## 3 System Design
### Acquisition in High-Sensitivity Mode
#### 3.4.3 Implementation  
The acquisition process, when operating in high-sensitivity mode, incorporates both the PCPS algorithm [23] and the GPDIT PDI strategy [7]. The acquisition algorithm computes the CAF using a CI time of 100 ms. The CAF is computed as the circular correlation between the received signal and a local replica of the Galileo pilot codes, using various trial Doppler frequencies. The circular nature of the correlation is ensured by the fact that the Galileo tiered codes have a duration of 100 ms. The outcome of up to 7 consecutive CAFs are merged using the GPDIT technique shown in Equation (3), resulting in a maximum total integration time of 700 ms. The acquisition uses a Constant False Alarm Rate (CFAR) detector. Table 4 shows the input parameters of the acquisition algorithm. Figure 5: Computation of an \(N\)-point FFT as a combination of \(P\) FFTs, each of \(N/P\) points. Table 5 shows the output parameters of the acquisition algorithm for cases where a positive acquisition is obtained. Table 6 shows the variables introduced in the description of the high-sensitivity algorithm. The proposed acquisition process is detailed in Algorithm 1. In the initial step, the acquisition buffers 700 ms of samples, equivalent to \(N\cdot N_{nc}\) samples, where \(N\) represents the number of samples used for CI (equivalent to 100 ms), and \(N_{nc}\) is 7, representing the maximum number of GPDIT non-coherent combinations used for detecting weak signals. For each GPDIT iteration, the algorithm takes 100 ms worth of samples (referred to as \(x[n]\)) from the acquisition buffer \(x_{IN}[n]\) and performs a Doppler frequency search on \(x[n]\), from the minimum Doppler frequency \(f_{min}\) to the maximum Doppler frequency \(f_{max}\) in \(f_{step}\) steps. For each tested Doppler frequency \(f_{d}\), the acquisition conducts Doppler wipeoff and performs a circular correlation between the received signal \(x[n]\) and Galileo pilot codes \begin{table}
\begin{tabular}{c c} \hline \hline
**Variable** & **Definition** \\ \hline \(x_{IN}[n]\) & Received GNSS signal input sample stream \\ \hline \(\bar{P}_{IN}\) & Input signal power estimation \\ \hline \(n_{nc}\) & Current non-coherent combination number \\ \hline \(N_{nc}\) & Maximum number of non-coherent combinations. This value is set to 7 \\ \hline \(N\) & Number of samples used for the coherent integration, representing 100 ms \\ \hline \([f_{min},f_{max}]\) & Doppler frequency span \\ \hline \(f_{d}\) & Tested Doppler frequency \\ \hline \(T_{s}\) & Sampling period \\ \hline \(R_{k}(\tau,f_{d})\) & \(k\)-th successive Cross-ambiguity function (CAF) \\ \hline \(Z_{\text{GPDIT}}(\tau,f_{d})\) & Result of the NCI using the GPDIT strategy as shown in Equation (3) [7] \\ \hline \(\Gamma_{\text{GLRT}}\) & Generalized Likelihood Ratio Test (GLRT) function with normalized variance \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Variables introduced in the description of the high-sensitivity algorithm. \begin{table}
\begin{tabular}{c c} \hline \hline
**Input Parameter** & **Definition** \\ \hline \(f_{min}\) & Minimum tested Doppler frequency \\ \hline \(f_{max}\) & Maximum tested Doppler frequency \\ \hline \(f_{step}\) & Doppler search step \\ \hline \(C[k]\) & FFT of the Galileo pilot signal to be detected, with a tiered code duration of 100 ms \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Input parameters of the high-sensitivity acquisition. \begin{table}
\begin{tabular}{c c} \hline \hline
**Output Parameter** & **Definition** \\ \hline \(f_{data}\) & Estimated Doppler frequency of the received signal \\ \hline \(\tau_{avg}\) & Estimated code phase \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Output parameters of the high-sensitivity acquisition. in the frequency domain \(C[k]\), obtaining the CAF \(R_{k}(\tau,f_{d})\). The CAF is used to compute the corresponding GPDIT iteration \(Z_{\text{GPDIT}}(\tau,f_{d})\).