/N_{0}=40\sim 60\) dBHz; this can be seen the variation trend of concern in Figure 5.  
Figure 5: The variation trend of \(\sigma_{time}\) versus \(C/N_{0}\). Figure 5 is only the signal arrival measurement time error of the ground terminal, the signal arrival measurement time error of the satellite can be calculated similarly, and it should be in the same order of magnitude. From the above analysis, it can be seen that the hardware device time delay difference, the Sagnac effect error, and the space propagation time delay difference together do not exceed 100 ps at most, which is much smaller than the signal arrival time measurement error and can be ignored.