## 3 MLE Parameter Estimation Model
### KF Model
The form of \(\xi_{i}\) can be written as:  
\[\xi_{i}(k)=\int_{(k-1)T}^{kT}\frac{\tau^{3-i}}{(3-i)!}Y(\tau)d\tau,\ \ \ i=1,2,3 \tag{28}\]  
\(\xi_{i}(k)\) is the remainder of Taylor expansion, which is used to represent model noise and describes the effects of random interference and model inaccuracy as described above. \(Y(t)\) is a zero-mean, white Gaussian noise process with a single-sided spectral density \(N_{y}\). The noise variance can be expressed as:  
\[E[Y^{2}(t)]=\sigma_{y}^{2}=\frac{N_{y}}{2T} \tag{29}\]  
and the value of \(E[\xi_{i}(k)\cdot\xi_{j}(k)]\) can be obtained:  
\[ \begin{array}{l}E[\xi_{i}(k)\cdot\xi_{j}(k)]\\ =E\Big{\{}\Big{[}\int_{(k-1)T}^{kT}\frac{t^{3-i}}{(3-i)!}Y(u)du\Big{]}\Big{[} \int_{(k-1)T}^{kT}\frac{\tau^{3-i}}{(3-i)!}Y(v)dv\Big{]}\Big{\}}\\ =\iint\limits_{uv}\frac{u^{3-i}}{(3-i)!}\frac{\tau^{3-i}}{(3-i)!}E[Y(u)\cdot Y( v)]\Big{\}}dudv\\ =\frac{\sigma_{v}^{2}\tau^{8-(i+i)}}{(3-i)!(3-i)!(7-(i+j))!}\end{array}  \tag{30}\]  
Then the variance matrix \(Q\) can be derived:  
\[ \begin{array}{l}Q=E[\xi(k)\xi^{T}(k)]\\ =E\Bigg{\{}\left[\begin{array}{c}\xi_{1}(k)\\ \xi_{2}(k)\\ \xi_{3}(k)\end{array}\right]\Big{[}\ \ \xi_{1}(k)\ \ \ \xi_{2}(k)\ \ \ \xi_{3}(k)\ \ \Big{]}\Bigg{\}}\\ =\sigma_{w}^{2}I\ \ \ \ T^{4}/20\ \ \ T^{3}/8\ \ \ T^{2}/6\\ T^{3}/8\ \ \ T^{2}/3\ \ \ T/2\\ T^{2}/6\ \ \ T/2\ \ \ \ 1\end{array}  \tag{31}\]  
where \(\sigma_{w}^{2}=\sigma_{y}^{2}T^{2}\). The forms of \(Q\) and \(R\) can be respectively denoted as [34]:  
\[Q=\sigma_{w}^{2}\left[ \begin{array}{ccc}T^{4}/20&T^{3}/8&T^{2}/6\\ T^{3}/8&T^{2}/3&T/2\\ T^{2}/6&T/2&1\end{array} \right] \tag{32}\]\[R=\sigma_{v}^{2}I \tag{33}\]  
where \(I\) is the unit matrix. The \({\sigma_{v}}^{2}\) in \(R\) is adjusted according to the value of \(A_{mp}\) estimated by MLE. For the \({\sigma_{v}}^{2}\) in the observation covariance matrix \(R\), we obtained a series of experience values based on multiple experiments during the algorithm experiments. According to the estimated value of the signal amplitude \(A_{mp}\) obtained in the MLE, these experience values are utilized to adjust the preset value of the \({\sigma_{v}}^{2}\). The KF algorithm obtains the filter value at the current time according to the state estimation value at the previous moment and the observation value at the current time. The entire carrier loop filtering process can be divided into two parts, the state estimation and state prediction. The output state estimate \(X_{k}\) of KF and the variance matrix \(P_{k}\) of the estimation error can be iteratively calculated by the following equations:  
\[X_{k,k-1}=\Phi X_{k-1} \tag{34}\]  
\[P_{k,k-1}=\Phi P_{k-1}\Phi^{T}+Q_{k-1} \tag{35}\]  
\[K_{k}=P_{k,k-1}H_{k}^{T}[H_{k}P_{k,k-1}H_{k}^{T}+R_{k}]^{-1} \tag{36}\]  
\[X_{kk}=X_{k,k-1}+K_{k}[Y_{k}-\mathrm{H}X_{k,k-1}] \tag{37}\]  
\[P_{k,k}=[I-K_{k}H]P_{k,k-1} \tag{38}\]  
From Equation (36) can be seen that when \(R_{k}\) is large, the corresponding \(K_{k}\) will be small, then the state estimate calculated by Equation (37) is small; when \(Q_{k}\) is small, one step prediction covariance matrix \(P_{k,k-1}\) calculated by Equation (35) will be small, which lead to a smaller state estimate \(X_{k-1}\) finally. From the above analysis, it can be seen that every update of the system state by KF is a compromise between the current system state uncertainty and the observation uncertainty. Therefore, in this paper, \(R_{k}\) and \(Q_{k}\) are obtained by real-time statistics of noise on historical observation and current observation to enhance the adaptability of KF to noise.