## 2 Open-Loop Tracking of GNSS Signals
The correlator outputs are  
\[I_{d,k}=\sqrt{P}R(\tau-\tau_{d})sinc(\pi(f_{d}-f_{k})T_{c})cos\Delta\varphi+n_{I} \tag{6}\]  
\[Q_{d,k}=\sqrt{P}R(\tau-\tau_{d})sinc(\pi(f_{d}-f_{k})T_{c})sin\Delta\varphi+n_{Q} \tag{7}\]  
where \(R(\cdot)\) is the autocorrelation function of the PRN code, \(T_{c}\) is the coherent integration time, \(\Delta\varphi\) is the carrier phase error. \(n_{I}\) and \(n_{Q}\) are the noise components which are still AGWN. Their variances are uniformly expressed as below  
\[ \begin{split}\sigma^{2}_{n,\pm}&=\frac{2}{N_{c}^{2 }}\sum_{n=0}^{N_{c}-1}E[n_{w}^{2}(t_{n})\cdot\frac{1\pm cos(4\pi(f_{IF}+f_{k})t _{n})}{2}]\\ &=\frac{N_{0}}{2T_{c}}\end{split}  \tag{8}\]  
where \(E[x]\) is the expectation of \(x\). \"+\" and \"\(-\)\" correspond to the variances of \(n_{I}\) and \(n_{Q}\) respectively and they are uniformly represented as \(\sigma^{2}_{n}\) in the following. The energy of the correlator outputs is computed based on I and Q values as below which makes up the entire image of the signal  
\[P(\tau_{d},f_{k})=I_{d,k}^{2}+Q_{d,k}^{2} \tag{9}\]  
The satellite signal is deemed to be in the grid with the maximum power peak in the batch correlator outputs as shown in Figure 1. The measurement of the incoming signal are calculated according to the code phase and Doppler frequency shift of the reference signal of the grid as below  
\[(\hat{\tau},\hat{f}_{D})=arg\max_{\tau_{d},f_{k}}P(\tau_{d},f_{k}) \tag{10}\]  
The estimation results \(\hat{\tau}\) and \(\hat{f_{D}}\) of the signal are used to form the final signal measurements. Different from the closed loop that there is a limitation in the power accumulation length owning to the requirement of the system stability, the power accumulation length of the open-loop tracking is not limited. It can be lengthened to hundreds of micro seconds and the sensitivity can be greatly improved. In this paper, the coherent integration length of 300 ms is taken as an example and the sensitivity is sufficient for the urban application [18]. For other values, although the detail parameters may change, the basic conception is not altered and the analysis results are the same. Nevertheless, there is a serious drawback for the open loop tracking method, which is the degraded measurement accuracy. As the measurement of open loop tracking is calculated from the parameters of the discrete search grid of the detected peak, the uncertainty in the grid will probably deteriorate the measurement accuracy. If the signal is detected in the false grid, the measurement error will be even further enlarged. Compared with the Doppler frequency measurement, the problem to code phase measurement is more serious. For Doppler frequency, considering the long coherent integration length of open loop tracking, the width of the frequency search grid is limited to an extremely small value due to the large power loss brought by the frequency mismatch [21]. If the coherent integration is as long as 300 ms as the statement above, the frequency search grid size is at most 5 Hz. Converted to velocity in the unit of m/s, taking the center frequency 1575.42 MHz of GPS L1 CA signal as an example, the uncertainty within the grid is at most 0.48 m/s. Furthermore, the false detection in the frequency dimension will almost never occur because even the neighboring grids of the right one contain little signal power. Accordingly, the effect of the discrete searching on the Doppler frequency measurement accuracy is a little. For the code phase measurement, the situation is really different. In the autocorrelation function of the PRN code, signal power is contained within the code phase delay from \(-1\) chip to 1 chip and it has nothing to do with the coherent integration length. Considering the code chip duration in distance is as long as hundreds of meters, the errors induced by the uncertainty in the grid and false detection will both exist and considerably degrade the accuracy.