## 4 Simulation Results
### Clock Phase Noise  
Two main types of clock phase noises are discussed in this section, namely Allan Deviation phase noise and the vibration induced phase noise [3]. In the simulation, the parameters for the TCXO are chosen as: h\({}_{0}\) = 1e-21, h\({}_{-1}\) = 1e-20, h\({}_{-2}\) = 2e-20, and the parameters of the OCXO is set to be: h\({}_{0}\) = 2.51e-26, h\({}_{-1}\) = 2.51e-23, h\({}_{-2}\) = 2.51e-22. In order to validate the equations provided in Section 3, the phase jitters induced by Allan Deviation phase noise (TCXO) both in the analysis using Equations (15) and (16) and a simulation are given in Figure 7. In this simulation, the master loop bandwidth is changed from 10 Hz to 20 Hz, and the slave loop bandwidth is 0.5 Hz. As depicted in the figure, phase error of the master loop is almost equivalent to that of slave loop, and they increases as the master loop bandwidth decreases. The differential loop phase error is very small when the master loop bandwidth is large enough, such as 15 Hz. More importantly, the analytical results are similar to the simulated results, which could illustrate the Equations (15) and (16) are right. Figure 6: Phase jitter induced by dynamic. Figure 7: Comparisons of the analytical and simulated results. Figures 8 and 9 show the phase jitters induced by Allan Deviation phase noise with the independent and assisting tracking loop structure respectively. The master loop is a 3 rd order loop with a fixed bandwidth of 15 Hz, and the slave loop is a 2 nd order loop with a variable a bandwidth ranged from 0.1 Hz to 10 Hz. The phase jitter of the independent tracking loop is shown in Figure 8. Due to the fixed bandwidth, the phase jitter of the master loop is a constant, which is shown in the Figure 9. In accord with the analysis in the previous section, the phase error of an independent loop becomes larger as the bandwidth decreases. As expected, the OCXO outperforms TCXO, especially for a narrow bandwidth, such as 1 Hz. Another phenomenon shown in this figure is that when the bandwidth is small than a threshold (2 Hz for TCXO and 1 Hz for OCXO), the phase jitter increases rapidly, which could result in a loss of loop. It also can be seen from the figure that the differential phase jitter is almost the same as that of the slave loop. That is because the phase jitter caused by the master loop is much smaller due to its relative large bandwidth. Figure 8: Phase jitter induced by Allan Deviation phase noise with an independent loop. Figure 9: Phase jitter induced by Allan Deviation phase noise with an assisted loop. Figure 9 shows the results of the assisting tracking loop. It is shown that the phase jitter of the two loops is almost equivalent. This matches the analysis presented above. Different from the independent loop structure, the differential phase jitter becomes smaller with the decrease of the bandwidth. It is almost zero when the slave loop bandwidth is smaller than 1 Hz. This means the slave loop could track the signal stably even with a small bandwidth, which is less than 1 Hz. This is because the results are closer to the theoretical results when the cut-off frequencies of the two loops become further apart. It can be also seen that in this assisting tracking loop structure, the differential phase jitter with the two types of oscillators are both very small, especially when the slave loop bandwidth is smaller than 1 Hz. That means the contributions of the two oscillators are almost identical in the differential phase error, the requirement of the oscillator in the receiver is greatly reduced. Based on the same method, vibration induced phase jitters of the independent and assisting tracking loop are analyzed and simulated in this paper. The oscillator's sensitivity is set to be typical value \(K_{\mathrm{g}}=1\)e-9 1/g, and a constant PSD of vibrations of \(G_{\mathrm{g}}=0.05\) g\({}^{2}\)/Hz is used with the frequency under 20 Hz ignored.