## 2 Frequency-Locked Detector Output \(C_{2\varphi}\)  
In the GPS receiver, the GPS signal received by the antenna is down converted to intermediate frequency through low noise amplifier, filters, and down-converter [15], the obtained signal \(S(t)\) can be written as follows:  
\[S(t)=AC(t)D(t)\cos[(\omega_{IF}+\omega_{Doppler})t+\phi_{0}]+n(t) \tag{1}\]  
where \(A\) is the signal amplitude, \(C(t)\) is PRN code of GPS, \(D(t)\) is the navigation data, \(\omega_{IF}\) is the intermediate frequency, \(\omega_{Doppler}\) is the Doppler frequency shift, \(\phi_{0}\) is the initial carrier phase, \(n(t)\) is the additive white Gaussian noise.  
According to [4; 16; 17], LOS (line of sight) signal through the front-end \(A/D\) sampling, quantization and integration, the in-phase, and quadrature-phase integrations at \(i\)th interval are \(I_{i}\) and \(Q_{i}\), which are shown as:  
\[\begin{array}{ll}I_{i}&\approx\sqrt{2\frac{C_{0}}{N_{0}}T_{coh}}R(\tau)\sin c (\tau\Delta f_{i}T_{coh})\cos(\tau\Delta f_{i}T_{coh}+\Delta\phi_{i})+n_{I,i} \\ &=\sqrt{2\frac{C_{0}}{N_{0}}T_{coh}}R(\tau)\sin c(\tau\Delta f_{i}T_{coh}) \cos(\Delta\theta_{i})+n_{I,i}\\ &=A_{i}\cos(\varphi_{i})\end{array} \tag{2}\]  
\[\begin{array}{ll}Q_{i}&\approx\sqrt{2\frac{C_{0}}{N_{0}}T_{coh}}R(\tau)\sin c (\tau\Delta f_{i}T_{coh})\sin(\tau\Delta f_{i}T_{coh}+\Delta\phi_{i})+n_{Q,i} \\ &=\sqrt{2\frac{C_{0}}{N_{0}}T_{coh}}R(\tau)\sin c(\tau\Delta f_{i}T_{coh}) \sin(\Delta\theta_{i})+n_{I,i}\\ &=A_{i}\sin(\varphi_{i})\end{array} \tag{3}\]  
where \(C/N_{0}\) is carrier to noise ratio, \(T_{coh}\) is coherent integration time, \(R(\tau)\) is the auto-correlation function of PRN code, \(\tau\) is the code phase delay between received and local replica, \(\Delta f_{i}\) and \(\Delta\phi_{i}\) are the frequency error between received and local replica and the initial phase error between received and local replica, \(\Delta\theta_{i}\) is total phase error that includes \(\Delta f_{i}\) and \(\Delta\phi_{i}\) in the \(i\)th intervals. \(n_{I,i}\), \(n_{Q,i}\) are independent white Gaussian noises corresponding to the \(I\) and \(Q\) branches, respectively. For the convenience of calculation, we take normalization processing of \(n_{I,i}\)-\(N(0,1),n_{Q,i}\)-\(N(0,1)\)[14], \(A_{i}\) and \(\varphi_{i}\) are signal amplitude and average phase estimation, containing noise component, relative to \(\sqrt{2\frac{C}{N_{0}}T_{coh}}R(\tau)\sin c(\tau\Delta f_{i}T_{coh})\) and \(\Delta\theta_{i}\). Detailed discussions of the effects of signal quantization are given in [18; 19].  
The output of frequency-locked detector during a period of time was proposed by [20], which is shown as:  
\[C_{2\varphi}=\frac{1}{M}\sum_{i=1}^{M}\frac{D_{i}}{P_{i}} \tag{4}\]where \(D_{i}=(Dot)^{2}-(Cross)^{2}\), \(P_{i}=(Cross)^{2}+(Dot)^{2}\), \(Dot+jCross=(I_{i}+jQ_{i})(\overline{I_{i-1}+jQ_{i-1}})\). \(M\) is the times of accumulation.  
Further, according to [PERSON] (1965), (4) can be simplified as:  
\[D_{i}=(Dot)^{2}-(Cross)^{2}=[(Dot)^{2}+(Cross)^{2}]\cos(2\varphi_{i}-2\varphi_ {i-1}) \tag{5}\]  
\[\begin{array}{rl}C_{2\varphi}&=\frac{1}{M}\sum\limits_{i=1}^{M}\frac{D_{i}}{ P_{i}}=\frac{1}{M}\sum\limits_{i=1}^{M}\frac{(Dot_{i})^{2}-(\overline{C} \overline{\log_{i}})^{2}}{(Dot_{i})^{2}+(\overline{C}\overline{\log_{i}})^{2} }\\ &=\frac{1}{M}\sum\limits_{i=1}^{M}\frac{[(Dot_{i})^{2}+(\overline{C}\overline{ \log_{i}})^{2}]\cos(2\varphi_{i}-2\varphi_{i-1})}{(Dot_{i})^{2}+(\overline{C} \overline{\log_{i}})^{2}}\\ &=\frac{1}{M}\sum\limits_{i=1}^{M}\cos(2\varphi_{i}-2\varphi_{i-1})\\ &=\frac{1}{M}\sum\limits_{i=1}^{M}\cos(2\Delta\varphi_{i})\end{array} \tag{6}\]  
where \(\varphi_{i-1}\) and \(\varphi_{i}\) are average phase estimation errors in the \((i-1)\)th and \(i\)th intervals, respectively. \(\Delta\varphi_{i}\) is the difference between \(\varphi_{i}\) and \(\varphi_{i-1}\).