# Comparison of L1 and L5 Bands GNSS Signals Acquisition
## 4 Application to GPS and Galileo Signals  
### Determination of the Integration Time  
Now, let us compare the different implementations with actual parameters. To have a good overview of the performance, several sensitivities are selected, from \(-140\) dBm to \(-160\) dBm with a step of \(5\) dBm to cover moderate to high sensitivities. However, as mentioned in the introduction and shown in Table 1, the expected power is not the same for the different signals. Therefore, to perform a fair comparison, the sensitivities should be adapted for each signal, and the values above would be for one signal only. Here the GPS L1 C/A has been chosen as reference. Since the GPS L5 signal has an expected power \(1.5\) dB above the L1 C/A signal power, the sensitivities considered for the L5 signal are \(-138.5\) dBm, \(-143.5\) dBm, \(-148.5\) dBm, \(-153.5\) dBm, and \(-158.5\) dBm. This allows us to do a fair comparison representative of the reality. The same will apply for the Galileo signals.  
Then, again to have a fair comparison, some elements must be considered specifically for each signal: The number of frequency bins and code bins, which influence the size of the search space and consequently influence the required Signal-to-Noise Ratio (SNR), probability of false alarm and probability of detection. This is illustrated in Figure 10, with \(N_{\text{FB}}\) the number of frequency bins, \(N_{\text{PCB}}\) the number of primary code bins, \(N_{\text{SCB}}\) the number of secondary code bins, \(N_{\text{CB}}=N_{\text{PCB}}N_{\text{SCB}}\) the number of code bins, \(N_{\text{CELL}}=N_{\text{FB}}N_{\text{CB}}\) the number of cells of the search space, \(P_{\text{FA,G}}\) and \(P_{\text{FA,C}}\) the global and cell probabilities of false alarm respectively, related by \(P_{\text{FA,G}}=1-\left(1-P_{\text{FA,C}}\right)^{N_{\text{CELL}}}\) or \(P_{\text{FA,C}}=1-\left(1-P_{\text{FA,C}}\right)^{1/N_{\text{CELL}}}\), and \(P_{\text{D}}\) the probability of detection [8].  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{**Implementation**} & \multicolumn{2}{c}{**Processing Time of One Frequency Bin \(T_{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{\text{ \text  
Finally, once the required SNR is estimated, the coherent integration time and the number of non-coherent accumulations can be evaluated, as shown in Figure 11. The implementation losses depend on the quantization, frequency step, code step, and potential sign transition. Following the outcomes of [32], a 3-bit quantization, a frequency step \(\delta f\) of \(2/(3T_{\text{C}})\), and a code step of \(1/2\) chip is considered, and since the full bit method and secondary code correlation is used there is no transition loss. Regarding the case of coherent integration only, Figure 11b shows that this is an iterative process. There is even a second iteration not shown in Figures 10 and 11, because increasing the coherent integration will decrease the frequency step (since \(\delta f=2/(3/T_{\text{C}})\)). This will increase the number of frequency bins \(N_{\text{FB}}\) (since \(N_{\text{FB}}=f_{\text{SS}}/\delta f=3f_{\text{SS}}T_{\text{C}}/2\) with \(f_{\text{SS}}\) the search space), which itself increases the required SNR after Figure 10, which consequently may change the coherent integration time after Figure 11b. Therefore, for a very accurate estimation in the case of coherent integration only, these iterations should be performed.