## 4 Augmented Performance Analysis
### Coherent Integration Time Parameter Selection and the SNR Gain  
The selection of coherent integration time is a key part of the design of GNSS receivers. It is the result of a compromise between various factors. On the one hand, to reduce noise and improve SNR, the integration time should be as long as possible. On the other hand, the integration time cannot be too long due to the attenuation of clock noise according to the analysis above. The selection of coherent integration time must consider both the noise and clock dynamic impact of the receiver. The coherent integral gain can be expressed as:  
\[G_{coh}=10 lg\Big{(}B_{pd}T_{coh}\Big{)} \tag{11}\]  
where \(B_{pd}\) and \(T_{coh}\) indicate the signal pre-detection bandwidth and coherent integration time, respectively. As the coherent integration time increases, the oscillator noise of the receiver accumulates to produce a frequency deviation, which causes the attenuation of the coherent integral gain. To avoid attention, non-coherent accumulation is adopted following coherent integration. The gain of non-coherent accumulation is as follows:  
\[G_{nc}=10 lgN_{nc}-L_{SQ} \tag{12}\]  
where \(N_{nc}\) represents the number of non-coherent accumulations. \(L_{SQ}\) is squaring loss, which exists because the mean of the output noise is not equal to zero due to the squaring operation. Squaring loss \(L_{SQ}\) can be expressed as follows:  
\[L_{SQ}=SNR_{coh}-SNR_{SQ} \tag{13}\]  
where \(SNR_{coh}\) is the SNR of the coherent integration signal, and \(SNR_{SQ}\) is the SNR of the non-coherent integration signal using the same duration, which is defined by:  
\[SNR_{SQ}=\frac{\pi}{\gamma(4-\pi)}\Big{(}e^{-\gamma/2}[(1+\gamma)I_{0}(\gamma /2)+\gamma I_{1}(\gamma/2)]-1\Big{)}^{2}. \tag{14}\]  
In the equation above, \(\gamma\) is defined as \(P^{2}/2\sigma_{n}^{2}\), \(P\) is a given autocorrelation amplitude, \(\sigma_{n}^{2}\) is the noise variance. \(I_{n}(\cdot)\) stands for \(n\) order modified Bessel function. Figure 8 shows squaring loss in non-coherent integration. It can be seen that a higher SNR before non-coherent accumulation corresponds to a lower squaring loss.  
Using the equations above, we compare the SNR gain under different combinations of coherent integration and non-coherent integration. The noise bandwidth before the  
Figure 8: The squaring loss of non-coherent integration.  
correlator is set to 2046 MHz. The range of the pre-detection SNR is from \(-48\) to \(-28\) dB. According to the analysis in Section 3.3, the total integration time is selected as 400 ms considering the influence of TCXO mentioned above. The simulation results are shown in Figure 9. It is obvious that a longer coherent integration time results in greater SNR gain. However, due to the influence of non-coherent loss, the gain of the 200 ms coherent integration time with two non-coherent integrations is almost the same as the direct 400 ms coherent integration, even in the ultra-low SNR stage. Therefore, it is suggested to use 200 ms coherent integration with two non-coherent combinations instead of 400 ms coherent integration, considering the frequency search step reduced along with the longer coherent integration.