## 4 Augmented Performance Analysis
### Acquisition Performance with LEO Assisted  
Using LEO to assist GNSS signal acquisition, one benefit is the reduction of the signal search range. The total number of search units in the two-dimensional search range of code phase and frequency can be expressed as:  
\[N_{cell}=\bigg{(}\frac{2f_{unc}}{f_{bin}}+1\bigg{)}\frac{2t_{unc}}{t_{bin}} \tag{15}\]  
where \(f_{unc}\) and \(t_{unc}\) are the uncertain ranges of the signal frequency and code phase, respectively. The searching step in the frequency and code domain are \(f_{bin}\) and \(t_{bin}\), respectively. It can be seen from Equation (15) that the signal acquisition time could be reduced when the number of searches in the uncertain range is small. The search step is typically set to \(1/2\) chip in the code phase. The relationship between the receiver position error \(Z\) and the code phase search number of times \(N\) is as follows:  
\[N=ceil\bigg{(}\frac{2\times Z}{(1/1023)\times 10^{-3}\times C}\bigg{)}. \tag{16}\]  
The typical standalone receiver's code search range for a single GPS satellite is 1023 chips. However, in the LEO-assisted acquisition method, according to the pseudo-range error of the Iridium satellite is \([-1000\) m, \(1000\) m\(]\), the chip deviation caused by the initial positioning error is about six chips, which means the chip search efficiency can be increased by about 170 times. Concerning frequency searching, a different coherent integration time corresponds to different frequency steps. According to the analysis of Section 3, in the LEO positioning-assisted GNSS receiver structure, the frequency error mainly comes from the accuracy of the receiver clock. For commonly used TCXO, the frequency error range is (\(-10\) Hz, \(10\) Hz). Figure 9: SNR performance under different integration combinations. In addition, the stability of the clock limits the coherent integration time to 400 ms. The frequency search step is 2.5 Hz corresponding to a coherent integration time of 200 ms. Figure 10 shows the relationship between Doppler shift deviation and frequency search times under different coherent integration times. It can be seen that a longer integration time leads to larger numbers of frequency searching. Therefore, it is necessary to consider the balance between the SNR gain and the time required to search the entire range of frequency uncertainty. Compared with 200 ms coherent integration, the frequency search step is halved in 400 ms coherent integration, resulting in quadrupled search time. On the other hand, the frequency error caused by the instability of the receiver and the satellite clock increases as the coherent integration time increases. The coherent integration attenuation increases accordingly. Therefore, as a result of compromise, the coherent integration time is set to 200 ms in the implementation. The LEO-assisted GNSS acquisition method not only can improve the efficiency of acquisition but also can improve the acquisition sensitivity. As we know, a long coherent integration time can be used to improve the detection probability in weak signal environments. In the acquisition, the longest coherent integration is limited to 10 ms for a standalone receiver because of the navigation bit reversal. The non-coherent integration method could be used to improve the SNR furtherly but with a square loss due to the analysis above. However, the LEO-assisted acquisition can extend the coherent integration time with aiding of the received navigation information. On the other hand, according to the preliminary estimation of the Doppler and code phase obtained from LEO assistance, a higher false alarm probability can be set to improve the detection probability. Due to non-coherent integration being used in the acquisition, the results of correlation obey the central chi-square distribution when there is no signal.