## 3 MLE Parameter Estimation Model
### MLE Cost Function
\begin{array}{l}\Lambda(\mu|Corr_{N})=-\frac{1}{2\sigma^{2}}(Corr_{N}- \overset{\wedge}{Corr_{N}})(Corr_{N}-\overset{\wedge}{Corr_{N}})^{H}-N\ln(2 \pi\sigma^{2})\\ =-\frac{1}{2\sigma^{2}}\Big{|}Corr_{N}-\overset{\wedge}{Corr_{N}}\Big{|}^{2}- N\ln(2\pi\sigma^{2})\\ =-\frac{1}{2\sigma^{2}}\sum\limits_{n=0}^{N-1}(real([PERSON]_{n})-A_{mp}\cos( \theta))^{2}-\frac{1}{2\sigma^{2}}\sum\limits_{n=0}^{N-1}(imag([PERSON]_{n})-A_{ mp}\sin(\theta))^{2}-N\ln(2\pi\sigma^{2})\\ =-\frac{1}{2\sigma^{2}}\sum\limits_{n=0}^{N-1}\Big{\{}|Corr_{n}|^{2}+A_{mp} ^{2}-2A_{mp}[real(Corr_{n})\cos(\theta)+imag(Corr_{n})\sin(\theta)]\Big{\}}-N \ln(2\pi\sigma^{2})\end{array}  \tag{13}\]  
where \(Corr_{n}=Corr(n)\), \(Corr_{N}=[Corr(0),\,Corr\,(1),\,\ldots\,\,\,.\,\,Corr\,(N-1)]\) is \(N\) consecutive coherent integration results. \(\mu=[A_{mp}\,\Delta f\,\Delta\varphi]^{T}\) represents the signal parameters to be estimated, \(n=0\,\ldots\,\,N-1\), \(\theta=2\pi\Delta f\pi T_{coh}+\Delta\varphi\), \(real(\cdot)\) and \(imag(\cdot)\) represent the real and imaginary part respectively. By finding the maximum value of Equation (13) to get the estimated value of \(\mu\), which can be expressed as:  
\[\frac{\partial\Lambda(\mu|Corr_{N})}{\partial\mu}=0 \tag{14}\]  
Deriving the partial derivative of \(\mu\):  
\[\frac{\partial\Lambda}{\partial A_{mp}}=-\frac{1}{\sigma^{2}}\sum\limits_{n= 0}^{N-1}\left\{A_{mp}-[real(Corr_{n})\cos(\theta)+imag(Corr_{n})\sin(\theta)]\right\} \tag{15}\]  
when \(\frac{\partial\Lambda}{\partial A_{mp}}=0\), the estimation value of \(A_{mp}\) is obtained:  
\[A_{mp}=\frac{1}{N}\sum\limits_{n=0}^{N-1}[real(Corr_{n})\cos(\theta)+imag(Corr _{n})\sin(\theta)] \tag{16}\]  
The signal amplitude \(A_{mp}\) estimated in the MLE is used to subsequently adjust the observation noise covariance matrix \(R\) in the KF. Since the items of \(\mid Corr_{n}\mid^{2}\), \(A_{mp}\)\({}^{2}\) and \(N\!In\! (2\pi\sigma^{2})\) in Equation (13)do not affect \(\Lambda\) partial derivative of \(\Delta f\) and \(\Delta\varphi\) respectively, we can remove such irrelevant items and the cost function is simplified as:  
\[M=\sum_{n=0}^{N-1}\left[real(Corr_{n})\cos(\theta)+imag(Corr_{n})\sin(\theta)\right] \tag{17}\]  
where \(\theta=2\pi\Delta f\Pi T_{coh}+\Delta\varphi\), \(real(\cdot)\) and \(imag(\cdot)\) represent the real and imaginary part respectively. We can find that if \(\theta\) is obtained, then \(A_{mp}\) can be calculated by Equation (16). According to the relationship between \(\theta\), \(\Delta f\) and \(\Delta\varphi\), \(\theta\) can be obtained as long as we get the value of \(\Delta f\) and \(\Delta\varphi\). Thus \(A_{mp}\) can be acquired by Equation (16), the rest is to estimate the value of \(\theta\), which is equivalent to get the value of \(\Delta f\) and \(\Delta\varphi\). Solving the problem of \(\theta\) in Equation (17), which the essence is to obtain the values of \(\Delta f\) and \(\Delta\varphi\). Getting the value of \(\Delta f\) and \(\Delta\varphi\) can be regarded as the process of solving a two-dimensional optimal problem. Therefore, the MLE solution of \(\Delta f\) and \(\Delta\varphi\) in \(\theta\) is transformed to solve the two-dimensional optimal solution by Equation (17).