## 3 V FAPLL Based on Maximum Likelihood Estimation
The Doppler frequency \(f_{d}\) is:  
\[f_{d}(k)=f_{0}+f_{1}k+\frac{f_{2}}{2}k^{2} \tag{15}\]  
where \(f_{0}\), \(f_{1}\), and \(f_{2}\) represent the Doppler frequency, first-order rate of change of Doppler frequency, and second-order rate of change of Doppler frequency, respectively. Then, the maximum likelihood cost function is as follows:  
\[ \begin{array}{c}L(f_{0}\,\ f_{1},\ f_{2}|g)=\frac{1}{N_{0}}\left\{\sum \limits_{k=0}^{N-1}g(k)\cdot C(k-\tau)\cos{(2\pi T\Big{(}f_{IF}k+f_{0}k+\frac {f_{1}}{2}\ k^{2}+\frac{f_{2}}{6}k^{3}\Big{)})}\right\}^{2}\\ \\ +\frac{1}{N_{0}}\left\{\sum\limits_{k=0}^{N-1}g(k)\cdot C(k-\tau)\sin{(2\pi T \Big{(}f_{IF}k+f_{0}k+\frac{f_{1}}{2}\ k^{2}+\frac{f_{2}}{6}k^{3}\Big{)})} \right\}^{2}\end{array}  \tag{16}\]  
The so-called maximum likelihood estimation of signal frequency performs a three-dimensional search on \(f_{0},f_{1},f_{2}\) to find a set of frequency estimates when the maximum likelihood function reaches the peak value. However, the three-dimensional search of frequency requires considerable computation. As shown in expression (19), the high dimensionality and nonlinearity of the cost function will entail a large amount of computation for the MLE algorithm, which is also the main disadvantage of this algorithm. The peak value of the cost function can be obtained through iterative calculation using the gradient method. For optimizing the maximum likelihood estimation method, we use the Levenberg-Marquardt algorithm. The LM algorithm belongs to optimization algorithm as well as an iterative maximum likelihood estimation method, and it is the most widely used nonlinear least squares algorithm and is an algorithm for finding the extreme value by using gradient, which belongs to the \"mountain climbing\" method. It has the advantages of both the gradient method and the Newton method. When \(\lambda\) is very small, the step size is equal to the Newton method step size, and when \(\lambda\) is large, the step size is approximately equal to the step size of the gradient descent method. The algorithm requires calculating the gradient matrix and Hessian matrix of the log-maximum likelihood function and using the gradient to find the extreme value. The update equation of the algorithm is as follows [27]:  
\[\hat{\theta}_{ML}^{i+1}=\hat{\theta}_{ML}^{i}-(H_{i}+d_{i})^{-1}G_{i}\;\;i=0, \;1,\;\ldots \tag{17}\]  
where \(\theta_{ML}\) represents the state vector, which is equal to \([g,f_{d}]^{T}\). \(G_{i}\) represents the gradient matrix, \(H_{i}\) is the Hessian matrix, and \(i\) represents the number of iterations. \[G_{i}=\left[\frac{\partial L(\theta|g_{N})}{\partial\theta}\right]_{\theta= \hat{\theta}^{i}} \tag{18}\]  
\[H_{i}=\left[\frac{\partial^{2}L(\theta|g_{N})}{\partial\theta^{2}}\right]_{ \theta=\hat{\theta}^{i}} \tag{19}\]  
where  
\[\frac{\partial L}{\partial\theta}=\int\limits_{\frac{\partial L}{\partial \theta}\int\limits_{f_{d}}}^{\frac{\partial L}{\partial f_{d}}}\;,\;\frac{ \partial^{2}L}{\partial\theta^{2}}=\int\limits_{\frac{\partial^{2}L}{\partial f _{d}\partial\tau}}^{\frac{\partial^{2}L}{\partial\theta^{2}\tau}}\frac{ \partial^{2}L}{\partial\tilde{\theta}_{d}^{2}\tilde{\theta}_{d}^{2}}\;\;j \tag{20}\]  
The value of the diagonal matrix \(d_{i}\) must ensure that \(H_{i}+d_{i}\) is always positive such that \((H_{i}+d_{i})^{-1}\) has a solution, the \((H_{i}+d_{i})^{-1}G_{i}\) term is the correction term for each iteration, and the function is equivalent to the discriminator in the traditional loop. In addition, the second-order derivative matrix--the Hessian matrix--is of great significance because it can determine whether the obtained solution is the minimum or maximum value. If the Hessian matrix is negative at the point corresponding to the solution, then this solution is the maximum. The Hessian matrix also defines the covariance matrix of the parameter estimates, and its relationship with the covariance matrix \(V(\theta)\) is \(V(\theta)=\left\{-E[H(\theta)]\right\}^{-1}\).