# Comparison of L1 and L5 Bands GNSS Signals Acquisition
## 4 Application to GPS and Galileo Signals
### Comparison of GPS L1 C/A and L5 Signals Acquisition
#### 4.2.3 Determination of Ratio of Complexity for Multiple Frequency Bins  
Now that the complexity considering only one frequency bin has been studied, let us consider the real case where it is the acquisition time that matters. For this, the relation between the processing time of one frequency bin and the acquisition time should be determined, i.e., how many frequency bins are searched until the detection of the signal. Of course, this depends on the Doppler frequency of the received signal; however, assuming a uniform distribution of the Doppler frequency over the search space, in average \(N_{\text{FB}}/2\) frequency bins are searched. Therefore, to include this in the evaluation of the complexity, the ratio of number of frequency bins searched between the different signals should be estimated, and since the number of frequency bins searched is directly proportional to the total number of bins, the ratio of total number of bins could be used. Two elements will thus be involved: (1) The width of the search space; and (2) the frequency step to browse the search space, which is inversely proportional to coherent integration time. Therefore, compared to the previous results, for the L5 signal with a coherent integration limited to 20 ms (as the L1 C/A signal), it is expected that the final ratio of complexity will be lower since the width of the search space is lower. However, for the L5 signal with longer coherent integration time, the frequency step will decrease, the number of frequency bins will increase and thus the final complexity will increase. This is shown in Table 11, where the ratio of number of frequency bins is around 0.8 when non-coherent integration is used regardless the assistance, and where the ratio increases with the sensitivity when coherent integration only is used. Applying these ratios to Table 10 gives Table 12 that shows the final complexity ratio, considering the processing time of one frequency bin, the number of frequency bins searched and the memory requirements. It can then be seen that without assistance, the L5 signal is much more complex to acquire. With assistance, the ratios of complexity are lower, but the L5 signal is still more complex to acquire, except for a few cases. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Parallel without Assistance**} & \multicolumn{4}{c}{**Serial with Assistance**} \\ \cline{2-9}
**Sensitivity** & \multicolumn{2}{c}{**Non-Coherent**} & \multicolumn{2}{c}{**Coherent Only**} & \multicolumn{2}{c}{**Non-Coherent**} & \multicolumn{2}{c}{**Coherent Only**} \\ \cline{2-9}  & **Average** & **Worst** & **Average** & **Worst** & **Average** & **Worst** & **Average** & **Worst** \\ \hline \(-140\) dBm & 84.2 & 109.3 & 40.9 & 47.4 & 4.2 & 5.5 & 1.7 & 1.9 \\ \(-145\) dBm & 97.6 & 111.1 & 52.1 & 45.6 & 4.9 & 6.3 & 2.1 & 2.1 \\ \(-150\) dBm & 105.4 & 106.8 & 34.6 & 25.0 & 6.0 & 6.2 & 1.6 & 1.1 \\ \(-155\) dBm & 102.4 & 101.6 & 15.3 & 9.5 & 5.9 & 5.9 & 0.7 & 0.4 \\ \(-160\) dBm & 98.9 & 98.2 & 5.6 & 3.2 & 5.7 & 5.8 & 0.3 & 0.2 \\ \hline \(-140\) dBm & 84.2 & 109.3 & 41.0 & 47.4 & 5.5 & 8.6 & 2.1 & 2.8 \\ \(-145\) dBm & 96.2 & 108.3 & 51.7 & 43.8 & 8.8 & 14.0 & 3.7 & 4.6 \\ \(-150\) dBm & 97.6 & 90.9 & 29.8 & 17.0 & 22.7 & 35.4 & 4.4 & 3.5 \\ \(-155\) dBm & 78.0 & 69.4 & 6.6 & 2.1 & 49.3 & 58.7 & 2.0 & 0.9 \\ \(-160\) dBm & 64.5 & 62.5 & 0.6 & 0.2 & 62.4 & 63.5 & 0.3 & 0.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Ratio of complexity (product of processing time ratio and memory ratio) between the L5 and L1 C/A signals. Top rows do not consider the input memory, while bottom rows consider the input memory.