## 3 MLE Parameter Estimation Model
### LM Algorithm  
The LM known as the damped least-squares (DLS) method, is one of the most effective algorithms to solve nonlinear least squares problems, which has both the advantages of gradient descent method and Gauss-Newton algorithm (GNA). The LM uses a gradient to find the maximum (minimum) value and obtains the optimal solution through iterative convergence. The (non-negative) damping factor \(\lambda\) is adjusted at each iteration. When \(\lambda\) is used as a smaller value, bringing the algorithm closer to the GNA, whereas if an iteration gives insufficient reduction in the residual, \(\lambda\) can be increased, giving a step closer to the gradient-descent direction. We use the LM algorithm to iterate to get the frequency and phase values in the paper. The duration of the coherent integration time is 1.6 ms, which is consistent with the LM iteration time. And the loop adjustment period of the carrier tracking loop is 25 ms. Since the LM iteration time is less than the loop adjustment period, the algorithm proposed in this paper can ensure the real tracking. Therefore, the LM algorithm is used to solve the two-dimensional optimal solution in Equation (17). The iterative optimal solution is given by the following equation:  
\[\overset{\wedge}{\mu}_{i+1}=\overset{\wedge}{\mu}_{i}+(H_{i}+\lambda)^{-1} \mathrm{G}_{i};i=0,1,2\ldots \tag{18}\]  
where subscript \(i\) represents the number of iterations, \(\overset{\wedge}{\mu}_{i}\) is an \(2\times 1\) state vector including \(\Delta f\) and \(\Delta\varphi\), \(\lambda\) is an \(2\times 2\) diagonal matrix, used to ensure that \(H_{i}+\lambda\) is positive definite and adjust the iterative convergence rate. \(H_{i}\) and \(G_{i}\) is \(2\times 2\) pseudo-Hessian matrix and \(2\times 1\) gradient vector respectively. \(H_{i}\) and \(G_{i}\) can be written as:  
\[H_{i}=\left[\frac{\partial^{2}M}{\partial\mu^{2}}\right]_{\mu=\mu_{i}} \tag{19}\]  
\[G_{i}=\left[\frac{\partial M}{\partial\mu}\right]_{\mu=\mu_{i}} \tag{20}\]  
where  
\[\frac{\partial M}{\partial\mu}=\left[\begin{array}{cc}\frac{\partial M}{ \partial\Delta f}&\frac{\partial M}{\partial\Delta\varphi}\\ \end{array}\right]^{T}\!,\frac{\partial^{2}M}{\partial\mu^{2}}=\left[\begin{array} []{cc}\frac{\partial^{2}M}{\partial\Delta f^{2}}&\frac{\partial^{2}M}{ \partial\Delta\varphi}\\ \frac{\partial^{2}M}{\partial\Delta\varphi\partial\Delta f}&\frac{\partial^{2} M}{\partial\Delta\varphi^{2}}\\ \end{array}\right] \tag{21}\]  
The entire LM algorithm iterative process is shown in Figure 4. The first step is to initialize the value of \(\overset{\wedge}{\mu}_{0}\) and \(\lambda\). Since we assume that the carrier is correctly tracked at the beginning then \(\overset{\wedge}{\mu}_{0}=[0\;0]\) and \(\lambda\) initialized to an experience value. Next, Equations (19) and (20) are utilized to get \(H_{i}\) and \(G_{i}\). Increasing \(\lambda\) to make \(H_{i}+\lambda\) positive definite. Then Equation (18) is used to update \(\overset{\wedge}{\mu}\) and Equation (17) is used to judge whether \(M(\overset{\wedge}{\mu}_{i+1})>M(\overset{\wedge}{\mu}_{i})\) is satisfied. If not, continue to increase \(\lambda\). If satisfied, judging the gradient vector \(G\) is less than 0.2 or greater than 0.8 (the judgment condition for \(G\) is set according to the actual test), then adjust \(\lambda\) according to the conditions respectively satisfied. The physical meaning of adjusting \(\lambda\) is that if the estimated value \(\overset{\wedge}{\mu}\) is closer to the optimal value of iteration, then increase \(\lambda\) to slow down the convergence rate, otherwise decrease \(\lambda\) to accelerate iterative convergence. The termination condition of the above iterative process is that the value of the gradient vector \(G\) falls below predefined threshold (Pre_Thres) or the number of iterations (Iter_Num) exceeds the set maximum (Iter_Max).