## 3 Impact Analysis of Phase Noise in Frequency Domain  
Taking a two-satellite formation satellite inter-satellite ranging system as an example, the application of a phase-noise power law spectral model in frequency domain analysis is analyzed. The core of the ranging system to generate the measurement value is the pseudo code and carrier tracking loop, and the loop tracking error corresponds to the accuracy of the measurement value. Here, we take the phase-locked loop of carrier tracking as an example. Let \(H(f)\) be the system transfer function of the phase-locked loop, as shown in Figure 6, \(E(f)=1-H(f)\) be the error transfer function, \(S_{psi}(f)\) be the phase noise power spectral density of the input signal, and \(S_{poi}(f)\) be the phase noise power spectral density of the local oscillator, then the total error spectrum can be expressed as [19]:  
\[S_{po}(f)=(S_{psi}(f)+S_{poi}(f))|E(f)|^{2}=S_{pi}(f)|E(f)|^{2} \tag{10}\]  
The error variance of phase tracking is expressed as:  
\[\sigma_{\theta}^{2}=\int_{0}^{\infty}S_{po}(f)\mathrm{d}f=\int_{0}^{\infty}S_ {pi}(f)|E(f)|^{2}\mathrm{d}f \tag{11}\]  
When the loop bandwidth is very narrow (within 5 Hz), the first two low-order noises (corresponding to high-frequency noises) in Equation (7) can be ignored, and the latter three types of phase noise dominate the final tracking accuracy. The carrier tracking loop \begin{table}
\begin{tabular}{c c c} \hline
**Frequency (Hz)** & \(\boldsymbol{L(f)}\) **(dBc/Hz)** & \(S_{\phi}(f)\) **(rad2/Hz)** \\ \hline
1\(\sim\)10 & \((-34-51)/2\) & \(5.62\times 10^{-5}\) \\
10\(\sim\)100 & \((-51-63)/2\) & \(2.00\times 10^{-6}\) \\
100\(\sim\)1 K & \((-63-75)/2\) & \(1.26\times 10^{-7}\) \\
1 K\(\sim\)10 K & \((-75-95)/2\) & \(3.16\times 10^{-9}\) \\
10 K\(\sim\)100 K & \((-95-101)/2\) & \(1.58\times 10^{-10}\) \\
100 K\(\sim\)1 M & \((-101-120)/2\) & \(8.91\times 10^{-12}\) \\
1 M\(\sim\)10 M & \(-120\) & \(1.00\times 10^{-12}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Phase noise parameters in each frequency offset range (segment method). Figure 5: Phase jitter calculated according to Table 1. (The frequency deviation within 10 KHz occupies more than 80% of the total error). of the inter-satellite link system uses a third-order phase-locked loop, and its error transfer function can be expressed as:  
\[|E(\omega)|^{2}=|1-H(\omega)|^{2}=\frac{\omega^{2n}}{\omega^{2n}+\omega_{L}^{2n}},n=3 \tag{12}\]  
Among Equation (12), \(\omega_{L}\) represents the loop bandwidth (Hz). By substituting Equation (12) into Equation (11), the tracking error of the third-order phase-locked loop can be obtained as:  
\[\sigma_{\theta}=\sqrt{2}\pi f_{0}[\frac{\pi^{2}h_{4}}{3\omega_{L}^{3}}+\frac{ \pi h_{3}}{3\sqrt{3}\omega_{L}^{2}}+\frac{h_{2}}{6\omega_{L}}]1/2\ (rad) \tag{13}\]  
According to Equation (13), when the phase noise power law spectrum parameters of the measured signal are known, the tracking error caused by the phase noise under a certain loop bandwidth parameter can be calculated. Assuming that the inter-satellite link system receives a signal with a carrier-to-noise ratio of 75 dBHz, the selected reference source is a constant temperature crystal oscillator (OCXO), and the power law spectrum parameters are shown in Table 3. Usually, the power law spectral coefficients can also be obtained by a fitting algorithm based on the measured phase noise curve or the phase noise index. The final tracking errors can be obtained as shown in Figure 7 below. It can be seen that the phase-locked loop presents a low-pass characteristic to thermal noise (described by the carrier-to-noise ratio index), and the smaller the loop bandwidth, the smaller the tracking error introduced by the thermal noise. The phase-locked loop presents a high-pass characteristic of the phase noise. The larger the loop bandwidth, the smaller the tracking error caused by the phase noise. Accordingly, it is necessary to comprehensively select the loop bandwidth according to the actual system measurement accuracy index.