## 2 Open-Loop Tracking of GNSS Signals  
This section reviews the main concept of the open-loop tracking of GNSS signals. The block diagram of the batch correlators based open-loop tracking method is illustrated in Figure 1. The input signal is transferred to multiple identical correlators at the same time. In the presence of Additive Gaussian White Noise (AGWN), the input signal is expressed as follow  
\[s(t_{n})=\sqrt{2P}D(t_{n}-\uptau)c(t_{n}-\uptau)cos(2\uppi(f_{IF}+f_{d})t_{n}+ \varphi_{0})+n_{w}(t_{n}) \tag{1}\]  
where \(t_{n}\) is the discrete sampling time: \(t_{n}=t_{0}+nT_{s}\) and \(T_{s}\) is the sampling period. \(P\) is the incoming signal power. \(D(\cdot)\) is the navigation message data or secondary code. \(c(\cdot)\) is the Pseudo Random Noise (PRN) primary code. \(\uptau\) is the code phase shift of the incoming signal. \(f_{IF}\) is the signal down-conversion frequency. \(f_{d}\) is the instantaneous value of the Doppler frequency shift of the received satellite signal. \(\varphi_{0}\) is the initial value of the carrier phase. \(n_{w}\) is the additive Gaussian white noise with the single band power density of \(N_{0}\). Since an orthogonal set of binary pseudorandom codes are generated for different satellites, the signals from different satellites can be processed independently from each other [21]. As a result, the input signal model here contains only one satellite for simplicity. Based on the architecture of batch correlators shown in Figure 1, open-loop tracking estimates the signal code phase and Doppler frequency shift via discrete searching. Each combination of the code phase and Doppler frequency shift makes up a discrete search grid in the time-frequency two-dimension searching space. Every correlator in the architecture corresponds to one of the search grids. The correlation space, namely the code phase delay between the adjacent correlators in code dimension is the size of code search grid. The frequency difference between those in frequency dimension is the frequency search grid size. In the correlator, the input signal is correlated with the local reference signal which is composed of the carrier and code sequence generated by the Numerically Controlled Oscillators (NCO). Correlation is performed to compute the outputs of the in-phase (I) and quadrature (Q) branches as below  
\[I_{d,k}=\frac{1}{N_{c}}\sum_{n=0}^{N_{c}-1}s(t_{n})c(t_{n}-\tau_{d})\cdot \sqrt{2}cos(2\pi(f_{IF}+f_{k})t_{n}) \tag{2}\]  
\[Q_{d,k}=\frac{1}{N_{c}}\sum_{n=0}^{N_{c}-1}s(t_{n})c(t_{n}-\tau_{d})\cdot \sqrt{2}sin(2\pi(f_{IF}+f_{k})t_{n}) \tag{3}\]  
where \(N_{c}\) is the number of signal samples during the interval of coherent integration. In order to ensure signal independence, I and Q correlation values must be accumulated over an integer number of primary code periods which is 1 ms for the GPS L1 CA signal and the length cannot extend the navigation data duration for the unknown transitions of navigation data when no external assisted information is introduced. However, for new GNSS signal such as GPS L5 with pilot channel, the accumulation length can be set as long as possible when the secondary code is synchronized [22]. With the help of  
Figure 1: The block diagram of the batch correlators based open-loop tracking method. Assisted-GPS (AGPS), the time of code phase transition induced by navigation data is known and the coherent integration can be expanded over the data duration by taking care of the transition [23]. \(\tau_{d}\) and \(f_{k}\) are the code phase and Doppler frequency shift of the corresponding search grid which are  
\[\tau_{d}=d\cdot\tau_{b} \tag{4}\]  
\[f_{k}=k\cdot f_{b} \tag{5}\]  
where \(\tau_{b}\) and \(f_{b}\) are the sizes of code and frequency search grid respectively.