## 3 System Design
### Doppler Prediction  
The high-sensitivity acquisition parallelizes the code phase search by performing a circular cross-correlation between the received GNSS signal and a local replica of the signal's tiered code. However, this correlation has to be performed once for each possible trial Doppler frequency [23]. In cases where a static GNSS receiver lacks prior knowledge of the received Doppler frequencies, it is obligated to conduct a search spanning from \(-5\) kHz to \(+5\) kHz. As shown in Section 3.4.1, when operating in high-sensitivity mode, the use of a CI time of 100 ms yields a Doppler search step of 5 Hz. Consequently, the acquisition process involves testing 2001 Doppler frequencies, conducting a frequency sweep across all possible frequencies within \(\pm 5\) kHz of the nominal carrier frequency in increments of 5 Hz. This leads to a significant acquisition latency. When the receiver is processing signals in real time, the acquisition latency cannot be arbitrarily large. The key factors limiting the maximum acceptable latency are the maximum allowable TTFF and the degradation over time of the parameters estimated during acquisition, such as timing synchronization and Doppler frequency. This degradation is caused by the receiver clock drift and the continuously changing Doppler frequency in the received signals. If the acquisition process is slow, by the time the receiver initiates tracking the detected signal, the Doppler frequency has already changed. This, coupled with local crystal oscillator inaccuracies, diminishes the accuracy of the Doppler frequency and code phase estimated during the acquisition process. The receiver employs the assistance data to predict the Doppler frequency of the incoming signals, leading to a reduction in the Doppler search space during acquisition. The Doppler search is performed considering potential inaccuracies in the Doppler prediction, and the presence of a Carrier Frequency Offset (CFO) in the analog front-end, originating from deviations in the accuracy of the local crystal oscillator. Despite the use of Doppler prediction, the latency of the high-sensitivity acquisition process may still remain large, decreasing the probability of successfully tracking the detected signals. To address this problem, the receiver offers an option to execute the acquisition process in two stages. The first stage performs a relatively large Doppler search around the predicted Doppler frequency. Then, in the second stage, the acquisition process is repeated, capturing new samples from the analog front-end and employing a very small Doppler search space centered on the Doppler frequency identified in the first stage. The execution of stage 2 is much faster than the execution of stage 1 because of the smaller Doppler search space. While this two-stage execution increases overall acquisition latency and may influence the probability of detection, stage 2 reduces the latency from sample capture to the initiation of the tracking process, thereby enhancing the accuracy of the parameters estimated during acquisition. The two-step acquisition process is shown in Table 7. \begin{table}
\begin{tabular}{c c} \hline \hline
**Stage** & **Algorithm** \\ \hline \multirow{4}{*}{Stage 1} & Capture samples from the analog front-end. \\  & Perform acquisition using a Doppler search space around the predicted Doppler frequency (the Doppler search space is large enough to account for any CFO and Doppler prediction inaccuracy). \\  & Capture samples from the analog front-end. \\  & Perform acquisition using a very small Doppler search space around the Doppler frequency estimated during stage 1. As a result, when executing stage 2, the system quickly transitions from capturing samples to starting the tracking process. \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Acquisition process in two steps.