## 3 Structure and Phase Error Analysis of Dual Antenna Tracking Loop
### Clock Phase Noise
Figure 4b shows the bandwidth values of the system with the approximated equation \(B_{\mathrm{L1}}-B_{\mathrm{L2}}\) and the numerical method. It can be found that the actual bandwidth is very close to the value from the approximation equation \(B_{\mathrm{L1}}-B_{\mathrm{L2}}\) when the slave loop bandwidth is less than 1 Hz. Thus, the approximate value \(B_{\mathrm{L1}}-B_{\mathrm{L2}}\) is used to compute the carrier phase jitters below for simplicity. According to Equation (18), phase jitter induced by the thermal noise at the output of the slave loop comprises two parts as well: the former is equivalent to the noise from the slave antenna at the output of the slave loop, and the latter is equivalent to the noise from the master antenna at the output of the sub-system \(H\!a(s)\), whose bandwidth is \(B_{\mathrm{L1}}-B_{\mathrm{L2}}\). Assuming that both ideal equivalent filter system gains are equivalent to 1, the thermal noise jitter of the slave loop output can be derived from Equations (5), (8) and (18):  
Figure 3: Equivalent structure of the slave tracking loop. \[ \begin{split}\sigma_{t2}^{2}&=\frac{N_{02}}{2\pi}\int_{0}^{ \infty}|{{{I}_{1}}(\omega)}|^{2}d\omega+\frac{N_{01}}{2\pi}\int_{0}^{\infty}|{{{I }_{d}}(\omega)}|^{2}d\omega\\ &=\frac{N_{02}}{2}\cdot 2{{{B}_{L2}}\cdot|{{{I}_{1-ideal}}(\omega)}| _{\delta}^{2}}+\frac{N_{01}}{2}\cdot 2{{{B}_{Ld}}\cdot|{{{I}_{d-ideal}}(\omega)}|_{ \delta}^{2}}\\ &=\frac{B_{L2}}{C/N_{02}}\Big{(}1+\frac{1}{2T_{coh2}C/N_{02}} \Big{)}+\frac{|{{{B}_{L1}}-{{B}_{L2}}}|}{C/N_{01}}(1+\frac{1}{2T_{coh1}C/N_{01 }})\end{split}  \tag{20}\]  
where \(N_{01}/2\) and \(N_{02}/2\) represent the equivalent PSD of the noise received from the master and slave antennas, and the \(B_{Ld}\) is the equivalent sub system bandwidth. Likewise, the differential phase between outputs of the two loops can be easily derived from Equation (18) and Figure 2 as follows:  
\[ \begin{split}\theta_{d}(s)&=\theta_{o1}(s)-\theta_ {o2}(s)\\ &=\theta_{o1}(s)-\theta_{i2}(s)H_{2}(s)-\theta_{o1}(s)H_{e2}(s)\\ &=\theta_{o1}(s)\big{(}1-H_{e2}(s)\big{)}-\theta_{i2}(s)H_{2}(s) \\ &=\theta_{i1}(s)H_{1}(s)H_{2}(s)-\theta_{i2}(s)H_{2}(s)\\ &=[\theta_{i1}(s)H_{1}(s)-\theta_{i2}(s)]H_{2}(s)\end{split}  \tag{21}\]  
Figure 4: Frequency responses and the bandwidth of the sub system. (**a**) Frequency responses of \(H_{1}(s)\), \(H_{e2}(s)\) and \(H_{d}(s)\); (**b**) Bandwidth of the sub system. The equivalent structure of the system is shown in Figure 5:  
Because the bandwidth of the slave loop is much smaller than that of the master loop, the effect of thermal noise from the assistance to the differential phase is mainly depends on the slave loop. Mathematically, the differential phase jitter induced by thermal noise can be derived as follows:  
\[ \begin{split}\sigma_{d}^{2}&=\frac{N_{02}}{2\pi}\int _{0}^{\infty}|H_{2}(\omega)|^{2}d\omega+\frac{N_{01}}{2\pi}\int_{0}^{\infty}|H_ {2}(\omega)|^{2}d\omega\\ &=\frac{N_{02}}{2}\cdot 2B_{L2}|H_{2-ideal}(\omega)|_{\delta}^{2}+ \frac{N_{01}}{2}\cdot 2B_{L2}|H_{2-ideal}(\omega)|_{\delta}^{2}\\ &=\frac{B_{L2}}{CN_{02}}\Big{(}1+\frac{1}{2T_{coh2}CN_{02}}\Big{)} +\frac{B_{L1}}{2B_{L1}CN_{01}}\Big{(}1+\frac{1}{2T_{coh1}CN_{01}}\Big{)}\cdot 2 B_{L2}\\ &=\frac{B_{L2}}{CN_{02}}\Big{(}1+\frac{1}{2T_{coh2}CN_{02}}\Big{)} +\frac{B_{L2}}{CN_{01}}\Big{(}1+\frac{1}{2T_{coh1}CN_{01}}\Big{)}\end{split}  \tag{22}\]  
It can be seen from the equation above, the assisted loop structure outperforms the independent loop in the accuracy of the differential phase.