# Weak and Dynamic GNSS Signal Tracking Strategies for Flight Missions in the Space Service Volume
## 4 INS-Assisted Open Loop Tracking Strategy
### Non-Coherent MLE Algorithm  
Hereinafter, a non-coherent MLE algorithm to estimate Doppler frequency error that is insensitive to carrier phase error is introduced. By computing the sum of the squares of \(I_{P}\) and \(Q_{P}\), we have the value of non-coherent integration (NI):\[V=I_{P}^{2}(k)+Q_{P}^{2}(k)=A^{2}[R(\delta\tau)]^{2}[D(k)]^{2}[\text{sinc}(kT \cdot\delta f)]^{2} \tag{29}\] \[=A^{2}[R(\delta\tau)]^{2}[\text{sinc}(kT\cdot\delta f)]^{2}\]  
It is obvious that NI brings about two positive features: First, the value of \(V\) is not affected by data bit \(D(k)\) regardless of whether \(D(k)\) equals +1 or \(-1\), so the influence of data bit transition is eliminated; Second, the value of \(V\) has nothing to do with carrier phase error \(\delta\phi_{0}\), and it changes with \(\delta f\). The two features induce us to construct a maximum likelihood cost function for Doppler estimation using the NI result, and the expression is as follows:  
\[L\left(\delta f\right)=\sum\limits_{k=1}^{N}{[I_{P}^{2}(k)+Q_{P}^{2}(k)]}=\sum \limits_{k=1}^{N}{A^{2}[R(\delta\tau)]^{2}[\text{sinc}(kT\cdot\delta f)]^{2}} \tag{30}\]  
where \(N\) represents the total number of IF data samples within a given CIT. According to the normal MLE solving method, \(\delta f\) reaches its MLE where the partial derivative of the likelihood cost function or its log-likelihood function [23] with respect to \(\delta f\) is 0. However, obviously, it is too complicated to write the analytic expression of the MLE of \(\delta f\). Thus, we rewrite the cost Function (30) as:  
\[L\left(\delta f\right)=\sum\limits_{k=1}^{N}{[I_{P}^{2}(k)+Q_{P}^{2}(k)]}=\sum \limits_{k=1}^{N}{[s(k)\cdot C\cdot\text{cos}]^{2}}+\sum\limits_{k=1}^{N}{[s(k )\cdot C\cdot\text{sin}]^{2}} \tag{31}\]  
For the sake of simplicity, we use \(C\) as \(C(k-\tau)\), \(\text{cos}\) as \(\text{cos}[2\pi(f_{IF}+\delta f)kT+\delta\phi]\), and \(\text{sin}\) as \(\text{sin}[2\pi(f_{IF}+\delta f)kT+\delta\phi]\). Then the gradient of the new cost function for the Doppler frequency is given by:  
\[ \begin{split} Gradient=\frac{\partial L}{\partial\delta f}=-4\pi T \sum\limits_{k=1}^{N}{[s(k)\cdot C\cdot\text{cos}]}\times\sum\limits_{k=1}^{N }{[k\cdot s(k)\cdot C\cdot\text{sin}]}\\ +4\pi T\sum\limits_{k=1}^{N}{[s(k)\cdot C\cdot\text{sin}]}\times \sum\limits_{k=1}^{N}{[k\cdot s(k)\cdot C\cdot\text{cos}]}=-4\pi T(I_{P} \overline{Q}_{P}-\overline{I}_{P}Q_{P})\end{split}  \tag{32}\]  
where:  
\[\left\{ \begin{array}{c}I_{P}=\sum\limits_{k=1}^{N}{s(k)\cdot C\cdot\text{ cos}}\\ Q_{P}=\sum\limits_{k=1}^{N}{s(k)\cdot C\cdot\text{sin}}\\ \overline{I}_{P}=\sum\limits_{k=1}^{N}{k\cdot s(k)\cdot C\cdot\text{cos}}\\ \overline{Q}_{P}=\sum\limits_{k=1}^{N}{k\cdot s(k)\cdot C\cdot sin}\end{array} \right. \tag{33}\]  
The Hessian [24] of the proposed cost function with regard to \(\delta f\) is given by:  
\[Hessian=\frac{\partial^{2}L}{\partial\delta f^{2}}=8\pi^{2}T^{2}(\overline{I} _{P}^{2}+\overline{Q}_{P}^{2}-I_{P}^{\ \ \ =\ \ }\overline{I}_{P}-Q_{P}\overline{\bar{Q}}_{P}) \tag{34}\]  
where:  
\[\left\{ \begin{array}{c}\overset{=}{\underset{I_{P}}{=}}=\sum\limits_{k=1}^{N }{k^{2}\cdot s(k)\cdot C\cdot\text{cos}}\\ \overset{=}{\underset{I_{P}}{=}}=\sum\limits_{k=1}^{N}{k^{2}\cdot s(k)\cdot C \cdot sin}\end{array} \right. \tag{35}\]Referring to [25], the frequency error is estimated with the gradient divided by the Hessian:  
\[\delta f=-\frac{Gradient}{Hessian}=\frac{I_{P}\overline{Q}_{P}-\overline{I}_{P}Q _{P}}{2\pi T(\overline{I}_{P}^{2}+\overline{Q}_{P}^{2}-I_{P}\overline{I}_{P}-Q _{P}\overline{\overline{Q}_{P}})} \tag{36}\]  
The calculation is completed in the block named MLE-based FLL discriminator in Figure 4. Finally, the coupling effect between Doppler frequency error and carrier phase error is cancelled. In addition, the code delay \(\tau\) is still estimated with CL scheme using a carrier-aided DLL. There are two early-minus-late (EML) type correlators with outputs:  
\[\left\{