## 4 Simulation Experiments
### Parameter Estimation Performance Analysis  
In this subsection, the parameter estimation performance for phase biases is evaluated through Monte Carlo simulations. As described earlier in Equation (19), the phase bias can be considered as a constant, and the estimation is obtained by observing it in a noisy environment. Several estimation methods are carried out and compared below. It has been mentioned that the existing phase compensation technique uses one observation directly as an estimate. This is referred to as the single point estimation in this paper. The robust estimation approach proposed in Section 3.2 obtains the estimate using a window size of \(L\) observation sequence. In this simulation, \(L\) is set to 30 for illustration. The averaging estimation is also considered in this simulation. To compare the three estimation methods mentioned above under different observation conditions, the noise term in Equation (19) is considered as Gaussian and non-Gaussian distributions, which can be modeled as follows:  
(1) Gaussian condition:  
\[\xi_{k}\sim\mathcal{N}(0,R) \tag{25}\]  
(2) Non-Gaussian condition:  
\[\xi_{k}\sim \begin{cases}\mathcal{N}(0,R),&\text{w.p.\ 0.9}\\ \mathcal{N}(0,100R),&\text{w.p.\ 0.1}\end{cases}  \tag{26}\]  
where \(\mathcal{N}(\mu,\Sigma)\) denotes the Gaussian probability density function with mean \(\mu\) and variance \(\Sigma\), w.p. means \"with probability\", and \(R\) denotes the nominal observation noise variance. When the arctangent discriminator is used, the noise variance can be calculated as [18]  
\[R=\sigma_{\xi}^{2}=\frac{1}{2T\cdot C/N_{0}}\left(1+\frac{1}{2T\cdot C/N_{0}}\right) \tag{27}\]  
where \(C/N_{0}\) denotes the the carrier-to-noise ratio. In this simulation, the coherent integration time is set to \(T=1\) ms. The non-Gaussian condition in Equation (26) is an outlier corrupted condition since ten percent of the observation noise values are drawn from the Gaussian distribution with severely increased variance. This condition can be used to test the robustness of the estimation methods. Note that a good estimation method should not be significantly disturbed by the potential observation outliers. The root mean square error (RMSE) is chosen as the performance metric, which is defined as  
\[\text{RMSE}\triangleq\sqrt{\frac{1}{M}\sum_{m=1}^{M}(\psi-\hat{\psi}^{(m)})^{ 2}} \tag{28}\]  
where \(\psi\) is the real phase bias value, \(\hat{\psi}^{(m)}\) is the estimated phase bias value at the \(m\)-th Monte Carlo run, and \(M=\) 10,000 represents the total number of Monte Carlo runs. Figure 2 shows the RMSE results obtained by different estimation methods in association with different signal strengths under the Gaussian and non-Gaussian conditions. It can be seen that the RMSE decreases as the \(C/N_{0}\) increases. However, the estimation accuracy of these estimation methods varies. It is shown that the single point estimation performs worst in both the Gaussian and non-Gaussian conditions, as expected. This is because this estimation approach cannot eliminate the effect of random noise and gross errors. From Figure 2a, the averaging estimation achieves the most accurate estimation in the Gaussian condition, as it is the optimal estimation under this circumstance. The robust estimation sacrifices a little accuracy for robustness. When \(C/N_{0}\) is 30-35 dB-Hz, the accuracy of the robust estimation is relatively lower than that of the averaging estimation. However, it is not easy to track such a weak signal when \(T=1\) ms. Therefore, the accuracy of the phase bias estimation is not important in this case. From Figure 2b, it is shown thatthe robust estimation has the best estimation accuracy under the non-Gaussian condition, indicating that it has excellent robustness resistant to observation outliers. Comparing Figure 2a,b, it can be seen that the single point estimation is significantly affected since only one observation is used, resulting in nontolerable estimation errors.