## 3 V FAPLL Based on Maximum Likelihood Estimation  
The IF signal model received by the receiver is expressed in Formula (7) [27]:  
\[g(t)=h(t)+n(t) \tag{7}\]  
where \(g(t)\) is the observed signal, \(h(t)\) is the real signal, and \(n(t)\) is the noise. Assuming the signal amplitude, code phase delay, Doppler shift, carrier phase, and other parameters change very slowly during the observation time interval so that they can be regarded as unknown constants during this time interval, the digital model of the signal is as follows:  
\[g(k)=A\cdot C(k-\tau)\cos(2\pi T(f_{IF}+f_{d})k+\varphi)+n(k) \tag{8}\]  
where for the meaning of the parameters \(A\), \(C(k)\), \(\tau\), \(T\), \(f\), \(\varphi\), please see \"List of symbols\". (\(A\), \(\tau\), \(f_{d}\), \(\varphi\)) are unknown signal parameters. The maximum likelihood estimation method is to find a value of \(\theta\) suitable for maximizing the possibility of obtaining the observed data, that is, if there is a sample of random vector \(X\) with observation quantity \(x\), find a solution of \(\theta\) such that the corresponding joint probability density function or probability function \(p(x;\theta)\) reach their maximum. For probability \(\text{P}(x|\theta)\), there are two cases: if \(x\) is a variable, \(\theta\) is invariant, and this function is termed the probability function and describes the probability for different sample points \(x\); on the contrary, if \(x\) is invariant, \(\theta\) is a variable, and this function is termed the likelihood function and describes the probability of having sample point \(x\) for different model parameters \(\theta\). If corresponding to the paper, \(\text{P}(r|A,\tau,f_{d},\varphi)\) represents that for a set of given sample values \([g(0),g(1),\dots,g(M)]\), find a set of values of the signal parameters \((A,\tau,f_{d},\varphi)\), so that the probability of the sample values is maximum, that is, find the optimal estimation of the signal parameters \((A,\tau,f_{d},\varphi)\). If the elements of \(g\) are all independent of each other, then the joint distribution  
\[\text{P}(r|A,\tau,f_{d},\varphi)=\prod_{i=1}^{M}p(r_{i}|A,\tau,f_{d},\;\varphi) \tag{9}\]  
is the probability density in Formula (9) in the paper. The product operation entails large amounts of computation, so it is common to take logarithms of the likelihood function, as shown below:  
\[\ln[\prod_{i=1}^{M}p(r_{i}|A,\ \tau,\ f_{d},\ \varphi)]=\sum_{i=1}^{M}\ln[p(r_{i}|A, \ \tau,\ f_{d},\ \varphi)] \tag{10}\]  
where \(\ln[\prod_{i=1}^{M}p(r_{i}|A,\ \tau,\ f_{d},\ \varphi)]\) is the Log--likelihood function, corresponding to \(U(A,\tau,f_{d},\varphi|r)\) in this paper. The joint probability density of the received signal at \(N\) sampling points is [27]:  
\[p(r|A,\tau,f_{d},\varphi)=\frac{1}{(\pi N_{0})^{N}}\exp{(-\frac{1}{N_{0}}(g- \hat{g})^{T}W(g-\hat{g}))} \tag{11}\]  
where \(\hat{g}\) denotes the estimated value of \(g\) obtained from Formula (8) in the absence of noise, and \(W\) denotes the weighting matrix of \(g\). Assuming that the observations and residuals are causally reversible, and the residuals are independent of each other, the log-likelihood cost function of the above joint probability density function can be defined as follows [27]:  
\[U(A,\tau,f_{d},\varphi|g)=-\frac{1}{N_{0}}\sum_{k=0}^{N-1}w_{k}|g(k)-\hat{g}( k)|^{2} \tag{12}\]  
The maximum likelihood estimate of the signal parameters can be obtained by maximizing the log-likelihood cost function \(U\) as follows [27]:  
\[\frac{\partial U(\theta|g)}{\partial\theta}=0 \tag{13}\]  
\[ \begin{array}{c}L(\tau,f_{d}|g)=\frac{1}{N_{0}}\left\{\sum\limits_{k=0}^{N-1 }g(k)C(k-\tau)\cos(2\pi T(f_{IF}+f_{d})k)\right\}^{2}\\ \\ +\frac{1}{N_{0}}\left\{\sum\limits_{k=0}^{N-1}g(k)C(k-\tau)\sin(2\pi T(f_{IF}+ f_{d})k)\right\}^{2}\end{array}  \tag{14}\]  
For the derivation of formula (14), please see the Appendix A.