## 3 System Design
### Acquisition in High-Sensitivity Mode  
#### 3.4.1 Theory of Operation  
The acquisition process makes it possible to identify faint GNSS signals. Upon detecting a signal, the acquisition offers an estimation of both the code delay and Doppler frequency sufficiently accurate to initiate the tracking loops. The acquisition process is a two-dimensional operation, involving the correlation of the received signal \(x_{IN}[n]\) with a local replica of the transmitted signal \(d[n]\) across various trial values of Doppler frequency \(f_{d}\) and time delay \(\tau\). The circular correlation between \(x_{IN}[n]\) and \(d[n]\) is referred to as the Cross Ambiguity Function (CAF). If we assume the absence of bit transitions in the received signal, the CAF can be expressed using Equation (1). In this equation, \(N\) represents the number of samples integrated coherently, and \(T_{s}\) denotes the sampling period. \[R_{xd}(\tau,f_{d})=\frac{1}{N}\sum_{n=0}^{N-1}x_{IN}[n]d[nT_{s}-\tau]e^{-j2\pi f _{d}nT_{s}} \tag{1}\]  
PDI techniques can be employed to overcome the limitation of extending CI indefinitely. These techniques involve combining multiple consecutive CAFs through a non-linear function, as depicted in Equation (2) [7]. In this equation, \(Z_{\rm X}\) is the result of the NCI ob \begin{table}
\begin{tabular}{c c} \hline \hline
**Assistance Data** & **Source** \\ \hline Reference date and time & Receiver embedded OS clock \\ \hline Reference user location & GNSS-SDR configuration file \\ \hline Galileo ephemeris data & \\ Galileo ionospheric data & XML files \\ Galileo UTC model & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Assistance data. \begin{table}
\begin{tabular}{c c} \hline \hline
**Objective** & **Assistance Data Used** \\ \hline Doppler frequency estimation & Reference date and time, reference user location, Galileo ephemeris data \\ \hline TOW estimation & Reference date and time \\ \hline Computation of the navigation solutions & Galileo ephemeris data, ionospheric data, and UTC model \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Purpose of the assistance data. tained using PDI techniques, \(R_{k}\) represents the \(k\)-th consecutive CAF, \(N_{nc}\) denotes the number of non-coherent combinations, \(f_{d}\) is the Doppler frequency, and \(\tau\) is the time delay. \[Z_{X}=f(\sum_{k=1}^{N_{nc}}R_{k}(\tau,f_{d})) \tag{2}\]  
When operating in high-sensitivity mode, the receiver described in this paper employs the GPDIT strategy [7] to acquire highly attenuated GNSS signals in real-time. The GPDIT strategy, illustrated in Equation (3), involves determining \(Z_{\text{GPDIT}}\) as the outcome of the NCI. \[Z_{\text{GPDIT}}=\sum_{k=1}^{N_{nc}}|R_{k}(\tau,f_{d})|^{2}+2|\sum_{k=2}^{N_{ nc}}R_{k}(\tau,f_{d})R_{k-1}^{*}(\tau,f_{d})| \tag{3}\]  
The GPDIT strategy is chosen for its effectiveness, as highlighted in [7]. This technique is deemed the most suitable in the presence of a frequency offset and demonstrates reasonable performance even in the face of phase noise resulting from the use of a TCXO. According to the findings in [7], GNSS signals with a \(C/N_{0}\) as low as 20 dB-Hz can be acquired, while maintaining a probability of false alarm \(P_{fa}<0.01\) and a probability of correct detection \(P_{d}\sim 0.8\). This is achieved through CI over a time span of 100 ms and employing GPDIT with up to \(N_{nc}=7\) non-coherent combinations. These parameters ensure a reliable acquisition of weak GNSS signals. While the GPDIT technique experiences significant degradation in the presence of data bits in the received signals [7], it proves effective when applied to pilot signals devoid of data and featuring predictable pilot codes. For this reason, the proposed high-sensitivity acquisition process acquires the Galileo pilot signals. For a stationary GNSS receiver the maximum Doppler frequency shift in the E1/L1 frequency band is around \(\pm 5\) kHz.