## 4 Results and Discussion  
### Optimization Results  
The mathematical model presented in Section 3 was solved using an AMD Ryzen 9 3900X 12-core processor. The time consumption of each step is listed in Table 2. To reduce the complexity associated with the solution of multi-objective problems and improve the convergence efficiency of the solution set, we used one of the most popular multi-objective optimization algorithms: the nondominated sorting genetic algorithm II (NSGA-II) [24]. As a characteristic of multi-objective optimization algorithms, an optimal solution set is usually obtained instead of a single optimal solution to balance the degree of optimization of each objective. The Pareto-optimal solution plane obtained in Step 1 of Section 3.2 is shown in Figure 3.  
During the operation of a mission, a smaller total readout displacement noise \(\delta x^{\text{total}}\) indicates a better detection of signals, and a larger feasible range of the beat-note frequency, which exists between \(f_{\text{upper}}\) and \(f_{\text{lower}}\), indicates a better offset frequency setting. In Figure 3a, the warmer the color of the scatter plot, the higher the value of \(\delta x^{\text{total}}\). Hence, the smaller the value of the beat-note frequency range, the smaller the value of the modulation depth; a larger value of \(f_{\text{USO}}\) corresponds to a smaller value of \(\delta x^{\text{total}}\). In Figure 3b, a \begin{table}
\begin{tabular}{c c c} \hline \hline
**Steps** & **Algorithm** & **Time Consumption** \\ \hline Step 1 & Nondominated sorting genetic algorithm II [24] & 240 s \\ Step 2 & Round & 0.005 s \\ Step 3 & Exhaustive enumeration & 1 s \\ Step 4 & Optimization & 0.03 s \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Time consumption of each step. Figure 2: Flowchart of the algorithm. warmer scatter color indicates a larger beat-note frequency interval; a larger beat-note frequency range corresponds to a larger \(\delta x^{\text{total}}\) value. Based on Figure 3, we can conclude that the value of the beat-note frequency range is inversely related to the magnitude of \(\delta x^{\text{total}}\). The two targets need to be reasonably balanced via optimization to increase the maximum feasible range of the beat-note frequency and decrease \(\delta x^{\text{total}}\). Figure 3: Pareto-optimal solution plane: (**a**) Scatter plot of the total readout displacement noise with the modulation depth; USO frequency; and beat-note frequency in a feasible range along the x-, y-, and z-axes, respectively. The total readout displacement noise \(\delta x^{\text{total}}\) is represented by the colored bar. (**b**) Scatter plot of the beat-note frequency in a feasible range with the modulation depth; USO frequency; and total readout displacement noise along the x-, y-, and z-axes, respectively. The beat-note frequency in a feasible range is represented by the colored bar. This planning problem was solved based on the optimization algorithm introduced in Section 3.2. The results of the 20 Pareto-optimal (feasible) solutions obtained from the final solution are listed in Table 3. The units of \(f_{\text{lower}},f_{\text{upper}},f_{\text{ADC}},f_{\text{PT}},\) and \(f_{\text{USO}}\) are MHz, and the total readout displacement noise \(\delta x^{\text{total}}\) has units of \(\text{pm}/\sqrt{\text{Hz}}\). It is worth noting that these are the optimal 20 solutions by balancing the two optimization objectives. The results in Table 3 show that, after utilizing the proposed algorithm, the maximum beat-note frequency feasible range is 23 MHz and yields a total readout displacement noise of 4.21 \(\text{pm}/\sqrt{\text{Hz}}\), which corresponds to the parameter combination (\(f_{\text{lower}},f_{\text{upper}},f_{\text{ADC}},\)\(f_{\text{PT}},f_{\text{USO}},\)\(m\)) = (3 MHz, 25 MHz, 92 MHz, 90 MHz, 4140 MHz, 0.44).